Fictitious Controls and Approximation of an Optimal Control Problem for Perona-Malik Equation

We discuss the existence of solutions to an optimal control problem for the Cauchy-Neumann boundary value problem for the evolutionary Perona-Malik equations. The control variable v is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution u_d in L2(Omega) 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 for linear parabolic equations 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.