Primal-dual programs for the constrained optimal impulse control: discounted model

This paper studies constrained optimal impulse control problems of a deterministic system described by a (semi)flow, where the performance measures are the discounted total costs including both the costs incurred with applying impulses as well as running costs. We formulate the relaxed problem and the associated primal convex programs in measures together with the dual programs, and establish the relevant duality results. As an application, we formulate and justify a general procedure for obtaining optimal $(J+1)$-mixed strategies for the original impulse control problems. This procedure is illustrated with a solved example.