Input Regularization for Integer Optimal Control in BV with Applications to Control of Poroelastic and Poroviscoelastic Systems

TL;DR

This paper introduces a regularization technique for integer optimal control problems involving PDEs, enabling convergence analysis without restrictive regularity assumptions, with applications to biomechanics and porous media flow.

Contribution

It proposes a mollification-based regularization method that bypasses regularity restrictions, supported by a $\Gamma$-convergence analysis and numerical validation.

Findings

Regularization improves convergence in irregular PDE control problems.

Homotopy aids in achieving better objective values.

Practical performance without regularization is comparable to with regularization.

Abstract

We revisit a class of integer optimal control problems for which a trust-region method has been proposed and analyzed in arXiv:2106.13453v3 [math.OC]. While the algorithm proposed in arXiv:2106.13453v3 [math.OC] successfully solves the class of optimization problems under consideration, its convergence analysis requires restrictive regularity assumptions. There are many examples of integer optimal control problems involving partial differential equations where these regularity assumptions are not satisfied. In this article we provide a way to bypass the restrictive regularity assumptions by introducing an additional partial regularization of the control inputs by means of mollification and proving a $\Gamma$-convergence-type result when the support parameter of the mollification is driven to zero. We highlight the applicability of this theory in the case of fluid flows through deformable porous media equations that arise in biomechanics. We show that the regularity assumptions are violated in the case of poro-visco-elastic systems, and thus one needs to use the regularization of the control input introduced in this article. Associated numerical results show that while the homotopy can help to find better objective values and points of lower instationarity, the practical performance of the algorithm without the input regularization may be on par with the homotopy.