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

¹ Institute of Mathematics and Scientific Computing, Karl-Franzens University of Graz, Heinrichstraße 36, 80101 Graz, Austria.
² School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, P.R. China.
³ RICAM, Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria.
⁴ Unité de Mathématiques Appliquées, ENSTA ParisTech, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France.

^** Corresponding author: zprao1986@gmail.com

Received: 20 April 2017
Accepted: 14 December 2018

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 in Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman equations on multi-domains, in Control and Optimization with PDE Constraints, Vol. 164 of International Series of Numerical Mathematics. Birkhäuser, Basel (2013) 93–116. 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.