Existence result and approximation of an optimal control problem for the Perona–Malik equation

Abstract

AbstractWe discuss some optimal control problem for the evolutionary Perona–Malik equations with the Neumann boundary condition. The control variable v is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution $$u_d\in L^2(\Omega )$$ u d ∈ L 2 ( Ω ) and the current system state. Since we cannot expect to have a solution of the original boundary value problem for each admissible control, 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.