Optimal control problem of evolution equation governed by hypergraph Laplacian

TL;DR

This paper studies an optimal control problem involving a hypergraph Laplacian, addressing challenges due to its set-valued nature by introducing an approximation method and analyzing convergence.

Contribution

It proposes an approximation approach using clique expansion to derive optimality conditions for control problems governed by hypergraph Laplacians.

Findings

Existence of optimal control established using a priori estimates.

Introduction of clique expansion as an approximation method.

Convergence of the approximation to the original problem analyzed.

Abstract

In this paper, we consider an optimal control problem of an ordinary differential inclusion governed by the hypergraph Laplacian, which is defined as a subdifferential of a convex function and then is a set-valued operator. We can assure the existence of optimal control for a suitable cost function by using methods of a priori estimates established in the previous studies. However, due to the multivaluedness of the hypergraph Laplacian, it seems to be difficult to derive the necessary optimality condition for this problem. To cope with this difficulty, we introduce an approximation operator based on the approximation method of the hypergraph, so-called ``clique expansion.'' We first consider the optimality condition of the approximation problem with the clique expansion of the hypergraph Laplacian and next discuss the convergence to the original problem. In appendix, we state some basic properties of the clique expansion of the hypergraph Laplacian for future works.