On optimal control problem for the Perona-Malik equation and its approximation

We discuss the existence of solutions to an optimal control problem for the Neumann boundary value problem for the Perona-Malik equations. The control variable \begin{document}$ v $\end{document} is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution \begin{document}$ u_d\in L^2(\Omega) $\end{document} and the current system state. We deal with such case of non-linearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of its approximation using the model with fictitious control in coefficients of the principle elliptic operator. We introduce a special family of regularized optimization problems and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and derive some optimality conditions for the approximating problems.