P. Colli†, G. Gilardi‡, A. Signori§, J. Sprekels¶
Abstract: In this note, we study the optimal control of a nonisothermal phase eld system of CahnHilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a CahnHilliard 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 ux 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 further mathematical di culties because the mass conservation of the order parameter is no longer satis ed. In this paper, we study the case that the double-well potential driving the evolution of the phase transition is given by the nondi erentiable double obstacle potential, thereby complementing recent results obtained for the di erentiable cases of regular and logarithmic potentials. Besides existence results, we derive rst-order necessary optimality conditions for the control problem. The analysis is carried out by employing the so-called deep quench approximation in which the nondi erentiable double obstacle potential is approximated by a family of potentials of logarithmic structure for which meaningful rst-order necessary optimality conditions in terms of suitable adjoint systems and variational inequalities are available. Since the results for the logarithmic potentials crucially depend on the validity of the so-called strict separation property which is only available in the spatially two-dimensional situation, our whole analysis is restricted to the two-dimensional case.
MSC: 35K20, 35K55, 49J50, 49J52, 49K20.
keywords: Optimal control, nonisothermal CahnHilliard equation, thermal memory, CahnHilliard equation with source term, CahnHilliard Oono equation.
DOI 10.56082/annalsarscimath.2023.1-2.175
†pierluigi.colli@unipv.it Dipartimento di Matematica, Universita Degli Studi di Pavia F. Casorati , and Research Associate at the IMATI C.N.R. Pavia, via Ferrata 5, I 27100 Pavia, Italy;
‡gianni.gilardi@unipv.it Dipartimento di Matematica, Universita Degli Studi di Pavia F. Casorati , and Research Associate at the IMATI C.N.R. Pavia, via Ferrata 5, I 27100 Pavia, Italy;
§andrea.signori@polimi.it Dipartimento di Matematica, Politecnico di Milano, via E. Bonardi 9, I-20133 Milano, Italy;
¶juergen.sprekels@wias-berlin.de Department of Mathematics, HumboldtUniversitat zu Berlin, Unter den Linden 6, D10099 Berlin, Germany, and Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D10117 Berlin, Germany
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application, Volume 15 no 1-2, 2023
