Ergodic control for constrained diffusions : Characterization using HJB equations

• Published in: SIAM journal on control and optimization Vol. 43; no. 4; pp. 1467 - 1492
• Main Authors: BORKAR, Vivek, BUDHIRAJA, Amarjit
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2004
• Subjects: Applied sciences; Computer science; control theory; systems; Control system synthesis; Control theory. Systems; Exact sciences and technology; Markov analysis; Mathematical programming; Operational research and scientific management; Operational research. Management science; Queuing; R&D; Research & development; Stochastic control theory; Viscosity; Markov process; Solution uniqueness; Discontinuous; Multivalued function; Discount; Existence of solution; Stochastic process; Queueing network; Uniqueness theorem; domains with corners; Partial differential equation; oblique Neumann problem; Markov chain; Control constraint; coupling; pseudoatom; constrained processes; HJB equation; Diffusion process; Cost function; Diffusion equation; ergodic control; Optimal control (mathematics); Discontinuity; Neumann problem; Bellman equation; Existence theorem; controlled reflected diffusions; Ergodicity; Markov model; Wedge; Constrained optimization; Polyhedron; Value function; optimal Markov control; Hamilton Jacobi equation; Optimal control; Viscosity solution; Discrete time; Stochastic control; viscosity solutions
• ISSN: 0363-0129; 1095-7138
• DOI: 10.1137/s0363012902417619

Recently in [A. Budhiraja, SIAM J. Control Optim., 42 (2003), pp. 532--558] an ergodic control problem for a class of diffusion processes, constrained to take values in a polyhedral cone, was considered. The main result of that paper was that under appropriate conditions on the model, there is a Markov control for which the infimum of the cost function is attained. In the current work we characterize the value of the ergodic control problem via a suitable Hamilton--Jacobi--Bellman (HJB) equation. The theory of existence and uniqueness of classical solutions, for PDEs in domains with corners and reflection fields which are oblique, discontinuous, and multivalued on corners, is not available. We show that the natural HJB equation for the ergodic control problem admits a unique continuous viscosity solution which enables us to characterize the value function of the control problem. The existence of a solution to this HJB equation is established via the classical vanishing discount argument. The key step is proving the precompactness of the family of suitably renormalized discounted value functions. In this regard we use a recent technique, introduced in [V. S. Borkar, Stochastic Process Appl., 103 (2003), pp. 293--310], of using the Athreya--Ney--Nummelin pseudoatom construction for obtaining a coupling of a pair of embedded, discrete time, controlled Markov chains.