On the Optimality of Lyapunov-based Feedback Laws for Constrained Difference Inclusions

State-feedback optimal control and cost evaluation problems for constrained difference inclusions are considered. Sufficient conditions, in the form of Lyapunov-like inequalities, are provided to derive an upper bound on the cost associated with the solution to a constrained difference inclusion with respect to a given cost functional. Under additional sufficient conditions, we determine the cost exactly without computing solutions. The proposed approach is extended to study an optimal control problem for discrete-time systems with constraints. In this setting, sufficient conditions for closed-loop optimality are given in terms of a constrained steady-state-like Hamilton-Jacobi-Bellman equation. Applications and examples of the proposed results are presented.