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

Abstract

We 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.

Variational Model with Nonstandard Growth Condition in Image Restoration and Contrast Enhancement

“…In other words, we aim to show that some optimal solutions to (31)-( 32) can be approximated by the solutions of (64)-(66). To this end, we make use of some results of the variational convergence of minimization problems [26,[29][30][31] and begin with some auxiliaries (see also Refs. [14,15,30,35,36] for other aspects of this concept).…”

“…To this end, we make use of some results of the variational convergence of minimization problems [26,[29][30][31] and begin with some auxiliaries (see also Refs. [14,15,30,35,36] for other aspects of this concept).…”

Asymptotic Behavior of Some Parabolic Equations and Application in Image Restoration

“…And, in [3] the authors proposed a novel parabolic equations for image restoration and enhacement.They proved the existence of solution, established a nonnegative weak solution obtained as limit of approximation and give some application in image processing. Also, in [15] we find some optimal control problem for the Perona-Malik equation. The authors obtained existence results and approximation of an optimal control problem.…”