Optimal temperature distribution for a nonisothermal Cahn-Hilliard system with source term

TL;DR

This paper investigates the optimal control of a nonisothermal Cahn-Hilliard system with a source term, incorporating thermal memory effects via Green-Naghdi theory, and establishes differentiability and optimality conditions for the control problem.

Contribution

It introduces a novel nonisothermal phase field model with thermal memory and source term, and derives the first-order optimality conditions for control.

Findings

Proved Fréchet differentiability of the control-to-state operator.

Established solvability of the adjoint systems.

Derived first-order necessary optimality conditions.

Abstract

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter, typical of the classic Cahn-Hilliard equation, is no longer satisfied. In this paper, we analyze the case that the double-well potential driving the evolution of the phase transition is differentiable, either (in the regular case) on the whole set of reals or (in the singular logarithmic case) on a finite open interval; nondifferentiable cases like the double obstacle potential are excluded from the analysis. We prove the Fr\'echet differentiability of the control-to-state operator between suitable Banach spaces for both the regular and the logarithmic cases and establish the solvability of the corresponding adjoint systems in order to derive the associated first-order necessary optimality conditions for the optimal control problem.

Full text

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Optimal temperature distribution for a nonisothermal Cahn–Hilliard system with source term

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centerPierluigi Colli*(1)*

e-mail: pierluigi.colli@unipv.it

Gianni Gilardi (1)

e-mail: gianni.gilardi@unipv.it

Andrea Signori*(2)*

e-mail: andrea.signori@polimi.it

Jürgen Sprekels*(3)*

e-mail: juergen.sprekels@wias-berlin.de

(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, Germany

and

Weierstrass Institute for Applied Analysis and Stochastics

Mohrenstrasse 39, D-10117 Berlin, Germany