Degenerate Stochastic Control Problems with ExponentialCosts and Weakly Coupled Dynamics: Viscosity Solutions and a Maximum Principle

This paper considers a class of optimization problems arising in wireless communication systems. We analyze the optimal control and the associated Hamilton–Jacobi–Bellman (HJB) equations. It turns out that the value function is a unique viscosity solution of the HJB equation in a certain function class. To deal with the fast growth condition of the value function in establishing uniqueness, we construct particular semiconvex/semiconcave approximations for the viscosity sub/supersolutions, and obtain a maximum principle on unbounded domains. The localized envelope function technique introduced in this paper permits an analysis of the uniqueness of viscosity solutions defined on unbounded domains in cases with very general growth conditions when combined with appropriate system dynamics. The optimization problem with state constraints is also considered.