Optimal Temperature Distribution for a Nonisothermal Cahn–Hilliard System with Source Term

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 Fréchet 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. Crucial for the whole analysis to work is the boundedness property stating that the order parameter attains its values in a compact subset of the interior of the effective domain of the nonlinearity. While this property turns out to be generally valid for regular potentials in three dimensions of space, it can be shown for the logarithmic case only in two dimensions.

Optimal Temperature Distribution for a Nonisothermal Cahn-Hilliard System in Two Dimensions With Source Term and Double Obstacle Potential

“…With essentially the same proof as that of [10,Thm. 4.1] (which needs no repetition here), we can show that the adapted optimal control problem ( P α ) has for every α > 0 at least one solution.…”

“…The state system (1)-( 5) was recently analyzed in [9] concerning wellposedness and regularity (see the results cited below in Section 2); in [10] the corresponding optimal control problem (P) has been analyzed for the simpler differentiable case when the indicator function 2) is replaced by either a regular function or by a logarithmic expression of the form…”

“…For such logarithmic nonlinearities, in [10] results concerning existence of optimal controls, Fréchet differentiability of the control-to-state operator, and meaningful first-order necessary optimality conditions (in terms of the associated adjoint state problem and variational inequality) have been derived, at least in the spatially two-dimensional situation. In this paper, we complement the results of [10] by investigating the optimal control problem for the nondifferentiable double obstacle case.…”

“…For such logarithmic nonlinearities, in [10] results concerning existence of optimal controls, Fréchet differentiability of the control-to-state operator, and meaningful first-order necessary optimality conditions (in terms of the associated adjoint state problem and variational inequality) have been derived, at least in the spatially two-dimensional situation. In this paper, we complement the results of [10] by investigating the optimal control problem for the nondifferentiable double obstacle case. While the existence of optimal controls is not too difficult to show, the derivation of first-order necessary optimality conditions is a much more challenging task, since the existence of appropriate Lagrange multipliers cannot be derived from the standard theory.…”

“…It turns out that this general strategy works with suitable modifications. Let us stress at this point that our approach makes use of the results obtained for the logarithmic case investigated in [10]; since in that case the derivation of differentiability properties of the associated control-to-state operator was only possible under the premise that the order parameter ϕ satisfies the so-called strict separation property (meaning that ϕ attains its values in a compact subset of (−1, 1)), and since this property could only be shown in the spatially two-dimensional case, our analysis does not apply to three-dimensional domains Ω.…”