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Approximation of an optimal control problem on a network with a perturbed problem in the whole space

Mohamed Camar-Eddine, Mériadec Chuberre, Mounir Haddou, Olivier Ley

TL;DR

This work analyzes a singular perturbation of a classical infinite-horizon optimal control problem in $\mathbb{R}^2$ aimed at concentrating trajectories on a network $\Gamma$ via a term $\frac{1}{\varepsilon}\nabla d$. It develops a framework linking the $\varepsilon\to 0$ limit to optimal control problems on networks, proving subsequential convergence of trajectories to $\Gamma$ and, in the Eikonal case, convergence of the value function to the network value function $V_{\Gamma}$. Under controllability and cost-structure assumptions, the perturbed value functions $V^{\varepsilon}$ converge locally uniformly to $\overline{V}\circ\phi_d$, with $\overline{V}=V_{\Gamma}$ on $\Gamma$, thereby connecting full-space control with network formulations. The results offer a principled route for approximating network-based optimal control problems via classical singular perturbations, and they reveal subtle phenomena near the junction, including potential nonuniqueness and semigroup failures in the limit dynamics.

Abstract

A classical optimal control problem posed in the whole space R^2 is perturbed by a singular term of magnitude $ε$^{-1} aimed at driving the trajectories to a prescribed network $Γ$. We are interested in the link between the limit problem, as $ε$ $\rightarrow$ 0, and some optimal control problems on networks studied in the literature. We prove that the sequence of trajectories admits a subsequential limit evolving on $Γ$. Moreover, in the case of the Eikonal equation, we show that the sequence of value functions associated with the perturbed optimal control problems converges to a limit which, in particular, coincides with the value function of the expected optimal control problem set on the network $Γ$.

Approximation of an optimal control problem on a network with a perturbed problem in the whole space

TL;DR

This work analyzes a singular perturbation of a classical infinite-horizon optimal control problem in aimed at concentrating trajectories on a network via a term . It develops a framework linking the limit to optimal control problems on networks, proving subsequential convergence of trajectories to and, in the Eikonal case, convergence of the value function to the network value function . Under controllability and cost-structure assumptions, the perturbed value functions converge locally uniformly to , with on , thereby connecting full-space control with network formulations. The results offer a principled route for approximating network-based optimal control problems via classical singular perturbations, and they reveal subtle phenomena near the junction, including potential nonuniqueness and semigroup failures in the limit dynamics.

Abstract

A classical optimal control problem posed in the whole space R^2 is perturbed by a singular term of magnitude ^{-1} aimed at driving the trajectories to a prescribed network . We are interested in the link between the limit problem, as 0, and some optimal control problems on networks studied in the literature. We prove that the sequence of trajectories admits a subsequential limit evolving on . Moreover, in the case of the Eikonal equation, we show that the sequence of value functions associated with the perturbed optimal control problems converges to a limit which, in particular, coincides with the value function of the expected optimal control problem set on the network .
Paper Structure (13 sections, 29 theorems, 180 equations, 3 figures)

This paper contains 13 sections, 29 theorems, 180 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of the problem and main results
  3. Study of the perturbed ODE \ref{['traj-eps']}
  4. Qualitative properties of the trajectories $X^{x,\alpha}$
  5. Dynamic of trajectories inside the branches
  6. Behavior of trajectories with locally constant control near the origin
  7. Convergence of the approximated control problem to the control problem on $\Gamma$
  8. Proof of Theorem \ref{['limit-vf']}
  9. Properties of $d_\Gamma$
  10. Proofs of Propositions \ref{['behav-O']} and \ref{['behav-1']}
  11. Proof of Proposition \ref{['behav-O']}
  12. Proof of Proposition \ref{['behav-1']}
  13. Proofs of Lemmas \ref{['controle-vitesse-123']} and \ref{['traj-y-tilde']}

Key Result

Theorem 2.1

Consider traj-eps where $f$ satisfies hyp-f-ell and $d$ is given by choix-d. For every $x\in\mathbb R^2$ and $\alpha\in \mathcal{A}$,

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (62)

  • Theorem 2.1: Convergence of trajectories
  • Theorem 2.2: Dynamic on $\Gamma$ outside $O$
  • Theorem 2.3: Convergence to the control problem on the network
  • Lemma 3.1: Well-posedness of \ref{['traj-eps']} for $\varepsilon >0$
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3: Invariant subsets and entry times
  • Remark 3.4
  • proof
  • ...and 52 more