Journal of Inequalities and Applications 
Volume 6 (2001), Issue 5, Pages 519-545
doi:10.1155/S1025583401000327

Viscosity solutions of two classes of coupled Hamilton-Jacobi-Bellman equations

Mingqing Xiao1 and Tamer Başar2

 1Department of Mathematics, University of California, Davis, One Shields Ave., Davis 95616, CA, USA
2Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, 1308 West Main Street, Urbana 61801, IL, USA
Received 12 August 1999; Revised 27 January 2000

Abstract

This paper studies viscosity solutions of two sets of linearly coupled Hamilton-Jacobi-Bellman (HJB) equations (one for finite horizon and the other one for infinite horizon) which arise in the optimal control of nonlinear piecewise deterministic systems where the controls could be unbounded. The controls enter through the system dynamics as well as the transitions for the underlying Markov chain process, and are allowed to depend on both the continuous state and the current state of the Markov chain. The paper establishes the existence and uniqueness of viscosity solutions for these two sets of HJB equations, whose Hamiltonian structures are different from the standard ones.