The optimal control of an evolutionary boundary value problem

We consider a nonlinear initial boundary value problem in a two-dimensional rectangle. We derive  variational  formulation of the problem which is in the form of an evolutionary variational inequality in a product Hilbert space. Then, we establish the existence of a unique weak solution  to the problem and prove the  continuous dependence of the solution with respect to some parematers. We proceed with the study of an associated control problem for which we prove the existence of an optimal pair. Finally, we consider a perturbed optimal control problem for which we prove a convergence result.

Keywords:

nonlinear boundary value problem , evolutionary inequality , weak solution , convergence results , Optimal control