A hybrid optimal control problem constrained with hyperelasticity and the global injectivity condition

TL;DR

This paper develops a hybrid optimal control framework for hyperelasticity problems with global injectivity constraints, relevant to cardiac mechanics, including mathematical analysis and numerical simulations.

Contribution

It introduces a novel optimal control approach for hyperelasticity with volume preservation and injectivity constraints, with rigorous analysis and simulation.

Findings

Mathematical analysis using $ ext{L}^p$-parabolic maximal regularity.

Derivation of optimality conditions for the control problem.

Numerical simulations demonstrating the approach's effectiveness.

Abstract

The purpose of this paper is to address a class of hybrid optimal control problems constrained with hyperelasticity and constant global volume. This type of problems can intervene for example in the mechanical aspects of cardiac activity. The time deformation of the heart tissue is modeled with the elastodynamics equations dealing with the displacement field as main unknown. These equations are coupled with a pressure whose time variations are aimed to be maximized. This pressure variable corresponds to a Lagrange multiplier associated with the so-called global injectivity condition, translating the fact that the total volume of the domain remains constant. We develop an optimal control approach in a general framework that covers in particular the maximization of the variations of this pressure, and also the time the maximum is reached, defining what we call a {\it hybrid} optimal control problem. Mathematical analysis based on the $\L^p$-parabolic maximal regularity is provided for the state equations and the rigorous derivation of optimality conditions. Numerical simulations for a toy-model illustrate the capacity of the approach.