Switching Point Optimization for Abstract Parabolic Equations

TL;DR

This paper investigates the optimization of switching points in semilinear parabolic equations, establishing differentiability properties and proposing gradient-based methods, with numerical validation and insights into global optimization challenges.

Contribution

It proves the continuous Fréchet differentiability of the switching-point-to-control map and introduces a framework for gradient-based optimization of switching points.

Findings

The switching-point-to-control map is continuously Fréchet-differentiable.

Gradient-based methods can be applied for local optimization.

Numerical experiments validate the theoretical results and highlight non-convexity issues.

Abstract

This work is concerned with a switching point optimization problem governed by a semilinear parabolic equation in abstract function spaces. It is shown that the switching-point-to-control mapping is continuously Fr\'echet-differentiable when considered with values in the dual of H\"older continuous functions in time. By treating the state equation in weak form based on the concept of maximal parabolic regularity, one can then show that the reduced objective is continuously differentiable w.r.t. the switching points, which allows to use gradient-based methods like the proximal gradient method for its minimization. Numerical experiments confirm our theoretical findings, but also illustrate that such a method will in general not be able to solve the problem to global optimality due to the non-convex nature of the switching-point-to-control map. We therefore give a precise characterization of the convex hull of set of feasible switching functions in terms of an extended formulation. The latter might be useful for a branch-and-bound approach for the computation of global minimizers, but this is subject to future research.