Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions

TL;DR

This paper establishes junction conditions for Hamilton-Jacobi-Bellman equations in finite horizon control problems on multi-domains, addressing both continuous and lower semi-continuous cases with new generalizations and stability results.

Contribution

It extends previous work by providing more general junction conditions and stability analysis for HJB equations on multi-domains with weaker assumptions.

Findings

Comparison principle guarantees uniqueness of solutions.

Characterization of the value function as a unique viscosity solution.

Results applicable to both continuous and lower semi-continuous costs.

Abstract

This paper deals with junction conditions for Hamilton-Jacobi-Bellman (HJB) equations for finite horizon control problems on multi-domains. We consider two different cases where the final cost is continuous or lower semi-continuous. In the continuous case we extend the results of "Hamilton-Jacobi-Bellman equations on multi-domains" by the second and third authors in a more general framework with switching running costs and weaker controllability assumptions. The comparison principle has been established to guarantee the uniqueness and the stability results for the HJB system on such multi-domains. In the lower semi-continuous case, we characterize the value function as the unique lower semi-continuous viscosity solution of the HJB system, under a local controllability assumption.