Lipschitz Continuity of the Value Function in Mixed-Integer Optimal Control Problems

TL;DR

This paper proves that the optimal value function in mixed-integer control problems on Banach spaces is locally Lipschitz continuous under certain conditions, providing theoretical insights into stability and sensitivity analysis.

Contribution

It establishes Lipschitz continuity of the value function for mixed-integer control problems with semilinear and linear dynamics, extending existing theory to more complex control settings.

Findings

Optimal value function is locally Lipschitz continuous under natural assumptions.

Results apply to problems with semilinear and linear dynamics, convex costs, and constraints.

Sharpness of results demonstrated through an example.

Abstract

We study the optimal value function for control problems on Banach spaces that involve both continuous and discrete control decisions. For problems involving semilinear dynamics subject to mixed control inequality constraints, one can show that the optimal value depends locally Lipschitz continuously on perturbations of the initial data and the costs under rather natural assumptions. We prove a similar result for perturbations of the initial data, the constraints and the costs for problems involving linear dynamics, convex costs and convex constraints under a Slater-type constraint qualification. We show by an example that these results are in a sense sharp.