Optimal control problems on stratified domains

Alberto Bressan,Yunho Hong

doi:10.3934/nhm.2007.2.313

Alberto Bressan, Yunho Hong

Open Access

https://doi.org/10.3934/nhm.2007.2.313

Copy DOI

Abstract

We consider a class of optimal control problems defined on a stratified domain. Namely, we assume that the state space $\mathbb{R}^N$ admits a stratification as a disjoint union of finitely many embedded submanifolds $\mathcal{M}_i$. The dynamics of the system and the cost function are Lipschitz continuous restricted to each submanifold. We provide conditions which guarantee the existence of an optimal solution, and study sufficient conditions for optimality. These are obtained by proving a uniqueness result for solutions to a corresponding Hamilton-Jacobi equation with discontinuous coefficients, describing the value function. Our results are motivated by various applications, such as minimum time problems with discontinuous dynamics, and optimization problems constrained to a bounded domain, in the presence of an additional overflow cost at the boundary.

Highlights

The theory of viscosity solutions was initially developed in connection with continuous solutions of Hamilton-Jacobi equations, whose coefficients are continuous
In a different direction, motivated by problems in optimal control, sufficient conditions for the optimality of a feedback synthesis have been established in [12], under assumptions that do not require the continuity of the value function
The space RN is decomposed as the disjoint union of finitely many submanifolds of different dimensions, and we assume that the dynamics of the system as well as the running cost are sufficiently regular when restricted to each given manifold, but may well differ from one manifold to the other

Summary

Introduction

The theory of viscosity solutions was initially developed in connection with continuous solutions of Hamilton-Jacobi equations, whose coefficients are continuous. Various authors have extended the theory in cases where the value function is discontinuous [2, 15]. In a different direction, motivated by problems in optimal control, sufficient conditions for the optimality of a feedback synthesis have been established in [12], under assumptions that do not require the continuity of the value function. A further line of investigation, more recently pursued in [14, 5], is the case where the coefficients of the H-J equation are themselves discontinuous. We study the value function for an infinite-horizon optimal control problem, on a structured domain. Hamilton-Jacobi equation, Viscosity solution, Optimal control theory

ALBERTO BRESSAN AND YUNHO HONG

The value function is defined as

Here the Hamiltonian function is defined as

In order to show that