Optimal control problems on stratified domains
Alberto Bressan, Yunho Hong · Networks and Heterogeneous Media · 2007
We consider a class of optimal control problems defined on a stratified domain. Namely, we assume that the statespace $\mathbb{R}^N$ 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. Weprovide conditions which guarantee the existence of an optimal solution, and study sufficient conditions foroptimality. These are obtained by proving a uniqueness result for solutions to a corresponding Hamilton-Jacobiequation with discontinuous coefficients, describing the value function. Our results are motivated by variousapplications, such as minimum time problems with discontinuous dynamics, and optimization problems constrained to abounded domain, in the presence of an additional overflow cost at the boundary.