Ergodic Control for Constrained Diffusions: Characterization Using HJB Equations

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.

[1]  P. Lions.  Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part I , 1983 .

[2]  R. M. Cox.  Stationary and discounted control of diffusion processes , 1984 .

[3]  Lawrence M. Wein,et al.  Scheduling networks of queues: Heavy traffic analysis of a simple open network , 1989, Queueing Syst. Theory Appl..

[4]  Vivek S. Borkar,et al.  Optimal Control of Diffusion Processes , 1989 .

[5]  M. K. Ghosh,et al.  Ergodic control of multidimensional diffusions, II: Adaptive control , 1990 .

[6]  L. F. Martins,et al.  Routing and singular control for queueing networks in heavy traffic , 1990 .

[7]  P. Dupuis,et al.  On Lipschitz continuity of the solution mapping to the Skorokhod problem , 1991 .

[8]  P. Dupuis,et al.  On oblique derivative problems for fully nonlinear second-order elliptic PDE’s on domains with corners , 1991 .

[9]  P. L. Lions.  Viscosity solutions and optimal control , 1992 .

[10]  F. P. Kelly,et al.  Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling , 1993, Queueing Syst. Theory Appl..

[11]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[12]  Gopal K. Basak,et al.  Ergodic control of degenerate diffusions , 1997 .

[13]  Jan A. Van Mieghem,et al.  Dynamic Control of Brownian Networks: State Space Collapse and Equivalent Workload Formulations , 1997 .

[14]  Amarjit Budhiraja,et al.  Simple Necessary and Sufficient Conditions for the Stability of Constrained Processes , 1999, SIAM J. Appl. Math..

[15]  P. Dupuis,et al.  Convex duality and the Skorokhod Problem. II , 1999 .

[16]  P. Dupuis,et al.  Convex duality and the Skorokhod Problem. I , 1999 .

[17]  P. Dupuis,et al.  ON POSITIVE RECURRENCE OF CONSTRAINED DIFFUSION PROCESSES , 2001, math/0501018.

[18]  H. Kushner.  Heavy Traffic Analysis of Controlled Queueing and Communication Networks , 2001 .

[19]  R. Atar,et al.  Stability Properties of Constrained Jump-Diffusion Processes , 2002, math/0501014.

[20]  V. Borkar.  Dynamic programming for ergodic control with partial observations , 2003 .

[21]  Amarjit Budhiraja.  An Ergodic Control Problem for Constrained Diffusion Processes: Existence of Optimal Markov Control , 2003, SIAM J. Control. Optim..

[22]  V. Borkar,et al.  A further remark on dynamic programming for partially observed Markov processes , 2004 .