Linear programming approach to optimal impulse control problems with functional constraints

TL;DR

This paper develops a linear programming framework for solving infinite-horizon optimal impulse control problems with multiple functional constraints, leveraging Markov decision process tools to establish existence of stationary optimal strategies.

Contribution

It introduces a novel linear programming approach for constrained impulse control problems and proves the existence of stationary optimal strategies under general conditions.

Findings

Established a linear programming formulation for the problem.

Proved existence of stationary optimal control strategies.

Extended the approach to systems with multiple constraints.

Abstract

This paper considers an optimal impulse control problem of dynamical systems generated by a flow. The performance criteria are total costs over the infinite time horizon. Apart from the main performance to be minimized, there are multiple constraints on performance functionals of a similar type. Under a natural set of compactness-continuity conditions on the system primitives, we establish a linear programming approach, and prove the existence of a stationary optimal control strategy out of a more general class of randomized strategies. This is done by making use of the tools from Markov decision processes.