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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.

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Sources

Crandall, Ishii and Lions, User's guide to viscosity solutions (standard reference, not scraped)
Barles and Souganidis, Convergence of approximation schemes for fully nonlinear second order equations (standard reference, not scraped)