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 v is taken as a distributed control. The optimal control prob-lem is to minimize the discrepancy between a given distribution ud is an element of L2(ohm) 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 approx-imation 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-p osed, 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.