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Solutions of projected contact dynamics are unique under the inherited choice assumptions

Statement

Assume Countable Choice and Dependent Choice. Let $d \in N$ and $T > 0$ , let $C \subseteq R^{d}$ be nonempty, closed, and convex, and let $f : [0, T] \times C \to R^{d}$ satisfy $∥ f (t, x) - f (t, y) ∥_{2} \leq L ∥ x - y ∥_{2}$ for all $t, x, y$ , with $L \geq 0$ . Any two absolutely continuous solutions of the projected dynamics of Projected contact dynamics and explicit projection Euler step with the same initial state agree on $[0, T]$ . For $d = 0$ , extend the solution convention directly: $C = R^{0} = {0}$ , $N_{C} (0) = {0}$ , and a solution is the sole map $x (t) = 0$ .

Facts & Assumptions

Given: Countable Choice, Dependent Choice, the set and forcing, and two absolutely continuous solution maps in the Statement.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is inherited through [F4]'s absolutely-continuous FTC.

[A2]

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain is inherited through [F4]'s absolutely-continuous FTC.

[F1]

For $d \geq 1$ , Projected contact dynamics and explicit projection Euler step: a solution satisfies $f (t, x (t)) - \dot{x} (t) \in N_{C} (x (t))$ for almost every $t$ .

[F2]

The convex normal-cone operator is maximal monotone and its resolvent is projection: if $n_{x} \in N_{C} (x)$ and $n_{y} \in N_{C} (y)$ , then $⟨ n_{x} - n_{y}, x - y ⟩ \geq 0$ .

[F3]

Gronwall's integral inequality with variable and constant coefficients: if continuous $u \geq 0$ satisfies $u (t) \leq u (0) + \int_{0}^{t} 2 L u (s) d s$ , then $u (t) \leq u (0) e^{2 L t}$ .

[F4]

The product of two absolutely continuous functions is absolutely continuous and Fundamental theorem of calculus for absolutely continuous functions: products of absolutely continuous real functions are absolutely continuous with the ordinary product derivative almost everywhere, and the integral of that derivative is the endpoint difference.

[F5]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: $∣ ⟨ a, b ⟩ ∣ \leq ∥ a ∥_{2} ∥ b ∥_{2}$ .

[F6]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $R^{0}$ is the singleton zero vector space, with zero inner product and norm.

Proof

1.1F6given

If $d = 0$ , then $C = {0}$ and both solution maps are identically zero. Coordinatewise absolute continuity and differentiation are vacuous, and the inclusion is $0 \in N_{C} (0) = {0}$ , so this agrees with the extended convention and uniqueness holds.

1.2F1F2given

Assume $d \geq 1$ and let $x, y$ be the two solutions and define $n_{x} = f (t, x) - \dot{x}$ and $n_{y} = f (t, y) - \dot{y}$ at almost every time where both inclusions hold. By [F1] these belong to $N_{C} (x)$ and $N_{C} (y)$ ; [F2] gives $⟨ n_{x} - n_{y}, x - y ⟩ \geq 0$ .

2.1A1A2F1F2F3F4F5step 1.2givenalgebra∎

The function $u (t) = ∥ x (t) - y (t) ∥_{2}^{2}$ is absolutely continuous by [F4]. At almost every $t$ , [F1] and step 1.2 give $u^{'} (t) = 2 ⟨ \dot{x} - \dot{y}, x - y ⟩ = 2 ⟨ f (t, x) - f (t, y), x - y ⟩ - 2 ⟨ n_{x} - n_{y}, x - y ⟩ \leq 2 L u (t),$ using the Lipschitz hypothesis and [F5]. Since $u (0) = 0$ , [F4] integrates this differential inequality to $u (t) \leq \int_{0}^{t} 2 L u (s) d s$ . Applying [F3] gives $u (t) = 0$ for every $t$ , so $x = y$ . This argument uses no choices beyond the two axioms already required by [F4].

Depends on

Used by

Existence and uniqueness for projected dynamics and convergence of projection Euler under countable and dependent choice Theorem

Dependency tree · two levels

60 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

David E. Stewart, Dynamics with Inequalities: Impacts and Hard Constraints (standard reference, not scraped)
Dimitri P. Bertsekas, 6.253 Convex Analysis and Optimization, Complete Lecture Notes (standard reference, not scraped)