On Integer Optimal Control with Total Variation Regularization on Multi-dimensional Domains

TL;DR

This paper studies integer-valued optimal control problems with total variation regularization on multi-dimensional domains, deriving optimality conditions and analyzing a trust-region algorithm's convergence to optimal solutions.

Contribution

It introduces first-order optimality conditions and proves convergence of a trust-region algorithm for integer control problems with TV regularization.

Findings

Derivation of first-order optimality conditions for the problem.

Proof of convergence of the sequential linear integer programming algorithm.

Analysis of the feasible set's topological properties in BV spaces.

Abstract

We consider optimal control problems with integer-valued controls and a total variation regularization penalty in the objective on domains of dimension two or higher. The penalty yields that the feasible set is sequentially closed in the weak-$^*$ and closed in the strict topology in the space of functions of bounded variation. In turn, we derive first-order optimality conditions of the optimal control problem as well as trust-region subproblems with partially linearized model functions using local variations of the level sets of the feasible control functions. We also prove that a recently proposed function space trust-region algorithm -- sequential linear integer programming -- produces sequences of iterates whose limits are first-order optimal points.