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

TL;DR

This work develops a rigorous convex-analytic framework for constrained discounted impulse control of deterministic systems described by a (semi)flow. By formulating the problem in terms of occupation measures and dual programs, the authors establish the absence of relaxation and duality gaps under semicontinuity and Slater-type conditions, and they derive a second convex program whose Bellman optimum $W^*_{ar{g}}$ yields a practical, dynamic-programming-based route to optimal strategies. They then present a general four-step procedure to obtain optimal $(J+1)$-mixed strategies for the original impulse-control problem, including a constructive illustration via a closed-form solution for a fluid-queue example. The results provide a principled method to compute structured optimal policies in constrained impulse control, with potential applicability to PDPs and related deterministic control problems.

Abstract

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.

Figures (1)

• Figure 1: Trajectories of the impulsively controlled dynamical system.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Proposition 1
• Lemma 1
• Definition 3
• Definition 4
• Lemma 2
• Definition 5
• Lemma 3
• Theorem 1