Optimal distributed control of a Allen-Cahn/Cahn-Hilliard system with temperature
Different kinds of optimal control problems for Allen-Cahn/Cahn-Hilliard equations and systems have been extensively studied, and numerous publications in form of research articles, surveys, preprints or monographs are available. The optimal control problem considered in the present article refers to the Allen-Cahn/Cahn-Hilliard system in the form described in a series of papers by Miranville \&Co, actually the system in: [\textit{A. Miranville} et al., Commun. Pure Appl. Anal. 19, No. 4, 2257--2288 (2020; Zbl 1437.35084)] \[ \frac {\partial u}{\partial t} -\Delta ^2u = \Delta( f(u+v) + f(u-v)),\text{ for }(x,t)\in \Omega_T, \] \[ \frac{\partial v}{\partial t} -\Delta v = \theta-f(u+v) + f(u-v),\text{ for }(x,t) \in \Omega_T, \] \[ \frac{\partial {\theta}v}{\partial t}- \Delta \theta = -\frac{\partial v}{\partial t} +\phi,\text{ for } (x,t) \in\Omega_T, \] \[u= v = \theta =0,\text{ on }\partial \Omega\times (0,T) , \] \[u(x,0) =u_0(x), v(x,0) = v_0(x),\theta(x,0)=\theta_0(x), \text{ for } x \in \Omega, \] where \(\Omega_T=\Omega\times (0,T)\), \(\Omega\subset\mathbb{R}\). Variable \(\phi\) denotes the control and one assumes it is admissible, \(\phi\in\Phi_{ad}\). If \((u,v,\theta)\) is a solution of the initial boundary value problem to the Allen-Chan/Cahn-Hilliard system, one defines a quadratic performance functional \(\mathcal J(\phi;u,v,\theta)\), and the goal is to minimize it for \(\phi\) in a suitable space of admissible controls. Some basics are reviewed in the second section of the paper. The third section is devoted to the well-posedness of the system. One transforms the system in an equivalent one by introducing a new variable \(\mu= -\Delta u+f(u+v) + f(u-v)\), and under some assumptions on initial data, one proves the existence of a weak solution for the transformed system. A next result states the existence of a strong solution for \(t\in[ 0, T]\) and two estimates are proved. In the fourth section one considers the system in a linearized form and one proves the existence of a unique solution. For this system, one shows a result on the Fréchet differentiability of the control-to state mapping \(S\). The last part of this section is devoted to an investigation on the first-order optimality condition for the stated optimal control problem.
first-order necessary optimality conditions
scientific article; zbMATH DE number 7498417
scientific article