Constrained optimal impulse control and inventory model

In this article, we consider the deterministic impulsively controlled system with infinite horizon and several discounted objective functionals. The constructed optimal control problem with functional constraints is reformulated as a Markov decision process, leading to (primal) convex and linear programs in the space of so-called occupation measures. We construct the dual programs and investigate the solvability of all the programs. Example of an inventory model illustrates the developed theory.

💡 Research Summary

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This paper addresses a deterministic impulse‑controlled system evolving over an infinite horizon with several discounted objective functionals, while imposing functional constraints on the performance measures. The authors reformulate the constrained optimal control problem as a Markov decision process (MDP) and then translate the MDP into linear and convex programs defined on the space of occupation measures.

The state space (X) is taken as a non‑empty Borel subset of a complete separable metric (Polish) space, allowing for both Euclidean and more abstract settings. The system dynamics are described by a flow (\varphi(x,t)) that satisfies the usual semigroup, measurability and right‑continuity properties. Impulses are modeled as pairs ((\theta,a)) where (\theta\ge0) is the waiting time until the next impulse and (a\in A) (a compact Borel action set) determines the jump map (l(x,a)). Continuous‑time running costs (C_g^j(x)) and impulse costs (C_I^j(x,a)) are non‑negative and lower semicontinuous. A discount factor (\alpha>0) yields the discounted total cost for each criterion (j=0,\dots,J).

To embed the impulse control problem into an MDP, the authors augment the state space with an absorbing “cemetery” state (\Delta) (cost‑free) and define the action space (B=