Integer Optimal Control with Fractional Perimeter Regularization

Motivated by many applications, optimal control problems with integer controls have recently received a significant attention. Some state-of-the-art work uses perimeter-regularization to derive stationarity conditions and trust-region algorithms. However, the discretization is difficult in this case because the perimeter is concentrated on a set of dimension $d - 1$ for a domain of dimension $d$. This article proposes a potential way to overcome this challenge by using the fractional nonlocal perimeter with fractional exponent $0<\alpha<1$. In this way, the boundary integrals in the perimeter regularization are replaced by volume integrals. Besides establishing some non-trivial properties associated with this perimeter, a $\Gamma$-convergence result is derived. This result establishes convergence of minimizers of fractional perimeter-regularized problem, to the standard one, as the exponent $\alpha$ tends to 1. In addition, the stationarity results are derived and algorithmic convergence analysis is carried out for $\alpha \in (0.5,1)$ under an additional assumption on the gradient of the reduced objective. The theoretical results are supplemented by a preliminary computational experiment. We observe that the isotropy of the total variation may be approximated by means of the fractional perimeter functional.