On the viscosity solutions of hybrid HJB equations arising in optimal control of loss networks

Abstract

Call admission and routing controls for loss (circuit-switched) networks with semi- Markovian, multi-class call arrivals and general connection durations, were formulated as optimal stochastic control problems in [12, 13]. Each of the resulting so-called (network) hybrid HJB equations corresponds to a collection of coupled first-order partial differential equations for which, when it exists, the continuously differentiable value function is a solution to the associated hybrid HJB equations. In general, the smoothness of the value functions and uniqueness of the solutions to the hybrid HJB equations may not hold. In this paper, viscosity solutions to a general class of hybrid HJB equations are developed and under mild conditions it is shown that the value function is continuous and, further, any continuous value function is the unique viscosity solution to the hybrid HJB equations.