Pipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 24 results · all verified · 21 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Monotone Hamilton–Jacobi–Bellman schemes and policy iteration

1 · Prerequisites

2 · Summary

This page develops three related finite-control HJB branches on the periodic unit torus and a self-contained second-order comparison chain. It imports the periodic grid and positive interpolation interfaces from semi-lagrangian-characteristic-transport, the sign-reflected upwind convention from hyperbolic-finite-difference-methods, and the first-order viscosity class and comparison result from hamilton-jacobi-equations-and-viscosity-solutions.

First-order Cauchy schemes

The first-order model has a nonempty compact control set, bounded jointly continuous coefficients that are uniformly Lipschitz in space and time, and continuous periodic initial data. Each numerical update uses a nonempty finite control set; a compact-control discretization must satisfy the explicit uniform Hamiltonian error on bounded momentum sets. The page defines both a positive multilinear-interpolation semi-Lagrangian Euler-footpoint update and a sign-selected upwind update. Their monotonicity, stability, smooth-test consistency, discrete initial traces, and mesh-node half-relaxed-limit passage are proved locally. The semi-Lagrangian branch requires $h^{2} / τ \to 0$ for its interpolation defect; the upwind branch carries its own CFL condition. The exact published first-order comparison hypotheses are checked for the periodic Hamiltonian before the relaxed limits are identified. No convergence rate is asserted.

Discounted finite-state policy iteration

The stationary branch defines a finite state/action discounted generator Bellman system with conservative off-diagonal rates and positive discount. Each exact policy matrix is proved inverse-positive. Howard iteration uses strict improvements, retains equal-value actions, breaks maximizing ties by the least declared index, and stops within the finite number of deterministic policies when its budget is large enough. A smaller budget returns an explicit exhaustion status without a solution claim.

Diagonal controlled diffusion

For every $d \geq 1$ , the second-order model has finite ordered controls, bounded periodic jointly Lipschitz $α, b, f$ , and uniform diagonal ellipticity $α_{j}^{a} \geq α_{*} > 0$ . The explicit centered-diffusion and sign-selected one-sided-drift row is admitted only under the stated diffusion-dominance and time-step CFL inequalities. In two dimensions the five-point pattern below displays its spatial neighbors; in general dimension the row uses only $i$ and $i \pm e_{j}$ , with periodic index wrapping.

\begin{tikzpicture}
  \draw[gray] (-1.5,0) -- (1.5,0);
  \draw[gray] (0,-1.5) -- (0,1.5);
  \fill (-1,0) circle (2pt);
  \fill (0,0) circle (2pt);
  \fill (1,0) circle (2pt);
  \fill (0,-1) circle (2pt);
  \fill (0,1) circle (2pt);
  \node[left] at (-1,0) {$U_{i-e_1}$};
  \node[above right] at (0,0) {$U_i$};
  \node[right] at (1,0) {$U_{i+e_1}$};
  \node[below] at (0,-1) {$U_{i-e_2}$};
  \node[above] at (0,1) {$U_{i+e_2}$};
\end{tikzpicture}

The diagram records the stencil’s five nodes in the $d = 2$ case; it does not encode the coefficients or replace the row-weight proof. The page proves monotonicity, smooth consistency, and the discrete initial trace, then establishes bounded-USC/LSC comparison for the full declared coefficient class. The second-order proof uses the local semiconvex differentiability and elliptic theorem-of-sums chain, followed by a separate parabolic doubling argument. Applying this comparison to the existing conditional half-relaxed-limit corollary gives unconditional convergence and existence along an explicit admissible mesh family. General anisotropic central stencils and solution-error rates are outside the claim.

The local Alexandrov, Jensen, and theorem-of-sums arguments explicitly retain the Axiom of Choice where their selections and measure suppliers require it. The jet definition, sup-convolution transfer, and parabolic jet interface remain choice-free. All proofs remain draft authoring; native independent reviews and gates remain required.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Finite-state discounted generator Bellman system

Definition

Let $S = {1, \dots, n}$ be a nonempty finite state set, let each state $i$ have a nonempty finite ordered action set $A_{i}$ , and let $c > 0$ . For $a \in A_{i}$ , choose real coefficients $q_{i j}^{a}$ satisfying $q_{i j}^{a} \geq 0 (j \neq i), \sum_{j = 1}^{n} q_{i j}^{a} = 0,$ and a reward $r_{i}^{a} \in R$ . The row $q_{i}^{a}$ is a conservative generator row. A deterministic policy is a map $π$ with $π (i) \in A_{i}$ . Its policy matrix and reward vector are $A_{i j}^{π} = c δ_{i j} - q_{i j}^{π (i)}, r_{i}^{π} = r_{i}^{π (i)} .$ The discounted finite-state Bellman system is $c V_{i} - \max_{a \in A_{i}} (\sum_{j = 1}^{n} q_{i j}^{a} V_{j} + r_{i}^{a}) = 0, i \in S .$ The policy evaluation equation for $π$ is $A^{π} V = r^{π}$ .

Here $δ_{i j}$ is $1$ when $i = j$ and $0$ otherwise.

Remarks

For any $λ > \max_{i, a} (- q_{i i}^{a})$ , define, for each $i$ and $a \in A_{i}$ , $p_{i j}^{a} = δ_{i j} + q_{i j}^{a} / λ$ . Each such row has nonnegative entries and sums to one. Substituting $q_{i j}^{a} = λ (p_{i j}^{a} - δ_{i j})$ gives the equivalent system $V_{i} = \max_{a \in A_{i}} (γ \sum_{j} p_{i j}^{a} V_{j} + \frac{r_{i}^{a}}{c + λ}), γ = \frac{λ}{c + λ} < 1.$ This is a finite discrete-time discounted Bellman system with state-dependent action sets; no stochastic process representation is needed by the policy-iteration proof.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Second-order elliptic semijets and their value-sensitive closures

Definition

Let $U \subseteq R^{m}$ be open, $u : U \to R$ , and $z \in U$ . Write $S^{m}$ for the real symmetric $m$ by $m$ matrices. The second-order elliptic superjet consists of pairs $(p, X) \in R^{m} \times S^{m}$ such that $u (z + h) \leq u (z) + p \cdot h + \frac{1}{2} h^{T} X h + o (∣ h ∣^{2}) (h \to 0) .$ Here the inequality means that its positive remainder divided by $∣ h ∣^{2}$ has limit zero. Denote this set by $J^{2, +} u (z)$ . Define $J^{2, -} u (z) = - J^{2, +} (- u) (z) .$ The closed superjet ${\overline{J}}^{2, +} u (z)$ consists of all limits $(p, X)$ for which there are $z_{n} \in U$ and $(p_{n}, X_{n}) \in J^{2, +} u (z_{n})$ satisfying $(z_{n}, u (z_{n}), p_{n}, X_{n}) ⟶ (z, u (z), p, X) .$ Define ${\overline{J}}^{2, -} u (z) = - {\overline{J}}^{2, +} (- u) (z)$ . The value convergence is part of the definition. The same definitions work for functions allowed to equal $- \infty$ away from the finite base point, with all approximating base values finite. A two-sided quadratic expansion means that both the upper and lower inequalities hold with the same $(p, X)$ ; it is also called a second-order Peano expansion. Matrix order means $X \leq Y$ iff $h^{T} X h \leq h^{T} Y h$ for every $h$ .

For space-time $z = (t, x)$ this is the full Euclidean jet, whose matrix also has time and mixed entries. No assertion about parabolic jets is included in this definition.

LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passOpen item page →

Convex supports, localization, and the Lipschitz dual gradient

Statement

Assume the Axiom of Choice. Let $f$ be finite convex, in the sense of Convex and strictly convex functions on Euclidean convex sets, on an open convex $U \subseteq R^{n}$ . At each $x \in U$ the subdifferential $\partial f (x) = {p : f (y) \geq f (x) + p \cdot (y - x) (y \in U)}$ is nonempty; it is locally bounded and its graph is locally closed. It is a singleton exactly when $f$ is differentiable, and then equals ${D f (x)}$ . For every closed Euclidean ball $K \subset U$ (Open ball, closed ball and sphere in a metric space) there is a finite globally Lipschitz convex $H$ (Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction) on $R^{n}$ equal to $f$ on $K$ . For $F (x) = H (x) + ∣ x ∣^{2} / 2$ , the conjugate $F^{*} (p) = \sup_{x} (p \cdot x - F (x))$ is finite and $C^{1}$ , and $g = D F^{*}$ is globally one-Lipschitz with $g (p) = x$ iff $p \in \partial F (x)$ .

Facts & Assumptions

Given: The hypotheses in the Statement.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

A convex function on an open convex set is locally Lipschitz.

[F2]

Hahn-Banach dominated extension theorem for real vector spaces.

[F3]

For $n \geq 1$ every bounded sequence in $R^{n}$ has a convergent subsequence.

[F4]

Semicontinuous extreme value theorem on compact Euclidean sets.

Proof

1.1A1F1F2givenconstruct

If $U$ is empty, all assertions are vacuous. If $n = 0$ and $U$ is nonempty, then $U = R^{0} = {0}$ , $\partial f (0) = {0}$ , and $f$ is differentiable with derivative zero. Every localization ball is ${0}$ ; set $H (0) = f (0)$ . Then $F^{*} (0) = - f (0)$ and $g (0) = 0$ , so the remaining assertions hold directly. Henceforth assume $n \geq 1$ . At $x \in U$ , the directional slope $d (v) = \lim_{t ↓ 0} [f (x + t v) - f (x)] / t$ exists and is finite: convexity makes these quotients monotone in $t$ , while local Lipschitz continuity bounds them. Positive homogeneity follows by rescaling $t$ . Convexity applied at the midpoint of $x + 2 t v$ and $x + 2 t w$ gives $d (v + w) \leq d (v) + d (w)$ . By [A1] and [F2], Hahn-Banach extends zero on ${0}$ to a linear functional $p \cdot v \leq d (v)$ . The chord inequality gives $d (y - x) \leq f (y) - f (x)$ , so $p \in \partial f (x)$ .

2.1A1F1F3step 1.1algebra

If $f$ is $L$ -Lipschitz on a ball and $x$ lies in its concentric half ball, testing the support inequality at $x \pm t e$ for every unit vector $e$ and sufficiently small $t > 0$ gives $∣ p \cdot e ∣ \leq L$ . Thus all its subgradients have norm at most $L$ . On compact subsets this follows by a finite cover. If $x_{j} \to x$ , $p_{j} \to p$ , and $p_{j} \in \partial f (x_{j})$ , pass to the limit in every support inequality using continuity of $f$ ; hence $p \in \partial f (x)$ . Differentiability gives $\partial f (x) = {D f (x)}$ by testing both signs of small increments. Conversely, if $\partial f (x) = {p}$ , the local bound and [F3] imply that every choice $q_{y} \in \partial f (y)$ converges to $p$ as $y \to x$ : otherwise [A1] selects a contrary sequence with a convergent subsequence whose limit is a different limiting subgradient. The inequalities $p \cdot h \leq f (x + h) - f (x) \leq q_{x + h} \cdot h$ then prove differentiability with derivative $p$ .

3.1step 1.1step 2.1construct

Set $H (z) = \sup {f (x) + p \cdot (z - x) : x \in K, p \in \partial f (x)}$ . The local bounds from step 2.1 make slopes and intercepts uniformly bounded. Thus this supremum is finite, convex and globally Lipschitz; support inequalities give $H \leq f$ on $U$ , and the terms at $z = x \in K$ give equality on $K$ . This definition uses all supports, without choosing one for each point.

4.1F4step 3.1algebra

Since $H$ is globally Lipschitz, $F (z) - p \cdot z \to + \infty$ as $∣ z ∣ \to \infty$ . Continuity and compact extreme values give a minimizer. The identity that $F - ∣ \cdot ∣^{2} / 2$ is convex implies strict convexity of $F$ and uniqueness of this minimizer, denoted $g (p)$ . Its minimizing property is exactly $p \in \partial F (g (p))$ . More precisely, $p \in \partial F (x)$ iff $p - x \in \partial H (x)$ . One direction is immediate. For the other, test the support inequality for $F$ at $x + t (y - x)$ , divide by $t > 0$ , and let $t ↓ 0$ ; convexity makes the directional slope of $H$ at $x$ no larger than the full chord slope $H (y) - H (x)$ , yielding $H (y) - H (x) \geq (p - x) \cdot (y - x)$ . Monotonicity of supports of $H$ , obtained by adding their two defining inequalities, therefore gives $(p - q) \cdot (g (p) - g (q)) \geq ∣ g (p) - g (q) ∣^{2}$ . Hence $g$ is one-Lipschitz.

5.1step 4.1algebra∎

The supremum defining $F^{*} (p)$ is finite and attained uniquely at $g (p)$ . Evaluating the two suprema at each other's maximizers gives $0 \leq F^{*} (p + k) - F^{*} (p) - g (p) \cdot k \leq (g (p + k) - g (p)) \cdot k \leq ∣ k ∣^{2}$ . This proves $D F^{*} = g$ , whose continuity follows from its Lipschitz bound. It also proves the asserted subgradient inverse relation.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Jensen maximum principle for locally semiconvex functions

Statement

Assume the Axiom of Choice. Let $f$ be finite and locally semiconvex near $0 \in R^{m}$ , $m \geq 1$ : on some Euclidean ball (Open ball, closed ball and sphere in a metric space), $f (x) + λ ∣ x ∣^{2} / 2$ is convex (Convex and strictly convex functions on Euclidean convex sets) for a finite $λ \geq 0$ . Suppose $0$ is a strict local maximum of $f$ . For every sufficiently small $r > 0$ and every $δ > 0$ , the set of points $x \in B_{r} (0)$ where $f (x) + q \cdot x$ has a local maximum for some $∣ q ∣ \leq δ$ contains a compact subset of positive Lebesgue outer measure. Consequently, given any Lebesgue null set $N$ , such a maximum can be chosen with $x \notin N$ , $∣ x ∣ < r$ , and $∣ q ∣ \leq δ$ .

Facts & Assumptions

Given: The function, strict maximum, and parameters in the Statement; assume The Axiom of Choice.

[A1]

Every family of nonempty sets has a choice function (The Axiom of Choice); this supplies the Countable Choice hypotheses of the cited outer-measure suppliers.

[F1]

Finite convex functions on open convex sets are locally Lipschitz (A convex function on an open convex set is locally Lipschitz).

[F2]

A Lipschitz map $R^{m} \to R^{m}$ sends cube-cover null sets (Measure zero and content zero in $R^{m}$ by countable and finite cube covers) to null sets (A Lipschitz map $R^{m} \to R^{m}$ sends null sets to null sets). Cube-cover nullity agrees with zero Lebesgue outer measure (Countable covers by closed boxes, by open boxes and by closed cubes all compute Lebesgue outer measure). Outer measure is monotone, and a nonempty open ball has positive outer measure because it contains a positive-volume half-open box (Assuming countable choice, Lebesgue outer measure is an outer measure that restricts to elementary volume).

[F3]

A continuous function on a compact Euclidean ball attains its maximum (Semicontinuous extreme value theorem on compact Euclidean sets).

Proof

1.1F1F3givenchoose

Shrink to a ball ${\overline{B}}_{R} \subseteq U$ on which $g (x) = f (x) + λ ∣ x ∣^{2} / 2$ is convex and $f (0) > f (x)$ for $0 < ∣ x ∣ \leq R$ . By [F1], $f$ is continuous. Given $0 < r < R / 4$ , the compact annulus $r \leq ∣ x ∣ \leq R$ has a strictly positive gap $c = f (0) - \max f$ . Fix $0 < η \leq δ$ with $η R < c / 2$ . For every $∣ q ∣ \leq η$ , all global maximizers of $f (x) + q \cdot x$ on ${\overline{B}}_{R}$ lie in $B_{r}$ , since their value at zero is $f (0)$ and annulus values are at most $f (0) - c + η R$ . Their maximizing pairs form a closed subset of the compact product ${\overline{B}}_{R} \times {\overline{B}}_{η}$ , so their set $K$ is compact and lies in $B_{r}$ .

2.1step 1.1algebra

At a maximizing pair $(x, q)$ , $f (z) \leq f (x) - q \cdot (z - x)$ on ${\overline{B}}_{R}$ . Put $v = - q + λ x$ . Expanding the quadratic gives $g (x + h) \leq g (x) + v \cdot h + λ ∣ h ∣^{2} / 2$ . Convexity gives the reverse support $g (z) \geq g (x) + v \cdot (z - x)$ : for $h = z - x$ and small $t > 0$ , convexity on the segment from $x - t h$ to $z$ yields $g (z) - g (x) \geq (g (x) - g (x - t h)) / t \geq v \cdot h - λ t ∣ h ∣^{2} / 2$ , and let $t ↓ 0$ . Thus $f$ is differentiable at $x$ , $p_{x} = D f (x) = - q$ , and $f (x) + p_{x} \cdot (z - x) - λ ∣ z - x ∣^{2} / 2 \leq f (z) \leq f (x) + p_{x} \cdot (z - x)$ for $z \in {\overline{B}}_{R}$ ; in particular the slope at each $x \in K$ is unique.

3.1step 2.1algebra

For distinct $x, y \in K$ , set $d = y - x$ . The support bounds in step 2.1 and $f (y) \leq f (x) + p_{x} \cdot d$ give $(p_{x} - p_{y}) \cdot h \leq λ ∣ d + h ∣^{2} / 2$ whenever $y + h \in B_{R}$ . If $p_{x} \neq p_{y}$ , take $h = ∣ d ∣ (p_{x} - p_{y}) / ∣ p_{x} - p_{y} ∣$ , which is admissible since $∣ y + h ∣ < 3 r < R$ . Then $∣ p_{x} - p_{y} ∣ ∣ d ∣ \leq 2 λ ∣ d ∣^{2}$ . Equal slopes need no estimate, so $x \mapsto p_{x}$ is $2 λ$ -Lipschitz on $K$ .

4.1A1step 1.1step 3.1F2algebra

For each coordinate set $P_{j} (z) = \inf_{x \in K} (p_{x}^{j} + 2 λ ∣ z - x ∣)$ . Compactness makes this finite; the triangle inequality makes it $2 λ$ -Lipschitz, and step 3.1 gives $P_{j} (x) = p_{x}^{j}$ on $K$ . Hence $P = (P_{1}, \dots, P_{m})$ is $2 λ \sqrt{m}$ -Lipschitz and, since each $∣ q ∣ \leq η$ has a maximizer in $K$ with $P (x) = - q$ , $P (K) \supseteq {\overline{B}}_{η}$ . If $K$ were null, [F2] would make $P (K)$ null, contradicting the positive outer measure of $B_{η}$ ; thus $K$ has positive outer measure. If $λ = 0$ , this same construction makes $P$ constant while its image contains ${\overline{B}}_{η}$ , a contradiction, so no strict-maximum case is missed. Countable Choice required by [F2] follows from [A1].

5.1A1step 4.1F2given∎

Since $K$ has positive outer measure, it cannot be contained in the given null set $N$ . Choose $x \in K ∖ N$ and its maximizing slope $q$ ; then $∣ x ∣ < r$ and $∣ q ∣ \leq η \leq δ$ . This proves both conclusions.

LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passOpen item page →

Critical images of Euclidean Lipschitz maps are null

Statement

Assume the Axiom of Choice. Let $g : R^{n} \to R^{n}$ be $L$ -Lipschitz in the Euclidean norm (Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction), where $n \geq 1$ . Write $B (x, r)$ for the Euclidean ball (Open ball, closed ball and sphere in a metric space). Let $C = {p \in R^{n} : g (p + h) = g (p) + D g (p) h + o (∣ h ∣) as h \to 0, \det D g (p) = 0} .$ Then $g (C)$ is null in $R^{n}$ .

Facts & Assumptions

Given: The Axiom of Choice, an integer $n \geq 1$ , and an $L$ -Lipschitz map $g : R^{n} \to R^{n}$ .

[A1]

Every family of nonempty sets has a choice function (The Axiom of Choice); in particular this supplies Countable Choice.

[F1]

A set in $R^{n}$ is null when it has closed-cube covers with arbitrarily small total volume (Measure zero and content zero in $R^{m}$ by countable and finite cube covers).

[F2]

Countable unions and subsets of null subsets of $R^{n}$ are null under Countable Choice (Subsets and countable unions of null subsets of $R^{m}$ are null).

[F3]

Zorn's lemma follows from the Axiom of Choice (Zorn's lemma).

[F4]

The Euclidean unit ball is measurable and has measure $v_{n}$ with $0 < v_{n} < \infty$ (Open ball, closed ball and sphere in a metric space, Euclidean balls have positive finite Lebesgue measure).

[F5]

Lebesgue measure is translation invariant and scales under positive dilations by the $n$ th power (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, For a nonzero real $c$ , dilation by $c$ multiplies Lebesgue outer measure by $∣ c ∣^{n}$ , and reflection in the origin preserves it).

[F6]

Lebesgue measure is countably additive on pairwise disjoint measurable sets and monotone (Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume, Measures are monotone).

[F7]

The product $N \times N$ is countable ( $N \times N \approx N$ ).

[F8]

The rational vectors $Q^{n}$ form a countable dense subset of $R^{n}$ by coordinatewise rational approximation, and every nonempty subset of the enumerating index set has a least element ( $Q$ is countably infinite, A product of two at most countable sets is at most countable, The rationals embed densely in the reals, The well-ordering principle).

[F9]

Finite sums of nonnegative terms are monotone, and a nonnegative series whose partial sums are bounded by $M$ converges with sum at most $M$ (Finite sums and finite products, by recursion, Laws of finite sums and finite products, Series in the nonnegative extended real line, A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum).

Proof

1.1A1F3F8givenconstruct

Fix $R \geq 1$ and $0 < ε < 1$ . For each $p \in C \cap B (0, R)$ , differentiability gives a nonempty set of radii $0 < r_{p} < 1$ such that $∣ g (p + h) - g (p) - D g (p) h ∣ \leq ε ∣ h ∣$ whenever $∣ h ∣ \leq 5 r_{p}$ . By [A1], choose one such radius for each $p$ . Divide this ball family into dyadic radius classes $[2^{- j - 1}, 2^{- j})$ , largest first. In each class use [F3] on the set of subfamilies disjoint from all previously selected balls; the union of any chain is an upper bound, so a maximal disjoint subfamily exists. The union of selected subfamilies is disjoint. It is countable by [F8]: enumerate $Q^{n}$ and assign to each ball the first rational point it contains; disjoint balls get distinct points. If a rejected ball has radius $r$ and meets a selected ball of radius $R_{s} > r / 2$ , then $r < 2 R_{s}$ and every point of the rejected ball is within $2 r + R_{s} < 5 R_{s}$ of the selected center. Thus the fivefold dilates of a countable disjoint subfamily $B (p_{i}, r_{i})$ cover $C \cap B (0, R)$ .

1.2givenconstructalgebra

Put $v_{n} = λ (B (0, 1))$ . For any selected ball $B (p_{i}, r_{i})$ , the image $g (B (p_{i}, 5 r_{i}))$ lies within distance $5 ε r_{i}$ of $g (p_{i}) + D g (p_{i}) B (0, 5 r_{i})$ . This affine image lies in a disk of radius $5 L r_{i}$ centered at $g (p_{i})$ in an affine subspace of dimension $q \leq n - 1$ . In orthonormal coordinates centered at $g (p_{i})$ and adapted to that subspace, the tube lies in a box with $q$ side lengths at most $10 L r_{i} + 10 ε r_{i}$ and $n - q$ side lengths $10 ε r_{i}$ . Subdivide each side into intervals of length at most $ε r_{i}$ . The number of resulting boxes is at most $c_{n} (1 + L / ε)^{q}$ ; each is contained in an axis-parallel cube of side at most $c_{n} ε r_{i}$ . Since $q \leq n - 1$ and $ε < 1$ , their total cube volume is at most $C_{n} ε (L + 1)^{n - 1} r_{i}^{n}$ for a finite constant $C_{n} \geq 1$ depending only on $n$ . This constructs actual null covers without a rotation-invariance theorem or an area formula.

2.1A1F4F5F6step 1.1algebra

The selected balls lie in $B (0, R + 1)$ and are pairwise disjoint. By [F5], each has measure $v_{n} r_{i}^{n}$ , and the containing ball has measure $v_{n} (R + 1)^{n}$ . Countable additivity and monotonicity [F6] give $\sum_{i} r_{i}^{n} \leq (R + 1)^{n}$ , using $v_{n} > 0$ . The Countable Choice hypotheses of [F5] and [F6] follow from [A1].

3.1A1F1F7F9step 1.1step 1.2step 2.1algebra

The fivefold balls in step 1.1 cover $C \cap B (0, R)$ , and step 1.2 supplies a finite cube cover of each corresponding image with total volume at most $C_{n} ε (L + 1)^{n - 1} r_{i}^{n}$ . By [A1], choose one such finite cover for each selected ball. Index these cubes by $(i, j)$ ; [F7] enumerates the resulting subset of $N^{2}$ . Every finite partial sum is at most $C_{n} ε (L + 1)^{n - 1} \sum_{i} r_{i}^{n} \leq C_{n} ε (L + 1)^{n - 1} (R + 1)^{n}$ by [F9] and step 2.1. For any cover budget $δ > 0$ , take $ε = \min {1 / 2, δ / [C_{n} (L + 1)^{n - 1} (R + 1)^{n}]} > 0$ ; the bound is then at most $δ$ . By [F1], $g (C \cap B (0, R))$ is null.

4.1A1F2step 3.1∎

The set $C$ is the union of $C \cap B (0, R)$ over integers $R \geq 1$ . Each image in this countable union is null by step 3.1; [F2] therefore gives that $g (C)$ is null. Its Countable Choice hypothesis follows from [A1].

TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passOpen item page →

Rademacher differentiability in Euclidean space

Statement

Assume the Axiom of Choice. For every open $U \subseteq R^{n}$ and locally Lipschitz map $u : U \to R^{m}$ (each point has a neighborhood on which $∣ u (y) - u (z) ∣ \leq L ∣ y - z ∣$ for some finite $L$ , Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction), $u$ is Frechet differentiable outside a null subset of $U$ . Here Frechet differentiability at $x$ means that a linear map $A : R^{n} \to R^{m}$ satisfies $u (x + h) = u (x) + A h + o (∣ h ∣)$ .

Facts & Assumptions

Given: The Axiom of Choice, an open set $U \subseteq R^{n}$ , and a locally Lipschitz map $u : U \to R^{m}$ .

[A1]

Every family of nonempty sets has a choice function (The Axiom of Choice); the implication AC $\Rightarrow$ DC $\Rightarrow$ Countable Choice is supplied by AC implies DC implies countable choice.

[F1]

An ACL representative whose coordinate derivatives are locally $L^{p}$ has those derivatives as its weak derivatives (ACL representatives recover their weak gradients by Fubini).

[F2]

A Lipschitz function on an interval is absolutely continuous ( $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation).

[F3]

An absolutely continuous function has a derivative almost everywhere and is the integral of that derivative (Fundamental theorem of calculus for absolutely continuous functions).

[F4]

Ball averages of a locally integrable function converge to its value almost everywhere (Lebesgue differentiation theorem on $R^{n}$ ).

[F5]

For $w \in W^{1, 1} (B)$ on a ball of radius $r$ , $∥ w - w_{B} ∥_{L^{1} (B)} \leq C (n) r ∥ D w ∥_{L^{1} (B)}$ (Poincare inequality on a ball).

[F6]

Countable unions and subsets of null subsets of $R^{n}$ are null under Countable Choice (Subsets and countable unions of null subsets of $R^{m}$ are null).

[F7]

Nullity is the closed-cube-cover condition in $R^{n}$ (Measure zero and content zero in $R^{m}$ by countable and finite cube covers).

[F8]

$B (x, r) = {y : ∣ y - x ∣ < r}$ , and its measure satisfies $λ (B (x, r)) = v_{n} r^{n}$ with $0 < v_{n} < \infty$ (Open ball, closed ball and sphere in a metric space, Euclidean balls have positive finite Lebesgue measure, For a nonzero real $c$ , dilation by $c$ multiplies Lebesgue outer measure by $∣ c ∣^{n}$ , and reflection in the origin preserves it, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

[F9]

A weak derivative is defined by the signed test-function identity, and $W^{1, 1} (B)$ consists of $L^{1}$ functions whose first weak derivatives lie in $L^{1}$ (Weak derivative of a locally integrable function, Integer-order Sobolev spaces and their norms).

[F10]

Rational vectors are countable and dense in $R^{n}$ by coordinatewise rational approximation; rational open balls form a countable basis ( $Q$ is countably infinite, A product of two at most countable sets is at most countable, The rationals embed densely in the reals).

[F11]

Under Countable Choice, Lebesgue measure on $R^{n}$ is complete and is the restriction of Lebesgue outer measure to the measurable sets (Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume). Thus every measurable set of measure zero has outer measure zero, hence is null in the closed-cube-cover sense by A subset of $R^{m}$ has Lebesgue outer measure zero if and only if it is null in the sense of countable closed-cube covers; every subset is null by [F6].

Proof

1.1A1F1F2F3F10F11givenconstruct

Fix a scalar component and a ball compactly contained in a neighborhood where it is $L$ -Lipschitz. Each coordinate section is Lipschitz, hence absolutely continuous by [F2], and has a derivative almost everywhere by [F3], bounded by $L$ . For measurability, write the limsup and liminf of the coordinate difference quotients as countable suprema and infima over rational increments; continuity of the section makes these equal the corresponding real-increment limits. Define $g_{i}$ by their common finite limit where it exists and by zero elsewhere. Fubini makes the measurable exceptional set have Lebesgue measure zero; [F11] converts it to closed-cube-cover nullity. The continuous function is an ACL representative, so [F1] gives $D_{i} u = g_{i}$ weakly and $u \in W_{l o c}^{1, \infty}$ .

2.1A1F4F6F10F11step 1.1construct

Outside a null set, $x$ is a strong Lebesgue point of $g = (g_{1}, \dots, g_{n})$ . For each rational vector $q \in Q^{n}$ , apply [F4] locally to the scalar functions $∣ g - q ∣$ and to the components of $g$ . On a fixed bounded Lipschitz ball these functions are integrable; extend them by zero to $R^{n}$ to apply [F4], and restrict to interior points, where sufficiently small ball averages are unchanged. For each application, the almost-everywhere conclusion provides a measurable Lebesgue-measure-zero set containing the exceptions. Convert that set to closed-cube-cover nullity by [F11], then remove the countable union of these cube-cover-null sets using [F6]. At each remaining point, for every rational $q$ , ${lim sup}_{r ↓ 0} \frac{1}{λ (B (x, r))} \int_{B (x, r)} ∣ g (y) - g (x) ∣ d y \leq 2 ∣ q - g (x) ∣ .$ Rational vectors approximate $g (x)$ by [F10], so this limit is zero. The rational balls whose closures lie in neighborhoods on which the chosen component is Lipschitz form a countable cover of $U$ : local Lipschitz continuity gives a suitable ball around each point, density of rational centers/radii gives a rational subball, and [A1] selects one Lipschitz neighborhood and constant for each such ball. Apply the local argument on these balls.

3.1F1F5F8F9step 1.1step 2.1algebra

Fix such a point $x$ , put $p = g (x)$ , and on $B (x, r) ⋐ U$ set $w (y) = u (y) - u (x) - p \cdot (y - x)$ . The function $w$ is bounded on this ball. By [F1] and linearity of the test-function identity, its weak gradient is $g (y) - p$ , which is bounded there; thus $w \in W^{1, 1} (B (x, r))$ by [F9]. Also $w (x) = 0$ and $w$ is $(L + ∣ p ∣)$ -Lipschitz. The strong Lebesgue-point property and [F8] give $\int_{B (x, r)} ∣ D w ∣ = o (r^{n})$ . Applying [F5] yields $\int_{B (x, r)} ∣ w - w_{B (x, r)} ∣ = o (r^{n + 1}) .$

4.1F8step 3.1algebra

Write $M = L + ∣ p ∣$ . If $M = 0$ , $w$ vanishes on every sufficiently small ball. Otherwise fix $ϵ > 0$ . If $∣ w_{B (x, r)} ∣ \geq ϵ r$ , then on $B (x, c r)$ , with $c = \min {1 / 2, ϵ / (2 M)}$ , the Lipschitz bound and $w (x) = 0$ give $∣ w - w_{B (x, r)} ∣ \geq ϵ r / 2$ . Since $λ (B (x, c r)) = v_{n} c^{n} r^{n}$ by [F8], the integral in step 3.1 would be at least $v_{n} c^{n} ϵ r^{n + 1} / 2$ , a contradiction for all sufficiently small $r$ . Hence $w_{B (x, r)} = o (r)$ . If some $y \in B (x, r / 2)$ had $∣ w (y) ∣ \geq ϵ r$ , then for small $r$ we would have $∣ w (y) - w_{B (x, r)} ∣ \geq ϵ r / 2$ . On $B (y, c^{'} r) \subset B (x, r)$ , where $c^{'} = \min {1 / 4, ϵ / (4 M)}$ , Lipschitz continuity gives $∣ w - w_{B (x, r)} ∣ \geq ϵ r / 4$ . Its integral is at least $v_{n} (c^{'})^{n} ϵ r^{n + 1} / 4$ , again contradicting step 3.1. Thus $\sup_{B (x, r / 2)} ∣ w ∣ = o (r)$ . For $h \neq 0$ , take $r = 3 ∣ h ∣$ , so $x + h \in B (x, r / 2)$ ; this yields the Frechet expansion $u (x + h) - u (x) - p \cdot h = o (∣ h ∣)$ .

5.1A1F6F7F10step 4.1∎

The countable rational-ball basis [F10] and the finitely many real components of $u$ produce one exceptional cube-cover-null set by [F6] and the definition [F7]. The component derivatives assemble into the Frechet derivative. The full Axiom of Choice is used through the stated ACL, Fubini, differentiation, null-union and Poincare suppliers; no reduction to Countable Choice is asserted.

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Periodic Cartesian grids and tensor-product multilinear interpolation

Definition

Fix an integer $d \geq 1$ and integers $N_{j} \geq 3$ , $1 \leq j \leq d$ . This is the unit-period specialization of the periodic tensor-grid definition in Periodic tensor grids and multilinear interpolation, with each period $ℓ_{j} = 1$ . Put $h_{j} = 1 / N_{j}$ and define the periodic unit torus as $T^{d} = R^{d} / Z^{d}$ ; functions on it are identified with $Z^{d}$ -periodic functions on $R^{d}$ . The Cartesian grid has nodes $x_{i} = (i_{1} h_{1}, \dots, i_{d} h_{d}) (m o d Z^{d}), i_{j} \in {0, \dots, N_{j} - 1} .$ Indices on nodal arrays are always reduced modulo $N_{j}$ in coordinate $j$ .

For a nodal array $V = (V_{i})$ and $x$ in the cell with lower corner $x_{i}$ , write ${\tilde{x}}_{j} = (i_{j} + θ_{j}) h_{j}$ in a periodic lift, where $0 \leq θ_{j} \leq 1$ . Write $I_{h}$ for the supplier's tensor-product multilinear interpolant. In the local coordinates above, it is $I_{h} V (x) = \sum_{ϵ \in {0, 1}^{d}} (\prod_{j = 1}^{d} θ_{j}^{ϵ_{j}} (1 - θ_{j})^{1 - ϵ_{j}}) V_{i + ϵ} .$ At cell faces either adjacent cell formula gives the same value, because the weight of the opposite face is zero. The formula is periodic and therefore defines $I_{h} V$ on $T^{d}$ .

In dimension two the four nodal weights on one cell are shown below. The dot is the evaluation point; the labels are exactly the four factors in the formula. The diagram fixes the cell convention only; no later proof relies on it.

Remarks

For the lower-left node $V_{00}$ the weight is $(1 - θ_{1}) (1 - θ_{2})$ ; the other three weights are $θ_{1} (1 - θ_{2})$ , $(1 - θ_{1}) θ_{2}$ , and $θ_{1} θ_{2}$ . In every dimension the weights are nonnegative and sum to one by Periodic multilinear interpolation is convex and L-infinity contractive. The next local lemma proves the smooth interpolation error used by the schemes.

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Inverse positivity of discounted finite-state policy matrices

Statement

For every policy $π$ in Finite-state discounted generator Bellman system, the matrix $A^{π} = c I - Q^{π}$ is nonsingular, $(A^{π})^{- 1}$ is entrywise nonnegative, and each diagonal entry of $(A^{π})^{- 1}$ is strictly positive.

Facts & Assumptions

Given: A finite state set, $c > 0$ , and policy generator coefficients with $q_{i j} \geq 0$ for $i \neq j$ and $\sum_{j} q_{i j} = 0$ .

[F1]

For the policy generator, $q_{i j} \geq 0$ off the diagonal, $\sum_{j} q_{i j} = 0$ , and $A = c I - Q$ (Finite-state discounted generator Bellman system).

[F2]

A real square matrix with trivial kernel is invertible (Invertible matrix theorem: invertibility, full pivot rank, RREF $I$ , trivial nullspace and unique solvability are equivalent).

[F3]

Every nonempty finite set of real numbers attains its maximum and minimum (Every nonempty finite set of reals has a maximum and a minimum).

Proof

1.1F1F2F3choosealgebra

Nonsingularity. Suppose $A z = 0$ for a nonzero real vector $z$ . The finite set ${∣ z_{j} ∣ : 1 \leq j \leq n}$ has a maximum by [F3]; choose an index $i$ attaining it, so $∣ z_{i} ∣ = \max_{j} ∣ z_{j} ∣ > 0$ . By the zero row sum, $(A z)_{i} = c z_{i} + \sum_{j \neq i} q_{i j} (z_{i} - z_{j})$ . If $z_{i} > 0$ , every term in the sum is nonnegative and $c z_{i} > 0$ , contradicting $(A z)_{i} = 0$ . If $z_{i} < 0$ , every sum term is nonpositive and $c z_{i} < 0$ , again a contradiction. Thus the kernel is zero, and [F2] makes the square matrix nonsingular.

2.1F1F3step 1.1choosealgebra

Inverse positivity. Let $y \geq 0$ and let $z = A^{- 1} y$ , which exists by step 1.1. If $z_{i} < 0$ at an index where $z$ is minimal, then $(A z)_{i} = c z_{i} + \sum_{j \neq i} q_{i j} (z_{i} - z_{j}) < 0$ , contradicting $y_{i} \geq 0$ . The finite set of coordinates of $z$ attains its minimum by [F3], so this argument proves $A^{- 1} y = z \geq 0$ for every $y \geq 0$ . Applying it to each standard basis vector shows that every entry of $A^{- 1}$ is nonnegative.

3.1F1step 2.1algebra∎

Positive diagonal. Fix $i$ and set $z = A^{- 1} e_{i} \geq 0$ by step 2.1. If $z_{i} = 0$ , then $(A z)_{i} = \sum_{j \neq i} q_{i j} (0 - z_{j}) \leq 0$ , while $(A z)_{i} = (e_{i})_{i} = 1$ . Hence $z_{i} = (A^{- 1})_{i i} > 0$ . This proves the claim.

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Quadratic sup convolution and transfer of upper jets

Statement

Let $m \geq 1$ and let $K \subseteq R^{m}$ be nonempty compact, let $v : K \to R$ be upper semicontinuous (so ${lim sup}_{y \to x} v (y) \leq v (x)$ for each $x \in K$ ), and extend $v$ by $- \infty$ off $K$ . For $λ > 0$ define $\hat{v} (ξ) = \max_{x \in K} {v (x) - \frac{λ}{2} ∣ x - ξ ∣^{2}} .$ This is finite, continuous and $λ$ -semiconvex on $R^{m}$ , meaning that $\hat{v} (ξ) + λ ∣ ξ ∣^{2} / 2$ is convex (Convex and strictly convex functions on Euclidean convex sets). If $(q, Y) \in J^{2, +} \hat{v} (η)$ , every maximizing point equals $y = η + q / λ$ , and $(q, Y) \in J^{2, +} v (y), v (y) = \hat{v} (η) + \frac{∣ q ∣^{2}}{2 λ} .$ The jets of $v$ are understood for its extended function, or for its original local domain if $y$ is interior to $K$ . These conclusions use no choice principle.

Facts & Assumptions

Given: $K, v, λ$ as stated.

[F1]

Upper semicontinuous functions on compact Euclidean sets attain their maxima and are bounded above (Semicontinuous extreme value theorem on compact Euclidean sets).

[F2]

Finite convex functions on open convex domains are locally Lipschitz (A convex function on an open convex set is locally Lipschitz).

[F3]

Upper jets have the quadratic one-sided expansion in Second-order elliptic semijets and their value-sensitive closures.

Proof

1.1F1F2algebra

The maximum exists by [F1], is finite above by $\max_{K} v$ , and is finite below by evaluating at any fixed point of nonempty $K$ . Expanding the square gives $\hat{v} (ξ) + λ ∣ ξ ∣^{2} / 2 = \sup_{x \in K} {v (x) - λ ∣ x ∣^{2} / 2 + λ x \cdot ξ}$ . Every expression in the supremum is affine in $ξ$ , so the supremum is convex; finiteness and [F2] give continuity and the asserted semiconvexity.

2.1step 1.1F3algebra∎

Fix a maximizer $y$ at $η$ . Evaluation at $y$ and the upper-jet inequality [F3] give, for small $h$ , $λ (y - η) \cdot h - λ ∣ h ∣^{2} / 2 \leq \hat{v} (η + h) - \hat{v} (η) \leq q \cdot h + h^{T} Y h / 2 + o (∣ h ∣^{2})$ . Put $h = t e$ for every unit $e$ and both signs of $t$ , divide by $∣ t ∣$ , and let $t \to 0$ ; then $λ (y - η) = q$ , so the maximizer is unique and the value identity follows. For $x$ near $y$ , set $ξ = η + x - y$ , so the penalty is unchanged; the same jet inequality gives $v (x) - v (y) \leq \hat{v} (η + x - y) - \hat{v} (η) \leq q \cdot (x - y) + (x - y)^{T} Y (x - y) / 2 + o (∣ x - y ∣^{2})$ . This is the transferred upper jet, and for an interior $y$ the extension and original function agree locally.

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Alexandrov second differentiability of locally semiconvex functions

Statement

Assume the Axiom of Choice. Let $U \subseteq R^{n}$ be open, $n \geq 1$ , and let $u : U \to R$ be locally semiconvex: around each point there is a Euclidean ball (Open ball, closed ball and sphere in a metric space) and $c \geq 0$ such that $x \mapsto u (x) + c ∣ x ∣^{2} / 2$ is convex in the sense of Convex and strictly convex functions on Euclidean convex sets. Outside a set null in the cube-cover sense of Measure zero and content zero in $R^{m}$ by countable and finite cube covers, there are $p \in R^{n}$ and a symmetric matrix $A$ with $u (x + h) = u (x) + p \cdot h + \frac{1}{2} h^{T} A h + o (∣ h ∣^{2}) (h \to 0) .$ The remainder is uniform in direction. In particular $u$ is twice differentiable in the Peano sense almost everywhere.

Facts & Assumptions

Given: The Axiom of Choice, an open set $U \subseteq R^{n}$ , $n \geq 1$ , and a locally semiconvex real function $u$ on $U$ .

[A1]

Every family of nonempty sets has a choice function (The Axiom of Choice); the implication AC $\Rightarrow$ DC $\Rightarrow$ Countable Choice is supplied by AC implies DC implies countable choice, discharging both choice hypotheses of the absolutely-continuous FTC in [F7].

[F1]

A locally Lipschitz map between finite-dimensional Euclidean spaces is Frechet differentiable outside a cube-cover-null set (Rademacher differentiability in Euclidean space).

[F2]

A Lipschitz map $R^{n} \to R^{n}$ sends null sets to null sets (A Lipschitz map $R^{m} \to R^{m}$ sends null sets to null sets).

[F3]

The image of the differentiability points with singular derivative under a Lipschitz self-map of $R^{n}$ is null (Critical images of Euclidean Lipschitz maps are null).

[F4]

For a finite convex function on an open convex set, the subdifferential is nonempty, locally bounded, has locally closed graph, and its singleton points are exactly its differentiability points; the localized convex extension and dual-gradient inverse also hold (Convex supports, localization, and the Lipschitz dual gradient).

[F5]

Every bounded sequence in $R^{n}$ has a convergent subsequence (For $n \geq 1$ every bounded sequence in $R^{n}$ has a convergent subsequence).

[F6]

Countable Choice is the choice function principle for countable families (The Axiom of Countable Choice ( ${A C}_{ω}$ )); under it, countable unions of null subsets of $R^{n}$ are null (Subsets and countable unions of null subsets of $R^{m}$ are null).

[F7]

Lipschitz functions on intervals are absolutely continuous, and absolutely continuous functions are integrals of their almost-everywhere derivatives ( $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation, Fundamental theorem of calculus for absolutely continuous functions).

[F8]

Nullity in $R^{n}$ is defined by arbitrarily small total volumes of closed-cube covers (Measure zero and content zero in $R^{m}$ by countable and finite cube covers).

Proof

1.1A1F1F2F3F4F6F8construct

First consider a finite convex function $f$ on an open ball. Localize on a smaller closed ball $K$ and use [F4] to obtain a finite globally Lipschitz convex $H = f$ on $K$ , then set $F = H + ∣ \cdot ∣^{2} / 2$ and $g = D F^{*}$ , which is globally one-Lipschitz and satisfies $g (q) = x$ exactly when $q \in \partial F (x)$ . By [F1], the nondifferentiability set $N$ of $g$ is null; [F2] makes $g (N)$ null, and [F3] makes $g (C)$ null for the differentiable singular-derivative set $C$ of $g$ . The locally Lipschitz function $F$ is differentiable outside another null set $M$ by [F1]. The union $M \cup g (N) \cup g (C)$ is null by [F6]. For $x$ in the interior of $K$ outside that union, set $p = D F (x)$ . Then $g (p) = x$ , $g$ is differentiable at $p$ , and $B = D g (p)$ is invertible. Full AC is used by the stated suppliers and to choose the localizations simultaneously.

2.1A1F4F5step 1.1algebra

As $y \to x$ , every $q \in \partial F (y)$ tends to $p$ , uniformly over all such subgradients: otherwise, by [A1], choose $y_{k} \to x$ and $q_{k} \in \partial F (y_{k})$ staying a fixed distance from $p$ ; local boundedness in [F4] and [F5] give a convergent subsequence, and the locally closed graph in [F4] puts its limit in $\partial F (x) = {p}$ , a contradiction. Since $g (q) = y$ , differentiability of $g$ at $p$ gives $y - x = B (q - p) + o (∣ q - p ∣)$ . Invertibility absorbs the remainder to give $∣ q - p ∣ \leq 2 ∥ B^{- 1} ∥ ∣ y - x ∣$ for nearby $y$ , hence $q = p + B^{- 1} (y - x) + o (∣ y - x ∣)$ uniformly for every $q \in \partial F (y)$ .

3.1F7step 2.1algebra

Fix a small $h$ . The section $t \mapsto F (x + t h)$ is Lipschitz on $[0, 1]$ , hence absolutely continuous by [F7]. Wherever its derivative exists, the two support inequalities in directions $h$ and $- h$ show that it equals $q \cdot h$ for every $q \in \partial F (x + t h)$ . By step 2.1, for every $ϵ > 0$ and sufficiently small $∣ h ∣$ , $∣ q - p - B^{- 1} (t h) ∣ \leq ϵ t ∣ h ∣$ for every $0 < t \leq 1$ and every such $q$ . Integrating the measurable scalar derivative using [F7] gives $∣ F (x + h) - F (x) - p \cdot h - \frac{1}{2} h^{T} B^{- 1} h ∣ \leq ϵ ∣ h ∣^{2} / 2.$ No measurable subgradient selection is used. Replace $B^{- 1}$ by its symmetric part, which has the same quadratic form.

4.1A1F6F8step 3.1construct∎

Subtract $∣ x ∣^{2} / 2$ to obtain the quadratic expansion for $f$ in the interior of $K$ . Exhaust the original ball by countably many smaller closed balls and discard their null exceptional sets using [F6, F8]. The family of rational balls whose closures lie inside a local semiconvexity ball is countable and covers $U$ ; by [A1] choose a corresponding finite convexity constant for each. Apply the convex conclusion to $u + c ∣ \cdot ∣^{2} / 2$ on each ball and subtract that quadratic. Countable unions remain null by [F6, F8], and the remainder is direction-uniform by step 3.1. Full AC is retained; no reduction to Countable Choice is asserted.

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Finite-budget Howard policy iteration for a discounted generator Bellman system

Definition

For the finite system of Finite-state discounted generator Bellman system, order each finite set $A_{i}$ by its declared indices and put $P = \prod_{i \in S} ∣ A_{i} ∣$ . The procedure takes a positive integer evaluation budget $K \in N$ , $K \geq 1$ (default $K = P$ ), and proceeds as follows.

Initialize $π_{0} (i)$ to the least-index action in $A_{i}$ .
For the current policy $π_{k}$ , solve exactly $A^{π_{k}} V^{k} = r^{π_{k}}$ .
Form the strict-improvement set $J_{k} = {i : \max_{a \in A_{i}} (Q_{i}^{a} V^{k} + r_{i}^{a}) > Q_{i}^{π_{k} (i)} V^{k} + r_{i}^{π_{k} (i)}},$ where $Q_{i}^{a} V = \sum_{j} q_{i j}^{a} V_{j}$ .
If $J_{k} = \emptyset$ , return $(converged, π_{k}, V^{k})$ .
If $J_{k} \neq \emptyset$ and $k + 1 = K$ , return $(budget_exhausted, π_{k}, V^{k})$ ; this status makes no Bellman-solution claim.
Otherwise, at each $i \in J_{k}$ switch to the least-index maximizing action in $A_{i}$ ; retain $π_{k} (i)$ at every state outside $J_{k}$ and continue.

The policy matrix solve is well-defined by Inverse positivity of discounted finite-state policy matrices. A budget of $P$ policy evaluations is sufficient by the theorem below.

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Periodic first-order controlled Hamilton–Jacobi Cauchy problem

Definition

Let $d, q \geq 1$ , $T > 0$ , and let $A$ be a nonempty compact subset of $R^{q}$ (a finite action set is included). For each $a \in A$ , let $b_{a} : T^{d} \times [0, T] \to R^{d}$ and $ℓ_{a} : T^{d} \times [0, T] \to R$ be bounded and jointly continuous in $(x, t, a)$ . Write $b_{a}, ℓ_{a}$ also for their periodic lifts to $R^{d} \times [0, T]$ . Assume there are constants $B, L < \infty$ such that $∣ b_{a} (x, t) ∣ \leq B, ∣ ℓ_{a} (x, t) ∣ \leq L,$ and for all $a \in A$ , $x, y \in R^{d}$ , and $t, s \in [0, T]$ , $∣ b_{a} (x, t) - b_{a} (y, s) ∣ + ∣ ℓ_{a} (x, t) - ℓ_{a} (y, s) ∣ \leq L (∣ x - y ∣ + ∣ t - s ∣) .$ Given continuous periodic initial data $u_{0} : T^{d} \to R$ , define $H (x, t, p) = \max_{a \in A} {- b_{a} (x, t) \cdot p - ℓ_{a} (x, t)} .$ The periodic first-order controlled Hamilton–Jacobi Cauchy problem is $u_{t} + H (x, t, D u) = 0 on T^{d} \times (0, T), u (x, 0) = u_{0} (x) .$ The signs are fixed by $u_{t} = \min_{a \in A} {b_{a} \cdot D u + ℓ_{a}}$ . Whenever the controls are discretized, use a sequence of finite nonempty sets $A_{k} \subseteq A$ and define $H_{k} (x, t, p) : = \max_{a \in A_{k}} (- b_{a} \cdot p - ℓ_{a}) .$ Define the Hamiltonian error by $η_{k} (R) : = \sup_{x, t, ∣ p ∣ \leq R} ∣ H_{k} (x, t, p) - H (x, t, p) ∣ .$ The sequence is admissible when $η_{k} (R) \to 0$ for every $R < \infty$ . Exact minimization over $A$ corresponds to $η_{k} = 0$ . Scheme convergence below uses this explicit control-grid error whenever the controls are discretized.

Remarks

The compact action set and continuity make each displayed maximum attained. The coefficient assumptions imply the exact first-order comparison bounds on the periodic extension to $R^{d}$ : $∣ H (x, t, p) - H (y, s, p) ∣ \leq L (1 + ∣ p ∣) (∣ x - y ∣ + ∣ t - s ∣), ∣ H (x, t, p) - H (x, t, p^{'}) ∣ \leq B ∣ p - p^{'} ∣ .$ Thus the published whole-space comparison theorem applies once bounded upper semicontinuous subsolutions and bounded lower semicontinuous supersolutions with the same pointwise initial ordering have been established.

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Periodic diagonal controlled-diffusion HJB Cauchy problem

Definition

Let $d \geq 1$ , $T > 0$ , and let $A$ be a nonempty finite ordered control set. For each $a \in A$ and $j = 1, \dots, d$ , let $α_{j}^{a}, b_{j}^{a}, f^{a} : T^{d} \times [0, T] \to R$ be bounded and periodic, jointly continuous in $(x, t)$ , and uniformly Lipschitz in $(x, t)$ . Assume a common constant $α_{*} > 0$ satisfies $α_{j}^{a} (x, t) \geq α_{*} for every a, j, x, t .$ The diagonal controlled-diffusion HJB Cauchy problem is $u_{t} - \max_{a \in A} {\sum_{j = 1}^{d} (α_{j}^{a} (x, t) u_{x_{j} x_{j}} + b_{j}^{a} (x, t) u_{x_{j}}) + f^{a} (x, t)} = 0 on T^{d} \times (0, T),$ with continuous periodic initial data $u (x, 0) = u_{0} (x)$ . In the convention $u_{t} + F (x, t, D u, D^{2} u) = 0$ , this has $F (x, t, p, X) = - \max_{a \in A} {\sum_{j} (α_{j}^{a} X_{j j} + b_{j}^{a} p_{j}) + f^{a}},$ which is nonincreasing in each diagonal Hessian entry. No off-diagonal diffusion or arbitrary anisotropic central stencil is covered.

Remarks

The lower bound $α_{*} > 0$ is part of this selected coefficient branch; the monotonicity proof below only needs the separate mesh diffusion-dominance and time-step inequalities. The definition itself asserts no solution existence or uniqueness. Local draft items now prove comparison for the full stated coefficient class and use it to prove unconditional convergence and existence along an admissible mesh family. Those new arguments remain subject to the native independent reviews and gates and have not received workflow acceptance.

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Positive partition of unity and interpolation error for periodic multilinear interpolation

Statement

For the interpolant $I_{h}$ of Periodic Cartesian grids and tensor-product multilinear interpolation, every cell weight is nonnegative and the weights sum to one by Periodic multilinear interpolation is convex and L-infinity contractive. Hence $V \leq W$ on grid nodes implies $I_{h} V \leq I_{h} W$ , and $∥ I_{h} V ∥_{\infty} \leq ∥ V ∥_{\infty}$ . If $ϕ \in C^{2} (T^{d})$ and $V_{i} = ϕ (x_{i})$ , then $∥ I_{h} V - ϕ ∥_{\infty} \leq \frac{1}{8} \sum_{j = 1}^{d} h_{j}^{2} ∥ \partial_{j j} ϕ ∥_{\infty} .$

Facts & Assumptions

Given: The periodic grid and tensor interpolant, nodal arrays $V, W$ , and a periodic $C^{2}$ function $ϕ$ .

[F1]

The local grid is the unit-period specialization of the periodic tensor grid, with $N_{j} \geq 3$ , and its interpolant has nonnegative tensor weights summing to one, hence is order-preserving and $L^{\infty}$ -contractive (Periodic Cartesian grids and tensor-product multilinear interpolation, Periodic multilinear interpolation is convex and L-infinity contractive).

[F2]

If the second derivative of a $C^{2}$ function is bounded by $M$ on an interval, its first-order Taylor remainder from an endpoint has magnitude at most $M ∣ x - a ∣^{2} / 2$ (A uniform derivative bound gives a uniform Taylor remainder bound).

Proof

1.1F1algebra

The supplier result in [F1] says each evaluation is a convex combination of the corner data. Applying this to $W - V \geq 0$ proves order preservation, and applying it to $V$ and $- V$ proves the sup-norm bound.

1.2F2algebra

In one coordinate, let $x = x_{i} + θ h$ with $0 \leq θ \leq 1$ and let $L_{h} ϕ (x) = (1 - θ) ϕ (x_{i}) + θ ϕ (x_{i} + h)$ . Taylor-expand both endpoints about $x$ . By [F2], the linear terms cancel after weighting and $∣ L_{h} ϕ (x) - ϕ (x) ∣ \leq \frac{h^{2}}{2} ∥ ϕ^{''} ∥_{\infty} ((1 - θ) θ^{2} + θ (1 - θ)^{2}) = \frac{h^{2}}{2} θ (1 - θ) ∥ ϕ^{''} ∥_{\infty} \leq \frac{h^{2}}{8} ∥ ϕ^{''} ∥_{\infty} .$ This calculation also holds on the periodic seam by taking a periodic lift of the cell.

2.1step 1.1step 1.2algebra∎

Write the tensor interpolant as the product of the one-coordinate interpolation operators $I_{j}$ . Each $I_{j}$ has sup-norm at most one by step 1.1. The identity $I_{1} \dots I_{d} - I = \sum_{j = 1}^{d} I_{1} \dots I_{j - 1} (I_{j} - I)$ and the one-dimensional bound of step 1.2 give the claimed sum of errors. This proves the statement.

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Elliptic theorem of sums with the full block matrix bound

Statement

Assume the Axiom of Choice. Let $k \geq 1$ , $n_{i} \geq 1$ , let $U_{i} \subseteq R^{n_{i}}$ be open, and let $u_{i} : U_{i} \to R$ be upper semicontinuous, meaning ${lim sup}_{y \to x} u_{i} (y) \leq u_{i} (x)$ at each $x \in U_{i}$ . Suppose $ϕ$ is $C^{2}$ near $z = (z_{1}, \dots, z_{k}) \in \prod_{i} U_{i}$ and $\sum_{i} u_{i} (x_{i}) - ϕ (x)$ has a local maximum at $z$ . Set $A = D^{2} ϕ (z)$ , and let $∥ A ∥$ be its Euclidean operator norm (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||). For every $ε > 0$ there are $X_{i} \in S^{n_{i}}$ such that $(D_{i} ϕ (z), X_{i}) \in {\overline{J}}^{2, +} u_{i} (z_{i}),$ and $- (ε^{- 1} + ∥ A ∥) I \leq diag (X_{1}, \dots, X_{k}) \leq A + ε A^{2} .$ Here all jets are full Euclidean jets. In particular one may take each $U_{i}$ to be a space-time domain of dimension $d + 1$ . For a lower semicontinuous function $v$ , applying the theorem to $- v$ and then negating its jet gives the corresponding closed subjet for $v$ .

Facts & Assumptions

Given: The data of the Statement and The Axiom of Choice.

[A1]

Every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

Jets and their value-sensitive closures are defined in Second-order elliptic semijets and their value-sensitive closures.

[F2]

Quadratic sup convolutions of compactly restricted upper semicontinuous functions are continuous and semiconvex; an upper jet $(q, Y)$ at $η$ transfers, without changing $Y$ , to the original function at $η + q / λ$ , with original value $\hat{v} (η) + ∣ q ∣^{2} / (2 λ)$ (Quadratic sup convolution and transfer of upper jets).

[F3]

A finite locally semiconvex function has a second-order Peano expansion outside a Lebesgue null set, under AC (Alexandrov second differentiability of locally semiconvex functions). At a strict maximum the maximum can be perturbed by an arbitrarily small affine term to occur arbitrarily close to that point outside any specified null set (Jensen maximum principle for locally semiconvex functions).

[F4]

For symmetric $A$ , $∣ h^{T} A h ∣ \leq ∥ A ∥ ∣ h ∣^{2}$ (A symmetric matrix operator norm is its largest absolute Rayleigh quotient). Every bounded finite-dimensional sequence has a convergent subsequence (For $n \geq 1$ every bounded sequence in $R^{n}$ has a convergent subsequence).

Proof

1.1F2F4algebraconstruct

First prove the quadratic compactly localized case. Translate to $z = 0$ , subtract $u_{i} (0)$ and the affine terms $D_{i} ϕ (0) \cdot x_{i}$ , and suppose $u_{i} (0) = 0$ and $w (x) : = \sum_{i} u_{i} (x_{i}) \leq x^{T} A x / 2$ on compact neighborhoods $K_{i}$ of zero. Extend the restrictions by $- \infty$ off $K_{i}$ , set $λ = ε^{- 1} + ∥ A ∥ > 0$ and $B = A + ε A^{2}$ , and write $h = x - ξ$ . The identity $∣ ε^{- 1 / 2} h - ε^{1 / 2} A ξ ∣^{2} \geq 0$ gives $x^{T} A x \leq ξ^{T} B ξ + λ ∣ x - ξ ∣^{2}$ . Define ${\hat{u}}_{i}$ by [F2] and $f (ξ) = \sum_{i} {\hat{u}}_{i} (ξ_{i})$ . The supremum separates over the finite compact product, so $f (ξ) = \sup_{x} {w (x) - λ ∣ x - ξ ∣^{2} / 2} \leq ξ^{T} B ξ / 2$ . Each ${\hat{u}}_{i} (0) \geq 0$ , while $f (0) \leq 0$ , so all are zero; also $f + λ ∣ ξ ∣^{2} / 2$ is convex.

2.1A1step 1.1F3algebrachoose

The function $F (ξ) = f (ξ) - ξ^{T} B ξ / 2 - ∣ ξ ∣^{4}$ has a strict maximum at zero and is locally semiconvex, since the Hessian of the subtracted polynomial is bounded on every fixed ball. Apply [F3] outside the null set where $f$ lacks a Peano expansion: for each $n$ , [A1] selects $∣ ξ_{n} ∣ < 1 / n$ , $∣ q_{n} ∣ \leq 1 / n$ so $F + q_{n} \cdot ξ$ has a local maximum at $ξ_{n}$ and $f$ has expansion $(p_{n}, Z_{n})$ there. Restricting to lines gives $p_{n} = B ξ_{n} + 4 ∣ ξ_{n} ∣^{2} ξ_{n} - q_{n}$ and $Z_{n} \leq B + 8 ξ_{n} ξ_{n}^{T} + 4 ∣ ξ_{n} ∣^{2} I$ ; convexity of $f + λ ∣ ξ ∣^{2} / 2$ gives $Z_{n} \geq - λ I$ . Thus $p_{n} \to 0$ and $Z_{n}$ is bounded. AC supplies the countable selections and is inherited from [F3].

3.1step 1.1step 2.1F1F2algebra

Separability makes $Z_{n}$ block diagonal: each coordinate restriction gives the corresponding Peano expansion of ${\hat{u}}_{i}$ , and summing those expansions gives a second expansion of $f$ with Hessian $diag (Z_{n, 11}, \dots, Z_{n, k k})$ . Uniqueness follows because a quadratic form that is $o (∣ h ∣^{2})$ vanishes on every direction and hence by polarization. Set $X_{n, i} = Z_{n, i i}$ . The transfer in [F2] gives $y_{n, i} = ξ_{n, i} + p_{n, i} / λ$ , $(p_{n, i}, X_{n, i}) \in J^{2, +} u_{i} (y_{n, i})$ , and $u_{i} (y_{n, i}) = {\hat{u}}_{i} (ξ_{n, i}) + ∣ p_{n, i} ∣^{2} / (2 λ)$ . Since $y_{n, i} \to 0$ , they eventually lie inside $K_{i}$ ; continuity of ${\hat{u}}_{i}$ gives $u_{i} (y_{n, i}) \to 0 = u_{i} (0)$ .

4.1step 2.1step 3.1F1F4algebra

By [F4], extract one subsequence along which the finitely many entries of $Z_{n}$ converge to $Z = diag (X_{i})$ . The inequalities in step 2.1 pass to the limit, giving $- λ I \leq Z \leq B$ . The convergent jets and values from step 3.1 imply $(0, X_{i}) \in {\overline{J}}^{2, +} u_{i} (0)$ by [F1]; this proves the localized quadratic case.

5.1step 4.1F1algebra

For the general case, translate and subtract affine functions as above. Taylor's formula gives $ϕ (x) = x^{T} A x / 2 + o (∣ x ∣^{2})$ , so for every sufficiently small $η > 0$ there are compact neighborhoods $K_{i}^{η}$ with $\sum_{i} u_{i} (x_{i}) \leq x^{T} (A + η I) x / 2$ . Apply the quadratic case from step 4.1 with $A_{η} = A + η I$ ; it gives closed jets $(0, X_{i}^{η})$ with $- (ε^{- 1} + ∥ A_{η} ∥) I \leq diag (X_{i}^{η}) \leq A_{η} + ε A_{η}^{2}$ . Because zero is interior to each $K_{i}^{η}$ , localization does not change these closed jets.

6.1A1step 5.1F1F4givenchoosealgebra∎

Take $η_{n} = 1 / n$ and, by [A1], choose for each sufficiently large $n$ one tuple of closed-jet matrices supplied by step 5.1. Use [F4] to extract a convergent subsequence of these block matrices. The limit obeys the required bounds because $∥ A + η I ∥ \to ∥ A ∥$ . For each closed jet $(0, X_{i}^{η})$ , [A1] selects an actual approximating jet with point, value, gradient, and matrix within $1 / n$ ; the defining value-sensitive closure [F1] then gives the limiting closed jet. Restoring the subtracted affine terms gives gradient $D_{i} ϕ (z)$ , and negating jets proves the lower-semicontinuous version.

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Explicit monotone diagonal-diffusion Bellman grid update

Definition

Use the model of Periodic diagonal controlled-diffusion HJB Cauchy problem and a periodic grid with integers $N_{j} \geq 3$ , $h_{j} = 1 / N_{j}$ . Choose an integer $M \geq 1$ , set $τ = T / M$ , and put $t_{n} = n τ$ for $n = 0, \dots, M$ . For a nodal array $V$ , define $D_{j j}^{h} V_{i} = \frac{V_{i + e_{j}} - 2 V_{i} + V_{i - e_{j}}}{h_{j}^{2}} .$ For a control $a$ , define $D_{j, a}^{u p} V_{i} = {\begin{cases} (V_{i} - V_{i - e_{j}}) / h_{j}, & b_{j}^{a} (x_{i}, t_{n}) \geq 0, \\ (V_{i + e_{j}} - V_{i}) / h_{j}, & b_{j}^{a} (x_{i}, t_{n}) < 0. \end{cases}$ The recurrence is $U_{i}^{n + 1} = U_{i}^{n} + τ \max_{a \in A} {\sum_{j = 1}^{d} [α_{j}^{a} (x_{i}, t_{n}) D_{j j}^{h} U_{i} + b_{j}^{a} (x_{i}, t_{n}) D_{j, a}^{u p} U_{i}] + f^{a} (x_{i}, t_{n})} .$ At each time level $n = 0, \dots, M - 1$ , admit the update only when, at every node, control, and coordinate, $∣ b_{j}^{a} (x_{i}, t_{n}) ∣ h_{j} \leq α_{j}^{a} (x_{i}, t_{n}),$ and $τ \max_{i, a} \sum_{j = 1}^{d} (\frac{2 α_{j}^{a} (x_{i}, t_{n})}{h_{j}^{2}} + \frac{∣ b_{j}^{a} (x_{i}, t_{n}) ∣}{h_{j}}) \leq 1.$ Initialize by $U_{i}^{0} = u_{0} (x_{i})$ and perform the updates for $n = 0, \dots, M - 1$ , evaluating coefficients at $(x_{i}, t_{n})$ . Return $U^{M}$ . No claim about an arbitrary anisotropic central stencil is included.

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Positive-interpolation semi-Lagrangian update for first-order HJB equations

Definition

Choose integers $N_{j} \geq 3$ , a number $M \geq 1$ of time steps, and $τ = T / M$ . Sample $u_{0}$ at the grid nodes of Periodic Cartesian grids and tensor-product multilinear interpolation and set $U_{i}^{0} = u_{0} (x_{i})$ . At each $n = 0, \dots, M - 1$ , compute $U_{i}^{n + 1} = \min_{a \in A_{k}} {I_{h} U^{n} (x_{i} + τ b_{a} (x_{i}, t_{n})) + τ ℓ_{a} (x_{i}, t_{n})}, t_{n} = n τ .$ The position is evaluated on the torus, so its coordinates wrap modulo one. The algorithm always uses a finite nonempty set $A_{k} \subset A$ . For a finite control model take $A_{k} = A$ ; for a compact control model, supply finite sets whose Hamiltonian errors $η_{k} (R)$ from Periodic first-order controlled Hamilton–Jacobi Cauchy problem tend to zero on every bounded momentum set. This is the declared finite-control approximation branch; it does not presume an unimplemented continuous minimization oracle. Return $U^{M}$ .

The update is the explicit Euler-footpoint semi-Lagrangian branch specified by the plan. It defines its footpoint and interpolant locally; no unauthored characteristic-flow estimate is presumed.

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CFL-monotone upwind update for first-order HJB equations

Definition

Use the model and coefficient notation of Periodic first-order controlled Hamilton–Jacobi Cauchy problem and the periodic grid of Periodic Cartesian grids and tensor-product multilinear interpolation. Choose an integer $M \geq 1$ , set $τ = T / M$ , and put $t_{n} = n τ$ for $n = 0, \dots, M$ . Choose a finite nonempty control set $A_{k} \subseteq A$ . For a finite control model take $A_{k} = A$ ; for compact controls use a sequence of finite nonempty sets with the stated uniform Hamiltonian error. For each control and grid node put $b_{a, j}^{+} = \max {b_{a, j}, 0}, b_{a, j}^{-} = \max {- b_{a, j}, 0},$ and define the explicit row $G_{i}^{a} (V) = V_{i} + τ [\sum_{j = 1}^{d} (\frac{b_{a, j}^{+} (x_{i}, t_{n})}{h_{j}} (V_{i + e_{j}} - V_{i}) + \frac{b_{a, j}^{-} (x_{i}, t_{n})}{h_{j}} (V_{i - e_{j}} - V_{i})) + ℓ_{a} (x_{i}, t_{n})] .$ The periodic HJB upwind update at time $t_{n}$ is $U_{i}^{n + 1} = \min_{a \in A_{k}} G_{i}^{a} (U^{n}) .$ At every step $n = 0, \dots, M - 1$ , require the CFL condition $τ \max_{a, i} \sum_{j = 1}^{d} \frac{∣ b_{a, j} (x_{i}, t_{n}) ∣}{h_{j}} \leq 1$ Initialize $U_{i}^{0} = u_{0} (x_{i})$ and perform the updates for $n = 0, \dots, M - 1$ . Return $U^{M}$ . Since the transport velocity in Periodic advection schemes and the finite-interval upwind row is $- b_{a}$ for this Hamiltonian sign convention, the displayed one-sided direction is its sign-reflected upwind row.

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Second-order parabolic viscosity subsolutions and supersolutions

Definition

Let $u$ be upper semicontinuous and $v$ lower semicontinuous on the periodic space-time cylinder. A second-order parabolic viscosity subsolution of Periodic diagonal controlled-diffusion HJB Cauchy problem is a function $u$ such that every $C^{1, 2}$ test function $ϕ$ for which $u - ϕ$ has a local maximum at $(x_{0}, t_{0})$ , $0 < t_{0} < T$ , satisfies $ϕ_{t} (x_{0}, t_{0}) - \max_{a \in A} {\sum_{j} (α_{j}^{a} ϕ_{x_{j} x_{j}} + b_{j}^{a} ϕ_{x_{j}}) (x_{0}, t_{0}) + f^{a} (x_{0}, t_{0})} \leq 0.$ Here $C^{1, 2}$ means that $ϕ_{t}$ , all first spatial derivatives, and all second spatial derivatives exist and are continuous near the contact point. A supersolution is lower semicontinuous and satisfies the reverse inequality at every local minimum. A viscosity solution is both. For the Cauchy problem, the relaxed initial conditions are ${lim sup}_{(y, s) \to (x, 0), s > 0} u (y, s) \leq u_{0} (x), {lim inf}_{(y, s) \to (x, 0), s > 0} v (y, s) \geq u_{0} (x) .$

Remarks

This extends the contact convention in Viscosity subsolutions and supersolutions of a first-order equation and of the Cauchy problem to tests with the spatial Hessian required by the diffusion operator. The definition does not assert a comparison principle, existence, or uniqueness; those are separate theorems.

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Localization of strict half-relaxed contacts to periodic mesh nodes

Statement

Let $V_{k}$ be locally bounded periodic interpolant extensions of finite nodal arrays on grids with $h_{k} \to 0$ and time steps $τ_{k} \to 0$ . Regard this sequence as the family $V_{ϵ} = V_{k}$ when $(k + 1)^{- 1} < ϵ \leq k^{- 1}$ , and form its upper and lower half-relaxed limits as in Half-relaxed limits of a locally bounded family. If $\overline{V} - ϕ$ has a strict local maximum at an interior point $z_{0} = (x_{0}, t_{0})$ for a smooth test $ϕ$ , then there are mesh indices $k_{m} \to \infty$ and grid points $z_{m} = (x_{i_{m}}, t_{n_{m}}) \to z_{0}$ such that $V_{k_{m}} (z_{m}) - ϕ (z_{m}) \to \overline{V} (z_{0}) - ϕ (z_{0})$ and $z_{m}$ is a local maximum of the corresponding nodal values minus $ϕ$ on a fixed closed space-time ball. The lower half-relaxed limit has the dual strict minimum property. The proof uses Countable Choice only for sequential recovery from the set-defined half-relaxed limit.

Facts & Assumptions

Given: The locally bounded family and interpolant extensions of the statement, a strict local contact, and the half-relaxed limits.

[F1]

The upper half-relaxed limit is the infimum over radii of local suprema, and Countable Choice gives a sequence of near-recovery pairs; the lower limit has the dual property (Half-relaxed limits of a locally bounded family, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F2]

A tensor interpolant is a convex combination of the nodal values in its cell, whose vertices are within $\sqrt{d} h_{\max}$ in a periodic lift (Periodic Cartesian grids and tensor-product multilinear interpolation, Positive partition of unity and interpolation error for periodic multilinear interpolation).

[F3]

A closed bounded Euclidean set is compact and every open cover of a compact metric space has a finite subcover (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Open cover, subcover, compact metric space, and compact subset of a metric space). Upper semicontinuous functions attain maxima on compact sets; lower semicontinuous ones attain minima (Semicontinuous extreme value theorem on compact Euclidean sets). A nonempty finite set of real values also attains its extrema (Every nonempty finite set of reals has a maximum and a minimum).

[A1]

Countable Choice is used only for the countable near-recovery selections in the proof (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1F3givenalgebra

Choose a closed ball $K$ about $z_{0}$ contained in the open cylinder where the contact is strict. For any smaller radius $r > 0$ , the annulus $K ∖ B_{r} (z_{0})$ is compact by [F3]; upper semicontinuity of $\overline{V} - ϕ$ makes its maximum strictly less than $\overline{V} (z_{0}) - ϕ (z_{0})$ . Denote the positive gap by $g_{r}$ . The lower contact case gives the corresponding positive gap above the strict minimum.

1.2F1F2A1choose

By [F1] and [A1], select pairs $(k_{m}, y_{m})$ with $k_{m} \to \infty$ , $y_{m} \to z_{0}$ , and $V_{k_{m}} (y_{m}) - ϕ (y_{m}) \to \overline{V} (z_{0}) - ϕ (z_{0})$ . The interpolated value $V_{k_{m}} (y_{m})$ is a convex combination of the nodal values in its cell by [F2], so at least one cell vertex has value no smaller. Choose the first such vertex in the lexicographic order on its binary cell label $ϵ \in {0, 1}^{d}$ , where at the first differing coordinate $0$ precedes $1$ . Its spatial distance from $y_{m}$ is at most $\sqrt{d} h_{k_{m}}$ ; the piecewise-constant time extension has a grid time within $τ_{k_{m}}$ . Thus the selected nodal points converge to $z_{0}$ . Continuity of $ϕ$ makes the nodal $V - ϕ$ values at least the recovery values minus a quantity tending to zero, so their lower limit is at least the upper contact value.

2.1F1F3step 1.1step 1.2algebra

For all sufficiently large $m$ , maximize the finite nonempty set of nodal values $U_{i}^{n} - ϕ (x_{i}, t_{n})$ inside $K$ and take its lexicographically first maximizing tuple $(i, n)$ , comparing integer coordinates in displayed order and putting the smaller first differing coordinate first. A maximizer exists by [F3]; its value is at least the recovery value from step 1.2. Fix $r > 0$ and let $A_{r} = K ∖ B_{r} (z_{0})$ . The strict gap $g_{r}$ from step 1.1 holds on $A_{r}$ . For every $z \in A_{r}$ , the set-based definition [F1] supplies a positive radius on which $V_{ϵ} (y) \leq \overline{V} (z) + g_{r} / 8$ whenever both $ϵ$ and $∣ y - z ∣$ are smaller than that radius. Continuity of $ϕ$ and density of the positive rationals allow the following predicate on pairs $(z, q)$ : $P_{r} (z, q)$ means $V_{ϵ} (y) \leq \overline{V} (z) + g_{r} / 8$ for every $0 < ϵ < q$ and $∣ y - z ∣ < q$ , and $∣ ϕ (y) - ϕ (z) ∣ < g_{r} / 8$ for every $∣ y - z ∣ < q$ . The family $U_{r} = {B (z, q / 2) : z \in A_{r}, q \in Q_{> 0}, P_{r} (z, q)}$ is an open cover of $A_{r}$ : for each center some real radius works by the relaxed-limit definition and continuity, and a smaller rational radius also works. This is the set of all qualifying rational pairs, not a choice of one radius for each $z$ . By [F3] it has a finite subcover; let $q_{*} > 0$ be the least radius in that finite subcover. Whenever $ϵ < q_{*}$ and $y \in A_{r}$ , a covering center gives $V_{ϵ} (y) - ϕ (y) \leq \overline{V} (z) - ϕ (z) + g_{r} / 4 \leq \overline{V} (z_{0}) - ϕ (z_{0}) - 3 g_{r} / 4.$ The recovery value in step 1.2 is eventually larger than this bound, so no maximizer on the grid in $K$ lies in $A_{r}$ . Since this holds for every $r > 0$ , the maximizing nodes converge to $z_{0}$ . Their values have lower limit at least the contact value by step 1.2; the upper-limit definition [F1], applied at these nodes, gives upper limit at most the same value. Thus their values converge to the upper half-relaxed contact value.

3.1F1F2F3A1step 1.1step 2.1choose∎

For the lower half-relaxed limit, [F1] and [A1] select near-recovery pairs with values approaching $\underline{V} (z_{0})$ . Convex interpolation supplies a cell vertex with value no larger than the interpolated value; take the first such vertex in the binary-cell lexicographic order. Minimize over the finite grid points in $K$ and take the lexicographically first minimizer. Fix $r > 0$ and use the strict lower-contact gap $g_{r}$ from step 1.1 on $A_{r}$ . For each $z \in A_{r}$ , the set-defined lower limit gives a positive radius on which $V_{ϵ} (y) \geq \underline{V} (z) - g_{r} / 8$ whenever both $ϵ$ and $∣ y - z ∣$ are smaller than that radius. Define $P_{r}^{-} (z, q)$ to mean this bound holds for every $0 < ϵ < q$ and $∣ y - z ∣ < q$ , and $∣ ϕ (y) - ϕ (z) ∣ < g_{r} / 8$ for every $∣ y - z ∣ < q$ . The family of all $B (z, q / 2)$ with $z \in A_{r}$ , rational $q > 0$ , and $P_{r}^{-} (z, q)$ covers $A_{r}$ ; as in step 2.1, it uses all qualifying pairs rather than a pointwise choice. A finite subcover from [F3] has least radius $q_{*} > 0$ , so every sufficiently fine approximation at every $y \in A_{r}$ satisfies $V_{ϵ} (y) - ϕ (y) \geq \underline{V} (z) - ϕ (z) - g_{r} / 4 \geq \underline{V} (z_{0}) - ϕ (z_{0}) + 3 g_{r} / 4.$ The lower recovery values from the selected vertex tend to $\underline{V} (z_{0}) - ϕ (z_{0})$ ; hence no minimizing node lies in $A_{r}$ once the mesh is sufficiently fine. As this holds for each $r > 0$ , the minimizing nodes approach $z_{0}$ . Their upper limit is at most the recovery contact value, while the set-defined lower limit [F1] gives lower limit at least that value, so they converge to it. This proves the dual statement.

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Monotone improvement and finite termination of exact Howard iteration

Statement

For exact policy evaluations, every strict switch in Finite-budget Howard policy iteration for a discounted generator Bellman system weakly increases the policy value at every state and strictly increases it at each switched state. No deterministic policy can recur after a strict switch. With budget $P = \prod_{i} ∣ A_{i} ∣$ , the algorithm terminates within $P$ policy evaluations at the unique solution of the Bellman system in Finite-state discounted generator Bellman system. A smaller budget may return budget_exhausted without a solution claim.

Facts & Assumptions

Given: A finite nonempty state set, finite ordered nonempty action sets, generator rows, rewards, $c > 0$ , exact policy solves, and the iteration of Finite-budget Howard policy iteration for a discounted generator Bellman system.

[F1]

At each nonterminal policy $π_{k}$ , the algorithm solves $A^{π_{k}} V^{k} = r^{π_{k}}$ and changes only states where another action has strictly greater Bellman row value; at such states it chooses a maximizing action (Finite-budget Howard policy iteration for a discounted generator Bellman system).

[F2]

Every policy matrix inverse is entrywise nonnegative with strictly positive diagonal (Inverse positivity of discounted finite-state policy matrices).

[F3]

The policy set has exactly $P = \prod_{i} ∣ A_{i} ∣$ possible deterministic policies; every finite nonempty set of real values attains its extrema (Finite-state discounted generator Bellman system, Every nonempty finite set of reals has a maximum and a minimum).

Proof

1.1F1givenalgebra

Let $π_{k + 1}$ follow a strict-improvement step from $π_{k}$ . At an unchanged state $i$ , $π_{k + 1} (i) = π_{k} (i)$ , so $(r^{π_{k + 1}} - A^{π_{k + 1}} V^{k})_{i} = 0$ by [F1]. At a switched state $i$ , the new action has $\sum_{j} q_{i j}^{π_{k + 1} (i)} V_{j}^{k} + r_{i}^{π_{k + 1} (i)} > c V_{i}^{k}$ , while the old policy equation gives $\sum_{j} q_{i j}^{π_{k} (i)} V_{j}^{k} + r_{i}^{π_{k} (i)} = c V_{i}^{k}$ . Therefore the vector $z = r^{π_{k + 1}} - A^{π_{k + 1}} V^{k}$ is nonnegative and is strictly positive at every switched state.

2.1F2step 1.1algebra

The new exact policy solve gives $A^{π_{k + 1}} (V^{k + 1} - V^{k}) = r^{π_{k + 1}} - A^{π_{k + 1}} V^{k} = z .$ Multiplying by the nonnegative inverse in [F2] yields $V^{k + 1} - V^{k} = (A^{π_{k + 1}})^{- 1} z \geq 0$ . At a switched state $i$ , the $i$ th coordinate is at least $(A^{π_{k + 1}})_{i i}^{- 1} z_{i} > 0$ , so the value increase is strict there.

3.1F1F3step 2.1algebra

If a policy recurred after at least one strict switch, exact evaluation of that same nonsingular policy matrix would reproduce its unique value vector, contradicting the componentwise nondecrease and the strict increase at each intervening switch from step 2.1. Thus all evaluated policies are distinct. There are only $P$ policies by [F3], so after at most $P$ evaluations the strict-improvement set is empty; the budget- $P$ algorithm cannot exhaust first.

4.1F3step 3.1algebra∎

At termination, no action row exceeds the current policy row at any state. The policy equation says that the current row equals $c V_{i}$ ; hence $c V_{i} = \max_{a} (\sum_{j} q_{i j}^{a} V_{j} + r_{i}^{a})$ at every state, so $V$ solves the Bellman system. For uniqueness, suppose $V, W$ are two solutions and let $d = V - W$ . Choose a state $i$ where $d_{i} = \max_{j} d_{j}$ , which exists by [F3]. For every action, $\sum_{j} q_{i j}^{a} d_{j} = \sum_{j \neq i} q_{i j}^{a} (d_{j} - d_{i}) \leq 0$ by the generator signs and row sum. The difference of the two maxima is at most the maximum of the rowwise differences, so $c d_{i} \leq \max_{a} \sum_{j} q_{i j}^{a} d_{j} \leq 0$ , whence $d_{i} \leq 0$ . Thus $\max_{j} (V_{j} - W_{j}) \leq 0$ . Interchanging $V, W$ gives the reverse inequality, so $V = W$ .

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Initial trace for the explicit diagonal-diffusion HJB scheme

Statement

Let $u_{0}$ be continuous and periodic, and use the explicit diagonal-diffusion scheme of Explicit monotone diagonal-diffusion Bellman grid update with mesh widths $h_{k, j} \to 0$ , time steps $τ_{k} \to 0$ , and its diffusion-dominance and CFL inequalities at every row. Then for every sequence of integers $n_{k} \geq 0$ with $t_{k} = n_{k} τ_{k} \to 0$ and $t_{k}$ within the computed horizon, $\sup_{i} ∣ U_{i}^{n_{k}} - u_{0} (x_{i}) ∣ ⟶ 0.$ The estimate is uniform over all finite control choices generated by the maximization in the scheme.

Facts & Assumptions

Given: The grid scheme, bounded coefficients of Periodic diagonal controlled-diffusion HJB Cauchy problem, continuous periodic initial data, and row conditions from the statement. For any finite transition tree, expectation means the finite sum of endpoint values weighted by products of row coefficients.

[F1]

The update expands using the central second difference and sign-selected one-sided drift; its row conditions are $∣ b_{j}^{a} ∣ h_{j} \leq α_{j}^{a}$ and $τ_{k} \sum_{j} (2 α_{j}^{a} / h_{j}^{2} + ∣ b_{j}^{a} ∣ / h_{j}) \leq 1$ (Explicit monotone diagonal-diffusion Bellman grid update).

[F2]

There are finite constants $B$ , $F$ , and $A_{*}$ with $∣ b^{a} (x, t) ∣ \leq B$ , $∣ f^{a} (x, t) ∣ \leq F$ , and $α_{j}^{a} (x, t) \leq A_{*}$ for every control and point; $u_{0}$ is continuous and periodic (Periodic diagonal controlled-diffusion HJB Cauchy problem).

[F3]

The periodic lift of $u_{0}$ is uniformly continuous on every compact cube; finitely many adjacent periodic lifts therefore share a modulus $ω_{0} (δ) \to 0$ (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous, Periodic Cartesian grids and tensor-product multilinear interpolation).

[F4]

The nodes and coordinate neighbors use the periodic Cartesian convention of Periodic Cartesian grids and tensor-product multilinear interpolation.

Proof

1.1F1F2F4givenalgebra

Fix one control row and abbreviate $α = α_{j}^{a}$ , $b = b_{j}^{a}$ , $h = h_{j}$ . If $b \geq 0$ , the two neighbor transition weights in coordinate $j$ are $τ (α / h^{2} - b / h)$ at $x_{i} - h e_{j}$ and $τ α / h^{2}$ at $x_{i} + h e_{j}$ . If $b < 0$ , they are $τ α / h^{2}$ at $x_{i} - h e_{j}$ and $τ (α / h^{2} - ∣ b ∣ / h)$ at $x_{i} + h e_{j}$ . Their sum is $τ (2 α / h^{2} - ∣ b ∣ / h)$ and their signed first moment is $τ b$ . The diffusion-dominance inequality in [F1] makes both neighbor coefficients nonnegative. The center coefficient is $1 - τ \sum_{j} (2 α_{j}^{a} / h_{j}^{2} - ∣ b_{j}^{a} ∣ / h_{j})$ , which is nonnegative by the CFL inequality in [F1]. Including the center, the row weights sum to one. Summing over coordinates gives mean displacement $τ b^{a} (x_{i}, t_{n})$ . The raw squared displacement is at most $τ \sum_{j} (2 α_{j}^{a} (x_{i}, t_{n}) - ∣ b_{j}^{a} (x_{i}, t_{n}) ∣ h_{j}) \leq 2 τ d A_{*} .$ Subtracting the squared mean cannot increase the second moment.

2.1F1F2step 1.1algebra

Unroll $n_{k}$ updates as a finite transition tree. At a node of the tree, the maximum in the scheme chooses one of finitely many control rows; that row is a finite probability distribution by step 1.1. The terminal value is thus a maximum of finite weighted averages of $u_{0} (X_{n_{k}})$ plus accumulated rewards. On every branch the sum of the drift means has magnitude at most $B t_{k}$ . The centered increments have conditional weighted mean zero. For two different times, the weighted cross term vanishes by first summing the later centered increment over all branches extending a fixed earlier path prefix. Expanding the square and using step 1.1 gives, uniformly over all finite policies, $E ∣ X_{n_{k}} - x_{i} ∣^{2} \leq 2 B^{2} t_{k}^{2} + 4 d A_{*} t_{k} .$ The accumulated reward has magnitude at most $F t_{k}$ .

3.1F2F3step 2.1algebra∎

Put $M_{0} = ∥ u_{0} ∥_{\infty}$ . For any $0 < δ < 1$ , split each finite endpoint average according to $∣ X_{n_{k}} - x_{i} ∣ \leq δ$ . On that part the change in $u_{0}$ is at most $ω_{0} (δ)$ by [F3]; on the complement it is at most $2 M_{0}$ . The complement weight is at most $E ∣ X_{n_{k}} - x_{i} ∣^{2} / δ^{2}$ . Therefore every policy average differs from $u_{0} (x_{i})$ by at most $ω_{0} (δ) + \frac{2 M_{0}}{δ^{2}} (2 B^{2} t_{k}^{2} + 4 d A_{*} t_{k}) + F t_{k} .$ This bound is independent of $i$ and of the policy; taking a finite maximum preserves it. First let $k \to \infty$ and then $δ ↓ 0$ to obtain the claimed uniform initial trace.

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Nonnegative row weights for the diagonal-diffusion HJB stencil

Statement

Under the two inequalities in Explicit monotone diagonal-diffusion Bellman grid update, every fixed-control row of its explicit update is an affine combination of the center and the two coordinate neighbors with nonnegative coefficients summing to one, plus $τ f^{a}$ . Therefore the max over controls is order-preserving.

Facts & Assumptions

Given: Positive diagonal coefficients $α_{j}^{a}$ , drift coefficients $b_{j}^{a}$ , mesh widths $h_{j}$ , and time step $τ$ satisfying the diffusion- dominance and explicit CFL conditions.

[F1]

The scheme uses $D_{j j}^{h}$ , the sign-selected $D_{j, a}^{u p}$ , and the two inequalities $∣ b_{j}^{a} ∣ h_{j} \leq α_{j}^{a}$ and $τ \sum_{j} (2 α_{j}^{a} / h_{j}^{2} + ∣ b_{j}^{a} ∣ / h_{j}) \leq 1$ (Explicit monotone diagonal-diffusion Bellman grid update).

Proof

1.1F1givenalgebra

Fix $i, a, j$ and abbreviate $α = α_{j}^{a} (x_{i}, t_{n})$ , $b = b_{j}^{a} (x_{i}, t_{n})$ , and $h = h_{j}$ . If $b \geq 0$ , the coefficient of $U_{i - e_{j}}$ in the update is $τ (α / h^{2} - b / h)$ and that of $U_{i + e_{j}}$ is $τ α / h^{2}$ . If $b < 0$ , the coefficient of $U_{i + e_{j}}$ is $τ (α / h^{2} - ∣ b ∣ / h)$ and that of $U_{i - e_{j}}$ is $τ α / h^{2}$ . Each is nonnegative by $∣ b ∣ h \leq α$ in [F1].

2.1F1step 1.1algebra

The center coefficient is $1 - τ \sum_{j} (\frac{2 α_{j}^{a}}{h_{j}^{2}} - \frac{∣ b_{j}^{a} ∣}{h_{j}}) .$ It is at least $1 - τ \sum_{j} (2 α_{j}^{a} / h_{j}^{2} + ∣ b_{j}^{a} ∣ / h_{j}) \geq 0$ by [F1]. For each coordinate, the two neighbor coefficients sum to $τ (2 α_{j}^{a} / h_{j}^{2} - ∣ b_{j}^{a} ∣ / h_{j})$ ; adding the center coefficient therefore gives one. Hence each fixed-control update is a convex combination plus its constant forcing.

3.1step 2.1algebra∎

A convex combination is order-preserving in its nodal inputs. The maximum of finitely many order-preserving control maps is order-preserving, proving the claim.

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Closed elliptic space-time jets test the parabolic diagonal HJB inequality

Statement

Let $u$ be a USC viscosity subsolution in the sense of Second-order parabolic viscosity subsolutions and supersolutions, and write the local space-time variable as $z = (t, x)$ . If $((a, p), H) \in {\overline{J}}^{2, +} u (t, x)$ is a closed elliptic jet in all $d + 1$ variables (Second-order elliptic semijets and their value-sensitive closures), and $X$ is the spatial principal block of $H$ , then $a - \max_{c \in A} {\sum_{j} (α_{j}^{c} (x, t) X_{j j} + b_{j}^{c} (x, t) p_{j}) + f^{c} (x, t)} \leq 0.$ For an LSC supersolution and a closed elliptic subjet the reverse inequality holds. All points are interior in time; local periodic lifts are understood.

Facts & Assumptions

Given: The stated viscosity subsolution or supersolution and jet.

[F1]

An ordinary elliptic superjet $(q, H)$ means an upper expansion by $u (z) + q \cdot h + \frac{1}{2} h^{T} H h + o (∣ h ∣^{2})$ . A closed jet is the limit of ordinary jets at points $z_{n} \to z$ , with function values and jet entries converging. Subjets have the reversed expansion (Second-order elliptic semijets and their value-sensitive closures).

[F2]

Every $C^{2}$ space-time polynomial is a $C^{1, 2}$ admissible test; the diagonal HJB residual is continuous in the point, time derivative, spatial gradient and spatial Hessian (Second-order parabolic viscosity subsolutions and supersolutions, Periodic diagonal controlled-diffusion HJB Cauchy problem).

Proof

1.1F1F2givenalgebra

First let the superjet be ordinary. For each $θ > 0$ , the definition of the remainder in [F1] gives a neighborhood where the polynomial $u (z) + q \cdot h + \frac{1}{2} h^{T} (H + θ I) h$ lies above $u (z + h)$ and agrees at $h = 0$ . It is an admissible upper test by [F2]. Its tested derivatives are $a, p, X + θ I_{d}$ , so the subsolution inequality holds for these entries. Continuity in [F2] and $θ ↓ 0$ give the stated inequality for $a, p, X$ .

2.1F1F2step 1.1algebra∎

For a closed superjet, apply step 1.1 at every interior point in its witnessing sequence from [F1]. All sufficiently late points are interior; coefficient continuity and convergence of the finite jet entries let the inequalities pass to the limit by [F2]. For an ordinary subjet, the polynomial with $H - θ I$ lies below the function locally; use the lower-test inequality and let $θ ↓ 0$ . Passing along the closed-subjet witnessing sequence proves the supersolution assertion.

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Initial trace for the positive-interpolation and upwind HJB schemes

Statement

Let $u_{0}$ be continuous and periodic, and let $B$ and $L$ bound the drift and running cost in Periodic first-order controlled Hamilton–Jacobi Cauchy problem. For the semi-Lagrangian scheme, if $t_{n} = n τ \to 0$ , $h_{\max} \to 0$ , and $h_{\max}^{} / τ \to 0$ , then $\sup_{i} ∣ U_{i}^{n} - u_{0} (x_{i}) ∣ ⟶ 0.$ For the upwind scheme, the same conclusion holds when $t_{n} \to 0$ , $h_{\max} \to 0$ , and its CFL condition holds. Both statements are uniform over all finite control policies and remain true for a finite control discretization.

Facts & Assumptions

Given: The two finite update rules, bounded coefficients, continuous periodic $u_{0}$ , and a time step satisfying the corresponding mesh condition. For each finite transition tree, $E$ denotes the finite sum of path values weighted by the products of their row weights.

[F1]

The semi-Lagrangian transition weights are the nonnegative tensor weights of Periodic Cartesian grids and tensor-product multilinear interpolation; they sum to one and each coordinate node lies within $h_{j}$ of the footpoint (Positive partition of unity and interpolation error for periodic multilinear interpolation).

[F2]

The upwind row weights under the CFL condition are its center weight $1 - τ \sum_{j} ∣ b_{j} ∣ / h_{j}$ and neighbor weights $τ b_{j}^{+} / h_{j}$ , $τ b_{j}^{-} / h_{j}$ (CFL-monotone upwind update for first-order HJB equations).

[F3]

The coefficient and reward bounds are $∣ b_{a} ∣ \leq B$ and $∣ ℓ_{a} ∣ \leq L$ , and $u_{0}$ is continuous and periodic (Periodic first-order controlled Hamilton–Jacobi Cauchy problem).

[F4]

The restriction of a continuous periodic lift of $u_{0}$ to the compact cube $[- 1, 2]^{d}$ is uniformly continuous; translating nearby torus lifts into this cube gives a common modulus $ω_{0} (δ) \to 0$ (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous, Periodic Cartesian grids and tensor-product multilinear interpolation).

Proof

1.1F1givenalgebra

For a semi-Lagrangian row at a node $x_{i}$ and action $a$ , view its interpolation weights as a finite probability vector on the cell nodes around $y = x_{i} + τ b_{a} (x_{i}, t)$ . In a periodic lift, the weighted mean of each coordinate is $(1 - θ_{j}) x_{i_{j}} + θ_{j} (x_{i_{j}} + h_{j}) = y_{j}$ , by the product formula in [F1]. The centered interpolation error $ξ$ therefore has weighted mean zero and satisfies $∣ ξ ∣^{2} \leq \sum_{j} h_{j}^{2}$ .

1.2F2givenalgebra

For an upwind row, the weights in [F2] are nonnegative and sum to one by CFL. The weighted mean displacement to the previous-time node is $τ b_{a}$ : in coordinate $j$ it is $τ (b_{j}^{+} / h_{j}) h_{j} - τ (b_{j}^{-} / h_{j}) h_{j} = τ b_{j}$ . Its centered error $ξ$ has mean zero and second moment at most $τ \sum_{j} ∣ b_{j} ∣ h_{j} \leq τ B \sum_{j} h_{j}$ , since subtracting the squared mean cannot increase the second moment.

1.3F1F2F3algebra

Unroll either update for the finite number $n$ of steps. At each stage the minimum chooses one of finitely many control rows, and each row is a finite weighted average; induction therefore represents the value as the minimum of finite weighted path costs $u_{0} (X_{n}) + \sum_{r < n} τ ℓ_{r}$ . Along any such controlled path the sum of the drift increments has norm at most $B t$ . The centered increments have conditional mean zero. For $r < s$ , summing the increment $ξ_{s}$ over all next-branch weights from each fixed path prefix through time $s - 1$ gives zero; hence the finite weighted cross term $E (ξ_{r} \cdot ξ_{s})$ is zero. Expanding the square therefore shows that their total second moment is at most $n \sum_{j} h_{j}^{2}$ for the semi-Lagrangian rows and $t B \sum_{j} h_{j}$ for the upwind rows. Thus the endpoint displacement obeys $E ∣ X_{n} - x_{i} ∣^{2} \leq 2 B^{2} t^{2} + 2 n \sum_{j} h_{j}^{2}$ for the semi-Lagrangian case and $E ∣ X_{n} - x_{i} ∣^{2} \leq 2 B^{2} t^{2} + 2 t B \sum_{j} h_{j}$ for the upwind case. Every path's accumulated reward has magnitude at most $L t$ .

2.1F3F4step 1.3algebra∎

Let $M_{0} = ∥ u_{0} ∥_{\infty}$ and let $ω_{0} (δ) \to 0$ be the uniform modulus from [F4]. For any $0 < δ < 1$ , split the finite weighted endpoint average into paths with $∣ X_{n} - x_{i} ∣ \leq δ$ and its complement. On the first set, $∣ u_{0} (X_{n}) - u_{0} (x_{i}) ∣ \leq ω_{0} (δ)$ ; on the complement, it is at most $2 M_{0}$ . The complement weight is at most $E ∣ X_{n} - x_{i} ∣^{2} / δ^{2}$ . By step 1.3 this tends to zero as $t_{n} \to 0$ under $h_{\max}^{} / τ \to 0$ for the semi-Lagrangian scheme, and under $h_{\max} \to 0$ for the upwind scheme. The reward contributes at most $L t_{n}$ ; the same two-sided bound holds after taking a minimum over policies. First let the mesh and $t_{n}$ tend to zero, then let $δ ↓ 0$ . This proves the asserted uniform initial trace.

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Smooth test functions suffice for the first- and second-order HJB viscosity notions

Statement

For the continuous first-order Hamiltonian of Periodic first-order controlled Hamilton–Jacobi Cauchy problem, requiring the subsolution or supersolution inequality only for smooth ( $C^{\infty}$ ) local test functions is equivalent to requiring it for all $C^{1}$ tests. For the diagonal second-order operator of Periodic diagonal controlled-diffusion HJB Cauchy problem, the same equivalence holds between smooth tests and all $C^{1, 2}$ tests in Second-order parabolic viscosity subsolutions and supersolutions. The result is local in space-time and applies to upper semicontinuous subsolutions and lower semicontinuous supersolutions.

Facts & Assumptions

Given: A local upper contact (or lower contact) of an upper semicontinuous (or lower semicontinuous) function with a $C^{1}$ or $C^{1, 2}$ test function, and a continuous PDE residual depending on the corresponding first or second derivatives.

[F1]

A compactly supported smooth bump can be chosen equal to one on a smaller ball and between zero and one; scaling and dividing by its positive integral gives a nonnegative unit-mass mollifier supported in any prescribed small ball (A smooth bump between concentric Euclidean balls, The mollifier family generated by a unit-mass smooth bump).

[F2]

Every continuous derivative is uniformly continuous on a compact metric set (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[F3]

A finite-valued upper semicontinuous function on a compact Euclidean set attains its maximum, and a lower semicontinuous function attains its minimum (Semicontinuous extreme value theorem on compact Euclidean sets).

[F4]

The first-order HJB residual is $ϕ_{t} + H (x, t, D ϕ)$ , while the second-order residual is $ϕ_{t} - \max_{a} {\sum_{j} (α_{j}^{a} ϕ_{x_{j} x_{j}} + b_{j}^{a} ϕ_{x_{j}}) + f^{a}}$ ; both are continuous in the indicated derivatives (Periodic first-order controlled Hamilton–Jacobi Cauchy problem, Periodic diagonal controlled-diffusion HJB Cauchy problem, Second-order parabolic viscosity subsolutions and supersolutions).

[F5]

If $g$ is continuous on a closed interval, its integral exists and the integral from a fixed endpoint has derivative $g$ (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive).

Proof

1.1F4algebra

Start with an upper contact at $z_{0} = (x_{0}, t_{0})$ for a $C^{1}$ or $C^{1, 2}$ test $ϕ$ . On a closed Euclidean space-time ball $K$ contained in the interior cylinder, define $\tilde{ϕ} (z) = ϕ (z) + ε (∣ x - x_{0} ∣^{4} + ∣ t - t_{0} ∣^{4})$ for $ε > 0$ . The added term and all derivatives relevant to the chosen test class vanish at $z_{0}$ , while $u - \tilde{ϕ}$ is strictly smaller than $u (z_{0}) - ϕ (z_{0})$ at every other point of the contact neighborhood. Thus the original and strictified tests have the same PDE residual at $z_{0}$ .

2.1F1F2F5step 1.1algebra

Choose a nonnegative unit-mass mollifier $ρ_{δ}$ supported in the ball of radius $δ$ from [F1], with $δ$ small enough that its convolution samples only points in the contact neighborhood. Choose a smooth cutoff equal to one on a neighborhood of $K$ and supported inside that contact neighborhood using [F1], multiply $\tilde{ϕ}$ by it, and extend the product by zero. Convolve this compactly supported function with $ρ_{δ}$ to obtain $ϕ_{δ}$ . Rewriting the convolution with the smooth kernel as $\int g (y) ρ_{δ} (z - y) d y$ shows by uniform difference quotients on the compact support that $ϕ_{δ}$ is $C^{\infty}$ . On $K$ it coincides with convolution of $\tilde{ϕ}$ alone. For a $C^{1}$ function $g$ , [F5] gives $\frac{g (z + h e_{j}) - g (z)}{h} = \int_{0}^{1} \partial_{j} g (z + θ h e_{j}) d θ$ when the indicated derivative exists. Uniform continuity of $\partial_{j} g$ on the compact neighborhood makes these difference quotients converge uniformly as $h \to 0$ . Integrating this uniform limit against $ρ_{δ}$ proves $\partial_{j} (ρ_{δ} * g) = ρ_{δ} * \partial_{j} g$ . For a second spatial derivative, apply this identity first to $g = \tilde{ϕ}$ and then to $g = \partial_{j} \tilde{ϕ}$ ; this uses only the continuous second derivative, not a third derivative. Apply it once for time and once for each first spatial derivative. Thus $D^{α} ϕ_{δ} = ρ_{δ} * D^{α} \tilde{ϕ}$ for exactly the derivatives required in the $C^{1}$ or $C^{1, 2}$ class. By [F2], $\sup_{z \in K^{'}} ∣ D^{α} ϕ_{δ} (z) - D^{α} \tilde{ϕ} (z) ∣ \leq \sup_{z \in K^{'}, ∣ y ∣ \leq δ} ∣ D^{α} \tilde{ϕ} (z - y) - D^{α} \tilde{ϕ} (z) ∣ ⟶ 0$ on each smaller compact ball $K^{'}$ . Hence the approximants converge in $C^{1}$ or $C^{1, 2}$ , respectively.

3.1F3step 1.1step 2.1

The upper semicontinuous function $u - ϕ_{δ}$ attains a maximum on $K$ by [F3]. The strict gap from step 1.1 and uniform convergence from the preceding approximation step imply that, for every fixed neighborhood of $z_{0}$ , all maximizing points lie in that neighborhood once $δ$ is sufficiently small; the boundary of $K$ is excluded for the same reason. At any such maximizing point $z_{δ}$ , adding the constant $u (z_{δ}) - ϕ_{δ} (z_{δ})$ makes $ϕ_{δ}$ a smooth upper test. Apply the assumed smooth-test inequality there.

4.1F3F4step 1.1step 2.1step 3.1∎

The maximizing points in step 3.1 approach $z_{0}$ uniformly over the maximizing set, and the required derivatives of $ϕ_{δ}$ converge uniformly to those of $\tilde{ϕ}$ on $K^{'}$ by step 2.1. Continuity of the PDE residual [F4] therefore lets the smooth-test inequalities pass to the residual of $\tilde{ϕ}$ at $z_{0}$ , which equals that of $ϕ$ by step 1.1. This proves the upper-test implication. For a lower contact, replace maxima by minima and reverse the inequalities; the same uniform gap and derivative convergence apply. Since smooth tests are themselves in both larger test classes, the reverse implications are immediate.

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Smooth consistency of the diagonal-diffusion Bellman stencil

Statement

Assume the coefficients are bounded and periodic and a periodic test function $ϕ$ is smooth with bounded derivatives $ϕ_{t t}$ , $ϕ_{x_{j} x_{j}}$ , and $ϕ_{x_{j} x_{j} x_{j} x_{j}}$ , and the coefficients are bounded. For the scheme of Explicit monotone diagonal-diffusion Bellman grid update, define the normalized local residual $R_{h, τ} (x_{i}, t_{n}) = \frac{ϕ (x_{i}, t_{n} + τ) - S_{h, τ} ϕ (\cdot, t_{n})_{i}}{τ} .$ Then $R_{h, τ} = ϕ_{t} - \max_{a \in A} {\sum_{j} (α_{j}^{a} ϕ_{x_{j} x_{j}} + b_{j}^{a} ϕ_{x_{j}}) + f^{a}} + O (τ + \max_{j} [α_{j, \max} h_{j}^{2} ∥ ϕ_{x_{j}^{4}} ∥_{\infty} + ∣ b_{j} ∣_{\max} h_{j} ∥ ϕ_{x_{j} x_{j}} ∥_{\infty}]),$ where $α_{j, \max} : = \sup_{a, x, t} α_{j}^{a} (x, t)$ and $∣ b_{j} ∣_{\max} : = \sup_{a, x, t} ∣ b_{j}^{a} (x, t) ∣$ . The bound is uniform over nodes and times before the terminal step. The $O (\cdot)$ is a smooth-test residual estimate, not a solution-error rate.

Facts & Assumptions

Given: The finite-control diagonal diffusion scheme, the coefficient bounds, and a periodic smooth test function with the stated derivative bounds.

[F1]

The time update is $U^{n + 1} = U^{n} + τ \max_{a} {\sum_{j} (α_{j}^{a} D_{j j}^{h} U + b_{j}^{a} D_{j, a}^{u p} U) + f^{a}}$ with the derivative conventions of the algorithm (Explicit monotone diagonal-diffusion Bellman grid update).

[F2]

A Taylor remainder of order $m$ is bounded by the $(m + 1)$ st derivative supremum times the segment length to that power divided by $(m + 1)!$ (A uniform derivative bound gives a uniform Taylor remainder bound).

[F3]

The continuous operator is $u_{t} - \max_{a} {\sum_{j} (α_{j}^{a} u_{x_{j} x_{j}} + b_{j}^{a} u_{x_{j}}) + f^{a}} = 0$ (Periodic diagonal controlled-diffusion HJB Cauchy problem).

Proof

1.1F1F2givenalgebra

Applying [F2] at $x_{i} \pm h_{j} e_{j}$ and cancelling the odd terms gives $D_{j j}^{h} ϕ (x_{i}, t_{n}) = ϕ_{x_{j} x_{j}} (x_{i}, t_{n}) + O (h_{j}^{2} ∥ ϕ_{x_{j}^{4}} ∥_{\infty}) .$ Applying [F2] to the one-sided difference in [F1] gives $D_{j, a}^{u p} ϕ (x_{i}, t_{n}) = ϕ_{x_{j}} (x_{i}, t_{n}) + O (h_{j} ∥ ϕ_{x_{j} x_{j}} ∥_{\infty})$ for either sign of $b_{j}^{a}$ . The coefficient bounds make these estimates uniform over the finite action and coordinate sets.

2.1F1F2step 1.1F3algebra

Taylor expansion in time gives $\frac{ϕ (x_{i}, t_{n} + τ) - ϕ (x_{i}, t_{n})}{τ} = ϕ_{t} (x_{i}, t_{n}) + O (τ ∥ ϕ_{t t} ∥_{\infty}) .$ Substitute the spatial estimates of step 1.1 into the update in [F1]. Since $∣ \max_{a} X_{a} - \max_{a} Y_{a} ∣ \leq \max_{a} ∣ X_{a} - Y_{a} ∣$ , the uniform spatial residual is bounded by the sum displayed in the statement. Combining with the time estimate gives its formula and its explicit smoothness dependence.

3.1step 2.1∎

The mesh consistency residual tends to zero when $τ \to 0$ and $h_{\max} \to 0$ with bounded coefficients. It supplies the local consistency input for the half-relaxed-limit result and makes no assertion of a universal convergence rate.

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Monotonicity, L-infinity stability, and consistency of the semi-Lagrangian HJB step

Statement

For the update $S_{h, τ}$ of Positive-interpolation semi-Lagrangian update for first-order HJB equations, if $V \leq W$ at all nodes then $S_{h, τ} V \leq S_{h, τ} W$ . If $∣ ℓ_{a} ∣ \leq L$ , then $∥ S_{h, τ} V ∥_{\infty} \leq ∥ V ∥_{\infty} + τ L .$ For $ϕ \in C^{2}$ in space and time with bounded derivatives, on a compact test region with $∣ D ϕ ∣ \leq P$ , $\frac{ϕ (t_{n} + τ, x_{i}) - S_{h, τ} ϕ (t_{n}, \cdot)_{i}}{τ} = ϕ_{t} (x_{i}, t_{n}) + H (x_{i}, t_{n}, D ϕ (x_{i}, t_{n})) + O (τ + \frac{h_{\max}^{}}{τ} + η_{k} (P)) .$ The term $η_{k} (P)$ is zero when the full finite control set is used.

Facts & Assumptions

Given: The positive-interpolation update, coefficient bounds $B, L$ , a periodic smooth test function, and the control-grid error $η_{k} (P)$ .

[F1]

The step is $S_{h, τ} V_{i} = \min_{a \in A_{k}} {I_{h} V (x_{i} + τ b_{a} (x_{i}, t_{n})) + τ ℓ_{a} (x_{i}, t_{n})}$ (Positive-interpolation semi-Lagrangian update for first-order HJB equations).

[F2]

The interpolant is order-preserving, contractive in $ℓ^{\infty}$ , and approximates a smooth function with error at most $\frac{1}{8} \sum_{j} h_{j}^{2} ∥ \partial_{j j} ϕ ∥_{\infty}$ (Positive partition of unity and interpolation error for periodic multilinear interpolation).

[F3]

The Hamiltonian and sign convention satisfy $H = \max_{a \in A} (- b_{a} \cdot p - ℓ_{a}) = - \min_{a \in A} (b_{a} \cdot p + ℓ_{a})$ , and $∣ b_{a} ∣ \leq B$ , $∣ ℓ_{a} ∣ \leq L$ (Periodic first-order controlled Hamilton–Jacobi Cauchy problem).

Proof

1.1F1F2F3algebra

If $V \leq W$ , [F2] gives $I_{h} V \leq I_{h} W$ at every footpoint. Adding the same running cost preserves the inequality for each action, and taking the minimum preserves it again; hence $S_{h, τ} V \leq S_{h, τ} W$ . Also $∣ I_{h} V ∣ \leq ∥ V ∥_{\infty}$ by the convex weights in [F2]. Every action value in [F1] therefore has absolute value at most $∥ V ∥_{\infty} + τ L$ , and so does their minimum.

1.2F1F2F3givenalgebra

Let $y_{a} = x_{i} + τ b_{a} (x_{i}, t_{n})$ in a periodic lift. By [F2], $I_{h} ϕ (t_{n}, \cdot) (y_{a}) = ϕ (t_{n}, y_{a}) + O (h_{\max}^{} ∥ D^{2} ϕ ∥_{\infty}) .$ Taylor expansion at $(x_{i}, t_{n})$ gives $ϕ (t_{n}, y_{a}) + τ ℓ_{a} = ϕ (t_{n}, x_{i}) + τ (b_{a} \cdot D ϕ + ℓ_{a}) (x_{i}, t_{n}) + O (τ^{2} B^{2} ∥ D^{2} ϕ ∥_{\infty}) .$ The minimum of finitely many entries changes by at most their largest perturbation. Replacing $A_{k}$ by $A$ changes the minimum of $b_{a} \cdot D ϕ + ℓ_{a}$ by at most $η_{k} (P)$ , by [F3]. A time Taylor expansion then yields the normalized residual in the statement, because $- \min_{a \in A} (b_{a} \cdot D ϕ + ℓ_{a}) = H$ .

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 prove order preservation, the stated sup-norm bound, and the local consistency error. The interpolation term is divided by $τ$ , so convergence of this residual requires the explicit condition $h_{\max}^{} / τ \to 0$ . No solution-error rate follows from these one-step estimates.

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Monotonicity, L-infinity stability, and consistency of the upwind HJB step

Statement

Assume the first-order HJB coefficients satisfy the hypotheses of Periodic first-order controlled Hamilton–Jacobi Cauchy problem. Under the CFL condition in CFL-monotone upwind update for first-order HJB equations, each one-control update is a convex combination of the current node and its one-sided neighbors plus $τ ℓ_{a}$ . Consequently the control-minimum update is order-preserving and $∥ U^{n + 1} ∥_{\infty} \leq ∥ U^{n} ∥_{\infty} + τ L .$ For a smooth periodic test function $ϕ$ with bounded derivatives through order two, the normalized local residual satisfies $\frac{ϕ (t_{n} + τ, x_{i}) - S_{h, τ} ϕ (t_{n}, \cdot)_{i}}{τ} = ϕ_{t} (x_{i}, t_{n}) + H (x_{i}, t_{n}, D ϕ (x_{i}, t_{n})) + O (τ + h_{\max} + η_{k} (P)),$ where $S_{h, τ}$ is the scheme map, $h_{\max} = \max_{j} h_{j}$ , $P$ bounds $∣ D ϕ ∣$ on the tested compact set, and $η_{k} (P) = 0$ for exact control minimization.

Facts & Assumptions

Given: The periodic upwind update, its CFL condition, bounded coefficient and running-cost bounds, and a smooth periodic test function.

[F1]

For fixed $a$ , the row is $G_{i}^{a} (V) = (1 - τ \sum_{j} \frac{∣ b_{a, j} ∣}{h_{j}}) V_{i} + τ \sum_{j} (\frac{b_{a, j}^{+}}{h_{j}} V_{i + e_{j}} + \frac{b_{a, j}^{-}}{h_{j}} V_{i - e_{j}}) + τ ℓ_{a},$ with periodic indices and $b^{+} = \max (b, 0)$ , $b^{-} = \max (- b, 0)$ ; the scheme takes the minimum of these rows (CFL-monotone upwind update for first-order HJB equations).

[F2]

For $g \in C^{2}$ with $∣ g^{''} ∣ \leq M$ , Taylor's theorem bounds a first-order increment by its linear term plus $M h^{2} / 2$ (A uniform derivative bound gives a uniform Taylor remainder bound).

[F3]

The HJB sign convention is $H (x, t, p) = - \min_{a \in A} (b_{a} (x, t) \cdot p + ℓ_{a} (x, t))$ (Periodic first-order controlled Hamilton–Jacobi Cauchy problem).

Proof

1.1F1givenalgebra

The CFL condition makes the center coefficient in [F1] nonnegative; all neighbor coefficients are nonnegative by $b_{a, j}^{\pm} \geq 0$ . Their sum is $1 - τ \sum_{j} ∣ b_{a, j} ∣ / h_{j} + τ \sum_{j} (b_{a, j}^{+} + b_{a, j}^{-}) / h_{j} = 1.$ Thus each row is a convex combination of nodal values plus $τ ℓ_{a}$ and is order-preserving. A pointwise minimum of order-preserving maps is order-preserving. Every row value lies in $[- ∥ U^{n} ∥_{\infty} - τ L, ∥ U^{n} ∥_{\infty} + τ L]$ , so the minimum has the same bound.

1.2F1F2F3givenalgebra

If $b_{a, j} \geq 0$ , the drift term in [F1] is $b_{a, j} (ϕ (x_{i} + h_{j} e_{j}) - ϕ (x_{i})) / h_{j}$ ; if $b_{a, j} < 0$ , it is $b_{a, j} (ϕ (x_{i}) - ϕ (x_{i} - h_{j} e_{j})) / h_{j}$ . In either case it equals $b_{a, j} ϕ_{x_{j}} (x_{i}) + O (∣ b_{a, j} ∣ h_{j} ∥ ϕ_{x_{j} x_{j}} ∥_{\infty})$ by [F2]. The time increment divided by $τ$ is $ϕ_{t} (x_{i}, t_{n}) + O (τ ∥ ϕ_{t t} ∥_{\infty})$ . The minimum is Lipschitz with respect to uniform perturbations of its entries, and replacing $A$ by $A_{k}$ changes the Hamiltonian by at most $η_{k} (P)$ . Combining these estimates and [F3] gives the stated residual.

2.1F1step 1.1step 1.2∎

The weights in [F1] show that the upwind step is a monotone, stable, consistent approximation with the stated error. This conclusion is local consistency only and asserts no solution-error rate.

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Half-relaxed limits of the monotone diagonal-diffusion HJB scheme

Statement

Let $h_{k, j} \to 0$ , $τ_{k} = T / M_{k} \to 0$ , and let $U_{k}$ be the finite solutions of Explicit monotone diagonal-diffusion Bellman grid update with its diffusion-dominance and CFL inequalities at every row. Extend each array multilinearly in space, constantly on each time interval, and by its initial interpolant for $- 1 < t < 0$ ; identify the mesh sequence with the parameter family convention of Half-relaxed limits of a locally bounded family. Its upper half-relaxed limit is a bounded upper semicontinuous viscosity subsolution, and its lower half-relaxed limit is a bounded lower semicontinuous viscosity supersolution of the periodic diagonal-diffusion Cauchy problem, with the respective relaxed initial inequalities equal to $u_{0}$ . This is a stability statement only; it does not claim that the two limits coincide.

Facts & Assumptions

Given: The scheme, its finite horizon and mesh assumptions in the statement, and a smooth test function at a strict interior contact.

[F1]

Every fixed-control row is order-preserving and commutes with constant shifts. Its nonnegative weights sum to one; therefore the max update is order-preserving and changes the sup norm by at most $τ_{k} F$ (Nonnegative row weights for the diagonal-diffusion HJB stencil, Explicit monotone diagonal-diffusion Bellman grid update).

[F2]

The normalized smooth-test residual tends uniformly on compact sets to $ϕ_{t} - \max_{a} {\sum_{j} (α_{j}^{a} ϕ_{x_{j} x_{j}} + b_{j}^{a} ϕ_{x_{j}}) + f^{a}}$ as $h_{k, \max}, τ_{k} \to 0$ (Smooth consistency of the diagonal-diffusion Bellman stencil).

[F3]

Strict upper/lower contacts of the half-relaxed limits localize to nodal maxima/minima on a fixed cylinder, with nodes converging to the contact (Localization of strict half-relaxed contacts to periodic mesh nodes).

[F4]

For this operator, smooth tests suffice to establish the viscosity inequality for all $C^{1, 2}$ tests (Smooth test functions suffice for the first- and second-order HJB viscosity notions).

[F5]

The initial nodal values converge uniformly to $u_{0}$ as $t_{n} \to 0$ for this scheme (Initial trace for the explicit diagonal-diffusion HJB scheme).

[F6]

The half-relaxed limits are the set-defined USC/LSC envelopes of the locally bounded interpolant family and satisfy $\underline{U} \leq \overline{U}$ (Half-relaxed limits of a locally bounded family).

[A1]

Countable Choice is used only through [F3] and the sequential recovery form of [F6] (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1F1F6givenalgebra

The row weights in [F1] give $∥ U_{k}^{n} ∥_{\infty} \leq ∥ u_{0} ∥_{\infty} + n τ_{k} F \leq ∥ u_{0} ∥_{\infty} + T F .$ Multilinear interpolation is a convex combination, so the same bound holds for the space-time extensions. They are locally bounded and their half-relaxed limits have the semicontinuity and order properties in [F6].

2.1A1F1F2F3F4step 1.1algebra

Let $\overline{U} - ϕ$ have a local maximum at $(x_{0}, t_{0})$ , with $0 < t_{0} < T$ and smooth $ϕ$ . Add $ε (∣ x - x_{0} ∣^{4} + ∣ t - t_{0} ∣^{4})$ to $ϕ$ ; this makes the contact strict without changing its derivatives used by the PDE at $(x_{0}, t_{0})$ , as in [F4]. Select nodal local maxima $(x_{k}, t_{k})$ from [F3] and put $c_{k} = U_{i_{k}}^{n_{k}} - ϕ (x_{k}, t_{k})$ . All coordinate neighbors used by the explicit stencil at the previous time lie in the fixed contact cylinder for sufficiently large $k$ , because their distance to $(x_{k}, t_{k})$ is at most $h_{k, \max} + τ_{k}$ . Thus $U_{j}^{n_{k} - 1} \leq ϕ (x_{j}, t_{k} - τ_{k}) + c_{k}$ at every stencil node. Monotonicity and shift invariance in [F1] give $U_{i_{k}}^{n_{k}} = S_{k} (U^{n_{k} - 1})_{i_{k}} \leq S_{k} (ϕ (\cdot, t_{k} - τ_{k}) + c_{k})_{i_{k}} = S_{k} (ϕ (\cdot, t_{k} - τ_{k}))_{i_{k}} + c_{k} .$ At the contact node $U_{i_{k}}^{n_{k}} = ϕ (x_{k}, t_{k}) + c_{k}$ , so the normalized residual is nonpositive. Passing to the limit by [F2] yields $ϕ_{t} - \max_{a} {\sum_{j} (α_{j}^{a} ϕ_{x_{j} x_{j}} + b_{j}^{a} ϕ_{x_{j}}) + f^{a}} \leq 0$ at $(x_{0}, t_{0})$ .

2.2A1F1F2F3F4step 1.1algebra

At a local minimum of $\underline{U} - ϕ$ , subtract $ε (∣ x - x_{0} ∣^{4} + ∣ t - t_{0} ∣^{4})$ from $ϕ$ to make the contact strict without changing its tested derivatives, as in [F4]. The minimizing nodes in [F3] give $U_{j}^{n_{k} - 1} \geq ϕ (x_{j}, t_{k} - τ_{k}) + c_{k}$ on the stencil. Order preservation reverses the update inequality, so the residual is nonnegative and its limit gives the supersolution inequality. By [F4], both inequalities extend from smooth tests to all tests in the declared $C^{1, 2}$ class.

3.1F5F6step 1.1algebra∎

The initial-trace estimate [F5] and uniform continuity of $u_{0}$ show that every space-time sequence with mesh tending to zero and time tending to zero has values tending to $u_{0}$ at its limiting spatial point; the spatial interpolation adds at most the modulus of $u_{0}$ at distance $\sqrt{d} h_{k, \max}$ . Therefore the upper and lower relaxed limits both have the prescribed relaxed initial trace. Boundedness and semicontinuity were proved in step 1.1, completing the stability statement.

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Convergence of the positive-interpolation and upwind first-order HJB schemes

Statement

Let $u_{0}$ and the coefficients satisfy Periodic first-order controlled Hamilton–Jacobi Cauchy problem. Consider a sequence of periodic grids with $h_{k, \max} \to 0$ , time steps $τ_{k} = T / M_{k} \to 0$ , $M_{k} \in N$ , and finite nonempty control sets $A_{k} \subseteq A$ whose Hamiltonian errors $η_{k} (R)$ tend to zero for each finite $R$ . For the positive-interpolation scheme additionally assume $h_{k, \max}^{2} / τ_{k} \to 0$ . For the upwind scheme assume its CFL condition at every step. Extend the initial interpolant constantly for $- 1 < t < 0$ and identify the mesh sequence with the parameter family $V_{ϵ} = V_{k}$ for $(k + 1)^{- 1} < ϵ \leq k^{- 1}$ in Half-relaxed limits of a locally bounded family. Then the respective piecewise multilinear-in-space, piecewise-constant-in-time extensions of the nodal solutions converge locally uniformly on $T^{d} \times [0, T)$ to the same bounded continuous viscosity solution of the Cauchy problem. This conclusion includes exact finite-control models ( $A_{k} = A$ ), where $η_{k} = 0$ . The proof does not assert a rate.

Facts & Assumptions

Given: One of the two scheme families and mesh conditions in the statement. Let $V_{k}$ denote its spatially multilinear and temporally piecewise-constant periodic extension on $R^{d} \times (- 1, T)$ , initialized as in the statement for $- 1 < t < 0$ .

[F1]

Each update is order-preserving and commutes with adding a constant to all nodal values; it is bounded by the preceding sup norm plus $τ_{k} L$ . Its smooth-test residual is $ϕ_{t} + H + o (1)$ , uniformly on compact test sets, with the respective conditions $h_{k, \max}^{2} / τ_{k} \to 0$ or CFL and with $η_{k} (P) \to 0$ (Monotonicity, L-infinity stability, and consistency of the semi-Lagrangian HJB step, Monotonicity, L-infinity stability, and consistency of the upwind HJB step).

[F2]

For any locally bounded family, the set-defined upper and lower half-relaxed limits are respectively upper and lower semicontinuous, satisfy $\underline{V} \leq \overline{V}$ , and have sequential recovery pairs under Countable Choice (Half-relaxed limits of a locally bounded family, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F3]

A strict upper (respectively lower) half-relaxed contact by a smooth test localizes to nodal local maxima (respectively minima) on a fixed space-time ball; the localizing nodes converge to the contact and their values recover the envelope (Localization of strict half-relaxed contacts to periodic mesh nodes).

[F4]

Testing a subsolution or supersolution with smooth tests is equivalent to testing with all $C^{1}$ tests for this first-order equation (Smooth test functions suffice for the first- and second-order HJB viscosity notions).

[F5]

The periodic HJB Hamiltonian obeys $∣ H (x, t, p) - H (y, s, p) ∣ \leq L (1 + ∣ p ∣) (∣ x - y ∣ + ∣ t - s ∣), ∣ H (x, t, p) - H (x, t, q) ∣ \leq B ∣ p - q ∣,$ so with $C = \max {1, L, B}$ it satisfies the positive-constant whole-space comparison bounds and comparison applies to bounded USC subsolutions and bounded LSC supersolutions on the periodic lift when their initial traces are ordered (Periodic first-order controlled Hamilton–Jacobi Cauchy problem, Comparison for first-order Hamilton--Jacobi equations).

[F6]

For the semi-Lagrangian branch the nodal values approach $u_{0}$ uniformly as $t_{n} \to 0$ under $h_{k, \max}^{2} / τ_{k} \to 0$ ; for the upwind branch the same holds under its CFL condition (Initial trace for the positive-interpolation and upwind HJB schemes).

[F7]

The convex interpolation weights give $∣ I_{h_{k}} u_{0} (x) - u_{0} (x) ∣ \leq ω_{0} (\sqrt{d} h_{k, \max})$ for a modulus $ω_{0} (δ) \to 0$ , by uniform continuity of the periodic initial data (Positive partition of unity and interpolation error for periodic multilinear interpolation, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[F8]

Every open cover of a compact subset of the lifted space-time domain has a finite subcover (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[A1]

Countable Choice is used in [F2] and [F3] only to select near-recovery pairs and contradiction subsequences; no choice is used in any finite row calculation (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1F1F2givenalgebra

By the stability part of [F1] and $M_{k} τ_{k} = T$ , all nodal values satisfy $∥ U_{k}^{n} ∥_{\infty} \leq ∥ u_{0} ∥_{\infty} + L t_{n} \leq ∥ u_{0} ∥_{\infty} + L T .$ The convex interpolation weights give the same bound for $V_{k}$ , so the family is locally bounded on $T^{d} \times [0, T)$ . Form its upper and lower half-relaxed limits $\overline{V}, \underline{V}$ as in [F2].

2.1A1F1F3F4step 1.1algebra

Let $\overline{V} - ϕ$ have a local maximum at an interior point $(x_{0}, t_{0})$ , with $ϕ$ smooth. Add $ε (∣ x - x_{0} ∣^{4} + ∣ t - t_{0} ∣^{4})$ to $ϕ$ ; this makes the contact strict without changing its first derivatives at $(x_{0}, t_{0})$ , as in [F4]. Choose the nodal local maxima $(x_{k}, t_{k})$ in [F3], with $(x_{k}, t_{k}) \to (x_{0}, t_{0})$ ; write $c_{k} = U_{i_{k}}^{n_{k}} - ϕ (x_{k}, t_{k})$ . For all sufficiently large $k$ , every node used by the previous-time update at $(x_{k}, t_{k})$ lies inside the fixed contact ball: the upwind neighbors are at distance at most $h_{k, \max}$ and each semi-Lagrangian interpolation vertex is at distance at most $τ_{k} B + \sqrt{d} h_{k, \max}$ . Hence each such previous-layer nodal value satisfies $U_{j}^{n_{k} - 1} \leq ϕ (x_{j}, t_{k} - τ_{k}) + c_{k}$ . Order preservation and constant-shift invariance from [F1] imply $U_{i_{k}}^{n_{k}} = S_{k} (U^{n_{k} - 1})_{i_{k}} \leq S_{k} (ϕ (\cdot, t_{k} - τ_{k}) + c_{k})_{i_{k}} = S_{k} (ϕ (\cdot, t_{k} - τ_{k}))_{i_{k}} + c_{k} .$ Since $U_{i_{k}}^{n_{k}} = ϕ (x_{k}, t_{k}) + c_{k}$ , the normalized residual is nonpositive. Taking $k \to \infty$ and using the consistency estimate in [F1] gives $ϕ_{t} (x_{0}, t_{0}) + H (x_{0}, t_{0}, D ϕ (x_{0}, t_{0})) \leq 0$ . By [F4], $\overline{V}$ is a viscosity subsolution for the full $C^{1}$ test class.

2.2A1F1F3F4step 1.1algebra

At a local minimum of $\underline{V} - ϕ$ , subtract $ε (∣ x - x_{0} ∣^{4} + ∣ t - t_{0} ∣^{4})$ from $ϕ$ to make the contact strict without changing its derivatives at $(x_{0}, t_{0})$ , as in [F4]. Use the minimizing nodes in [F3]. All previous-layer stencil nodes lie in the contact ball, so $U_{j}^{n_{k} - 1} \geq ϕ (x_{j}, t_{k} - τ_{k}) + c_{k}$ . Order preservation and constant-shift invariance now yield the reverse update inequality, hence the normalized consistency residual is nonnegative. Passing to the limit gives the viscosity supersolution inequality. The smooth-test equivalence [F4] extends it to every $C^{1}$ lower test.

2.3F2F5F6F7step 1.1algebra

The initial trace estimate [F6] and the interpolation error [F7] show that both half-relaxed limits equal $u_{0}$ on $t = 0$ . They are bounded by step 1.1 and have the required semicontinuity by [F2]. Apply comparison [F5] to the periodic lifts of $\overline{V}$ and $\underline{V}$ : it gives $\overline{V} \leq \underline{V}$ . The reverse inequality is part of [F2], so they coincide with a continuous viscosity solution having initial data $u_{0}$ .

3.1F2F8step 2.3algebra∎

Fix $T^{'} < T$ and let $K = [0, 1]^{d} \times [0, T^{'}]$ in the periodic lift; it is compact by [F8]. Fix $ε > 0$ and write $u$ for the common continuous limit from step 2.3. For $z \in K$ and $q \in Q_{> 0}$ , put $P (z, q)$ when the set-defined upper and lower envelopes in [F2] give $∣ V_{k} (y) - u (z) ∣ \leq ε / 3 for every integer k with 1 / k < q and every y with ∣ y - z ∣ < q,$ and continuity gives $∣ u (y) - u (z) ∣ < ε / 3$ whenever $∣ y - z ∣ < q$ . The family of all balls ${B (z, q / 2) : z \in K, q \in Q_{> 0}, P (z, q)}$ covers $K$ : each point has a positive real radius from the two envelope definitions and continuity, and a smaller rational radius still works. This is the family of all qualifying pairs, so no radius is selected for every point. By [F8] take a finite subcover and let $q_{*} > 0$ be its least radius. For $1 / k < q_{*}$ , every $y \in K$ lies in one of these balls and therefore $∣ V_{k} (y) - u (y) ∣ < 2 ε / 3 < ε$ . This proves uniform convergence on the fundamental box through $T^{'}$ , hence on the torus and every compact subset of $T^{d} \times [0, T)$ .

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Comparison for periodic diagonal controlled-diffusion HJB equations

Statement

Assume the Axiom of Choice and the exact finite-control coefficient hypotheses of Periodic diagonal controlled-diffusion HJB Cauchy problem, in any dimension $d \geq 1$ . Let $u$ be bounded upper semicontinuous and $v$ bounded lower semicontinuous on $T^{d} \times (0, T)$ ; assume their interior $C^{1, 2}$ -test inequalities are those of Second-order parabolic viscosity subsolutions and supersolutions, using the local maximum/minimum contact convention of Viscosity subsolutions and supersolutions of a first-order equation and of the Cauchy problem. Suppose the raw relaxed initial traces are ordered at every $x \in T^{d}$ : ${lim sup}_{\begin{array}{c} (y, t) \to (x, 0) \\ t > 0 \end{array}} u (y, t) \leq {lim inf}_{\begin{array}{c} (y, t) \to (x, 0) \\ t > 0 \end{array}} v (y, t) .$ Then $u \leq v$ on $T^{d} \times (0, T)$ . In particular this applies to the model's common continuous initial datum and its two relaxed initial inequalities. Defining the values at $t = 0$ by their respective raw upper and lower envelopes gives the same order there. No continuity, spatial Lipschitz bound, or differentiability of $u, v$ is assumed.

Facts & Assumptions

Given: The exact coefficient class, bounded semicontinuous sub- and supersolutions, ordered relaxed traces, and the Axiom of Choice.

[A1]

Every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

The model has finite nonempty controls, $α_{j}^{c} \geq α_{*} > 0$ , and common bounds and a common joint space-time Lipschitz constant $L$ for $α, b, f$ (Periodic diagonal controlled-diffusion HJB Cauchy problem).

[F2]

At a local maximum of $w_{1} (z_{1}) + w_{2} (z_{2}) - ϕ (z_{1}, z_{2})$ for USC functions on open Euclidean sets and $C^{2}$ $ϕ$ , for every $μ > 0$ there are closed elliptic superjets with gradients $D_{i} ϕ$ and Hessians $H_{i}$ satisfying $diag (H_{1}, H_{2}) \leq D^{2} ϕ + μ (D^{2} ϕ)^{2}$ (Elliptic theorem of sums with the full block matrix bound).

[F3]

The closed elliptic space-time jets of the subsolution and supersolution satisfy their parabolic HJB inequalities using the time-gradient component and spatial Hessian block (Closed elliptic space-time jets test the parabolic diagonal HJB inequality).

[F4]

USC functions attain maxima on compact Euclidean sets. Closed bounded sets are compact, and bounded sequences have convergent subsequences (Semicontinuous extreme value theorem on compact Euclidean sets, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, For $n \geq 1$ every bounded sequence in $R^{n}$ has a convergent subsequence). Torus compactness follows by lifting to the compact fundamental cube and taking the quotient. The declared Axiom of Choice permits countable maximizing-point selections (The Axiom of Choice).

[F5]

The flat unit-torus distance is $D (x, y) = \min_{k \in Z^{d}} ∣ \tilde{x} - \tilde{y} + k ∣$ , where $\tilde{x}, \tilde{y}$ are any lifts to $R^{d}$ ; when $D (x, y) < 1 / 2$ , the minimizing lift is locally unique and $D^{2}$ is smooth there, as proved in step 1.1 below.

Proof

1.1F5algebraconstruct

For any fixed lifts, a lattice minimizer exists: only finitely many integer vectors can give a norm no larger than the norm at $k = 0$ . If a minimizing difference $r$ has $∣ r ∣ = D (x, y) < 1 / 2$ , then for every nonzero $k \in Z^{d}$ , $∣ k ∣ \geq 1$ and $r \cdot k \geq - ∣ r ∣ ∣ k ∣$ : for $r \neq 0$ this follows by expanding $∣ ∣ k ∣ r + ∣ r ∣ k ∣^{2} \geq 0$ , and for $r = 0$ it is immediate. Consequently $∣ r + k ∣^{2} - ∣ r ∣^{2} = ∣ k ∣^{2} + 2 r \cdot k \geq ∣ k ∣ (∣ k ∣ - 2 ∣ r ∣) > 0.$ Thus that minimizing difference is unique. Choose continuous local lifts near $x, y$ and keep the minimizing lattice vector fixed; their resulting difference $r^{'}$ stays of norm less than $1 / 2$ in a sufficiently small neighborhood. The same inequality for every nonzero $k$ shows that this difference remains the unique minimizer throughout that neighborhood. Hence there $D^{2} = ∣ r^{'} ∣^{2}$ , a smooth quadratic in the local coordinates, justifying [F5].

1.2A1F4givenchooseassume-contra

Define $u (x, 0)$ as the raw upper trace and $v (x, 0)$ as the raw lower trace. Boundedness makes them finite. For example, if $x_{k} \to x$ and $u (x_{k}, 0)$ approaches its limsup, the raw upper-trace definition and [A1] let us select interior $(y_{k}, t_{k})$ with $∣ y_{k} - x_{k} ∣ + t_{k} < 1 / k$ and $u (y_{k}, t_{k}) > u (x_{k}, 0) - 1 / k$ . Thus $y_{k} \to x$ , $t_{k} \to 0$ , and ${lim sup}_{k} u (x_{k}, 0) \leq u (x, 0)$ . The lower-trace argument is dual. For a sequence with positive times tending to zero, the raw trace definition gives the USC/LSC inequalities directly; splitting a mixed sequence into positive-time and initial-edge subsequences proves the extensions are USC/LSC on the closed initial edge. The assumed order gives $u (x, 0) \leq v (x, 0)$ . Assume for contradiction that $u (x_{*}, t_{*}) - v (x_{*}, t_{*}) > 0$ at some $0 < t_{*} < T$ . Choose $S$ with $t_{*} < S < T$ , then choose $η, γ > 0$ sufficiently small that $m : = u (x_{*}, t_{*}) - v (x_{*}, t_{*}) - η t_{*} - γ / (S - t_{*}) > 0.$ Put $P (t) = η t + γ / (S - t)$ . If $u (x, t) - v (y, s) - P (t) \geq 0$ , boundedness bounds $P (t)$ above by $∥ u ∥_{\infty} + ∥ v ∥_{\infty}$ , so $t \leq S - c$ for some $c > 0$ .

2.1A1F4F5step 1.2algebra

Using [F5], define $M_{ε} = \max_{x, y \in T^{d}, 0 \leq t < S} {u (x, t) - v (y, t) - P (t) - D (x, y)^{2} / (2 ε)} .$ The maximum exists by step 1.2 and [F4]: the expression is USC, and its positive superlevel sets stay in a compact time interval bounded away from $S$ . Also $M_{ε} \geq m$ . Write $M_{0} = \max_{x, 0 \leq t < S} (u (x, t) - v (x, t) - P (t)) \geq m$ , also attained. At any maximizing triple $(x_{ε}, y_{ε}, t_{ε})$ , boundedness gives $D (x_{ε}, y_{ε})^{2} = O (ε)$ . Compactness and semicontinuity imply $M_{ε} \to M_{0}$ : for every sequence $ε ↓ 0$ , a subsequence of maximizers tends to $(x, x, t)$ , and the limsup of the unpenalized expression is at most $M_{0}$ , whereas $M_{ε} \geq M_{0}$ . Moreover $0 \leq D (x_{ε}, y_{ε})^{2} / (4 ε) \leq M_{2 ε} - M_{ε} ⟶ 0,$ by evaluating the $2 ε$ maximum at an $ε$ maximizer. These conclusions hold uniformly over maximizing triples, by the same sequential compactness argument. No sequence of these maximizers can have $t_{ε} \to 0$ : then $x_{ε}, y_{ε} \to x$ along a subsequence and semicontinuity and the initial order give limsup of their maximum values at most $- P (0) < 0$ , contradicting $M_{ε} \geq m$ .

3.1A1F4F5step 1.2step 2.1algebra

For fixed sufficiently small $ε$ , introduce independent times and maximize $Φ_{ε, δ} (x, t, y, s) = u (x, t) - v (y, s) - P (t) - D (x, y)^{2} / (2 ε) - (t - s)^{2} / (2 δ)$ over $x, y \in T^{d}$ , $0 \leq t < S$ , $0 \leq s \leq S$ . Maxima exist by step 1.2 and [F4] and are at least $M_{ε} \geq m$ . Boundedness gives $∣ t - s ∣ = O (\sqrt{δ})$ and $t \leq S - c$ uniformly. For fixed $ε$ , compactness shows that as $δ ↓ 0$ every limit of maximizing quadruples is a synchronized maximizer of $M_{ε}$ : the limsup of the expression without the temporal penalty is at most $M_{ε}$ , while the maximum is at least $M_{ε}$ . Hence the temporal penalty tends to zero too. By step 2.1 and [F5], for all sufficiently small $ε$ these limiting times are positive and the spatial distance is less than $1 / 2$ . Consequently for each such $ε$ and all sufficiently small $δ$ , both times are strictly in $(0, S)$ and $D (x, y)^{2} = ∣ r ∣^{2}$ is smooth near the maximizer in local lifts with $r = x - y$ and $∣ r ∣ = D (x, y)$ . Thus all contacts used next are interior in the original cylinder.

4.1F2F5step 3.1algebra

At such a maximizer apply [F2] to $u (t, x)$ and $- v (s, y)$ with $ϕ (t, x, s, y) = ∣ x - y ∣^{2} / (2 ε) + (t - s)^{2} / (2 δ) + P (t), μ = ε .$ It gives a closed upper jet of $u$ with gradient $(a, p)$ and Hessian $H$ , and a closed lower jet of $v$ with gradient $(b, p)$ and Hessian $K$ , where $p = r / ε, a = (t - s) / δ + P^{'} (t), b = (t - s) / δ, a - b = P^{'} (t) \geq η .$ Negating the jet of $- v$ gives the lower jet of $v$ directly from its defining expansion and witnessing sequence. Let $X, Y$ be the spatial principal blocks of $H, K$ . The full Hessian of $ϕ$ has no space-time mixed entries. Its spatial block is $ε^{- 1} B$ with $B = (\begin{array}{cc} I & - I \\ - I & I \end{array})$ and $B^{2} = 2 B$ . Restricting the upper matrix inequality in [F2] to vectors with zero temporal components therefore gives $(\begin{array}{cc} X & 0 \\ 0 & - Y \end{array}) \leq (3 / ε) B .$ The temporal Hessian, including $P^{''} (t)$ and the potentially large $1 / δ$ , contributes nothing to this spatial restriction.

5.1F1F3step 4.1algebra

For each control $c$ write $G_{c} (x, t, p, X) = \sum_{j} (α_{j}^{c} (x, t) X_{j j} + b_{j}^{c} (x, t) p_{j}) + f^{c} (x, t)$ . By [F3], $a \leq \max_{c} G_{c} (x, t, p, X)$ and $b \geq \max_{c} G_{c} (y, s, p, Y)$ . Finiteness gives $η \leq a - b \leq \max_{c} {G_{c} (x, t, p, X) - G_{c} (y, s, p, Y)} .$ For $σ_{j}^{c} = \sqrt{α_{j}^{c}}$ , rationalization and [F1] give $∣ σ_{j}^{c} (x, t) - σ_{j}^{c} (y, s) ∣ \leq \frac{L}{2 \sqrt{α_{*}}} (∣ r ∣ + ∣ t - s ∣) .$ Evaluate the spatial matrix inequality of step 4.1 on each vector $(σ_{j}^{c} (x, t) e_{j}, σ_{j}^{c} (y, s) e_{j})$ and sum over $j$ to obtain $\sum_{j} (α_{j}^{c} (x, t) X_{j j} - α_{j}^{c} (y, s) Y_{j j}) \leq \frac{3}{ε} \sum_{j} ∣ σ_{j}^{c} (x, t) - σ_{j}^{c} (y, s) ∣^{2} \leq \frac{3 d L^{2}}{4 α_{*} ε} (∣ r ∣ + ∣ t - s ∣)^{2} .$ The drift difference is bounded by $C (∣ r ∣ + ∣ t - s ∣) ∣ r ∣ / ε$ , and the forcing difference by $L (∣ r ∣ + ∣ t - s ∣)$ , uniformly over the finite controls, using [F1]. Hence a constant $C$ independent of $ε, δ$ satisfies $η \leq C {\frac{(∣ r ∣ + ∣ t - s ∣)^{2}}{ε} + \frac{(∣ r ∣ + ∣ t - s ∣) ∣ r ∣}{ε} + ∣ r ∣ + ∣ t - s ∣} .$

6.1A1F4step 2.1step 3.1step 5.1discharge-contradiction∎

First let $δ ↓ 0$ for fixed $ε$ , selecting a convergent subsequence of maximizing quadruples by [A1] and [F4]. Since $ε$ is fixed, every term containing $∣ t - s ∣ / ε$ vanishes. The remaining spatial pair is a maximizer of $M_{ε}$ , so step 5.1 gives $η \leq C^{'} (D (x_{ε}, y_{ε})^{2} / ε + D (x_{ε}, y_{ε}))$ . Now let $ε ↓ 0$ . Both terms vanish by step 2.1, contradicting $η > 0$ . The assumed positive gap is impossible, so $u \leq v$ at every interior point. The initial-edge assertion is its assumed order.

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Conditional convergence of the diagonal-diffusion scheme under comparison

Statement

For a sequence of diagonal-diffusion grid schemes satisfying the hypotheses of Half-relaxed limits of the monotone diagonal-diffusion HJB scheme, suppose in addition that the exact periodic Cauchy problem Periodic diagonal controlled-diffusion HJB Cauchy problem has a comparison principle for bounded upper semicontinuous subsolutions and bounded lower semicontinuous supersolutions with ordered relaxed initial traces. Then the upper and lower half-relaxed limits coincide, define the unique bounded continuous viscosity solution, and the spatially interpolated numerical solutions converge locally uniformly on $T^{d} \times [0, T)$ to it. This corollary is conditional on the stated second-order comparison property; it does not supply or assert that property.

Facts & Assumptions

Given: The half-relaxed stability statement and the comparison hypothesis in the corollary.

[F1]

The upper limit is a bounded USC subsolution and the lower limit is a bounded LSC supersolution with the same initial data, and $\underline{U} \leq \overline{U}$ (Half-relaxed limits of the monotone diagonal-diffusion HJB scheme).

[A1]

The stated comparison principle applies to this periodic coefficient class for the bounded semicontinuous sub- and supersolutions in [F1], with their relaxed initial ordering. This is an explicit conditional hypothesis, not a result proved on this page (Periodic diagonal controlled-diffusion HJB Cauchy problem, Second-order parabolic viscosity subsolutions and supersolutions).

[F2]

The set definitions of the relaxed limits imply that if both envelopes equal a continuous function on a compact set, then a finite subcover of nearby upper and lower bounds proves uniform convergence there (Half-relaxed limits of a locally bounded family, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

Proof

1.1A1F1algebra

Apply [A1] to the upper subsolution and lower supersolution supplied by [F1]. Their common initial trace gives the required ordering, hence $\overline{U} \leq \underline{U}$ . Together with the reverse inequality in [F1], this proves equality.

2.1A1F1step 1.1

The common function is both upper and lower semicontinuous by [F1], hence continuous. It is both a viscosity subsolution and supersolution, so is a viscosity solution with the prescribed initial data. Comparison in [A1] implies uniqueness.

3.1F1F2step 1.1step 2.1∎

Fix $T^{'} < T$ and let $K = [0, 1]^{d} \times [0, T^{'}]$ in the periodic lift; this box is compact by Heine--Borel. Write $u$ for the common limit from step 2.1 and fix $ε > 0$ . For $z \in K$ and rational $q > 0$ , include $B (z, q / 2)$ in a family when for every integer $k$ with $1 / k < q$ and every $y$ with $∣ y - z ∣ < q$ , $∣ V_{k} (y) - u (z) ∣ < ε / 3$ , and continuity gives $∣ u (y) - u (z) ∣ < ε / 3$ . The set definitions in [F2] guarantee that these balls cover $K$ : each point has a positive real radius, and a smaller rational radius works. The family consists of all qualifying pairs, so no radius is selected point by point. By [F2], a finite subcover exists; take the least of its finitely many rational radii. The triangle inequality then gives $∣ V_{k} (y) - u (y) ∣ < 2 ε / 3 < ε$ uniformly on $K$ for all sufficiently large $k$ . This proves uniform convergence on the fundamental box through $T^{'}$ , hence on the torus and every compact subset of $T^{d} \times [0, T)$ . Thus the extensions converge locally uniformly.

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Unconditional convergence and existence for the periodic diagonal-diffusion HJB scheme

Statement

Assume the Axiom of Choice. For the full model of Periodic diagonal controlled-diffusion HJB Cauchy problem, let $h_{k, j} = 1 / N_{k, j} \to 0$ , $N_{k, j} \geq 3$ , and $τ_{k} = T / M_{k} \to 0$ . Assume the explicit scheme Explicit monotone diagonal-diffusion Bellman grid update satisfies its diffusion-dominance and CFL inequalities at every row, and initialize it by $U_{k, i}^{0} = u_{0} (x_{k, i})$ . Its multilinear-in-space, constant-on-time-interval extensions converge locally uniformly on $T^{d} \times [0, T)$ to the unique bounded continuous viscosity solution of the model Cauchy problem. Such admissible mesh sequences exist for every model in this coefficient class, so the conclusion also proves existence of that viscosity solution. Comparison and uniqueness apply among all bounded continuous viscosity solutions with the prescribed initial trace. There is no additional comparison hypothesis and no claimed solution-error rate.

Facts & Assumptions

Given: The exact coefficient class, continuous periodic initial data, and mesh family in the statement; assume the Axiom of Choice.

[A1]

Every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

The existing local discrete half-relaxed proof gives a bounded USC subsolution $\overline{U}$ and bounded LSC supersolution $\underline{U}$ , ordered as $\underline{U} \leq \overline{U}$ , with the prescribed relaxed initial traces. It includes the proven stencil monotonicity, smooth-test consistency, mesh contact localization, smooth-test equivalence and discrete initial estimate (Half-relaxed limits of the monotone diagonal-diffusion HJB scheme, Half-relaxed limits of a locally bounded family).

[F2]

The exact coefficient class has comparison for those bounded semicontinuous functions and ordered relaxed initial traces (Comparison for periodic diagonal controlled-diffusion HJB equations).

[F3]

Under precisely [F1] and a comparison principle of the strength [F2], the existing conditional corollary identifies both envelopes with a bounded continuous viscosity solution and proves locally uniform convergence of the spatial interpolants through every $T^{'} < T$ (Conditional convergence of the diagonal-diffusion scheme under comparison).

[F4]

There are common finite coefficient bounds $α_{j}^{a} \leq A^{*}$ , $∣ b_{j}^{a} ∣ \leq B^{*}$ , with $α_{j}^{a} \geq α_{*} > 0$ . All explicit updates are finite maxima of finite expressions and are defined on any finite admissible grid (Periodic diagonal controlled-diffusion HJB Cauchy problem, Explicit monotone diagonal-diffusion Bellman grid update).

[F5]

For $x \geq 0$ , $\max {1, ⌈ x ⌉}$ is the least positive integer at least $x$ (Ceilings give the least positive integer meeting a real threshold).

Proof

1.1A1F1F2F3givenalgebra

Apply [F1] to the given mesh family. Comparison [F2] applies, because [F1] supplies exactly its bounded USC/LSC functions, interior viscosity inequalities and ordered relaxed initial traces. It gives $\overline{U} \leq \underline{U}$ , hence equality by the reverse inequality in [F1]. The conditional implication [F3] now has its premise proved locally, and gives the claimed bounded continuous viscosity solution and convergence on each compact cylinder $T^{d} \times [0, T^{'}]$ , $T^{'} < T$ . Comparing any two bounded continuous solutions in both orders using [F2] proves uniqueness. Full AC is inherited from [F2], and also supplies the Countable Choice used by [F1].

2.1F4F5step 1.1constructalgebra∎

To ensure the existence conclusion is nonvacuous, choose uniform grids $N_{k, j} = N_{0} + k$ for every $j$ , where $N_{0} = \max {3, ⌈ B^{*} / α_{*} ⌉}$ is an integer by [F5]. Put $N = N_{0} + k$ and $M_{k} = \max {1, ⌈ T d (2 A^{*} N^{2} + B^{*} N) ⌉} + k, τ_{k} = T / M_{k} .$ Then $∣ b_{j}^{a} ∣ h_{j} \leq B^{*} / N \leq α_{*} \leq α_{j}^{a}$ , while $τ_{k} \sum_{j} (2 α_{j}^{a} / h_{j}^{2} + ∣ b_{j}^{a} ∣ / h_{j}) \leq (T / M_{k}) d (2 A^{*} N^{2} + B^{*} N) \leq 1.$ Thus every row is admissible, $h_{k, j} \to 0$ , and $τ_{k} \leq T / k \to 0$ . The finite recurrence in [F4] produces the numerical arrays on this explicit mesh family. Step 1.1 gives their limit and therefore existence for every stated datum and coefficient family.

5 · Examples, counterexamples and false statements

None yet.