Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential

Pierluigi Colli; Gianni Gilardi; Andrea Signori; Jürgen Sprekels

doi:10.3934/dcdss.2022210

Article Contents

2023, Volume 16, Issue 9: 2305-2325. Doi: 10.3934/dcdss.2022210

This issue Previous Article The Caginalp phase field systems with logarithmic nonlinear terms Next Article On the regularized Moore-Gibson-Thompson equation

Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential

1.
Dipartimento di Matematica "F. Casorati", Università di Pavia and Research Associate at the IMATI – C.N.R. Pavia, via Ferrata 5, I-27100 Pavia, Italy
2.
Dipartimento di Matematica, Politecnico di Milano, via E. Bonardi 9, I-20133 Milano, Italy
3.
Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099, Berlin and Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany

^*Corresponding author: Pierluigi Colli
^*Corresponding author: Pierluigi Colli

Dedicated to the memory of Professor Gunduz Caginalp

Received: October 2022

Early access: January 2023

Published: September 2023

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Abstract

In this paper, we investigate optimal control problems for a nonlinear state system which constitutes a version of the Caginalp phase field system modeling nonisothermal phase transitions with a nonconserved order parameter that takes thermal memory into account. The state system, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. It consists of two nonlinearly coupled partial differential equations that govern the phase dynamics and the universal balance law for internal energy, written in terms of the phase variable and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. We extend recent results obtained for optimal control problems in which the free energy governing the phase transition was differentiable (i.e., of regular or logarithmic type) to the nonsmooth case of a double obstacle potential. As is well known, in this nondifferentiable case standard methods to establish the existence of appropriate Lagrange multipliers fail. This difficulty is overcome utilizing of the so-called deep quench approach. Namely, the double obstacle potential is approximated by a family of (differentiable) logarithmic ones for which the existence of optimal controls and first-order necessary conditions of optimality in terms of the adjoint state variables and a variational inequality are known. By proving appropriate bounds for the adjoint states of the approximating systems, we can pass to the limit in the corresponding first-order necessary conditions, thereby establishing meaningful first-order necessary optimality conditions also for the case of the double obstacle potential.

Keywords:

Phase field model,
thermal memory,
double obstacle potential,
optimal control,
first-order necessary optimality conditions,
adjoint system,
deep quench approximation.

Mathematics Subject Classification: Primary: 49J20, 49K20; Secondary: 35K55, 35K51, 49J50.

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References

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