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

Yaroslav Kohut; Olha Kupenko

doi:10.3934/mcrf.2022045

Article Contents

2023, Volume 13, Issue 4: 1466-1483. Doi: 10.3934/mcrf.2022045

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

Yaroslav Kohut^1, , and
Olha Kupenko^1,2,

1.
Oles Honchar Dnipro National University, Dnipro, Ukraine
2.
Dnipro University of Technology "Dnipro Polytechnics", Dnipro, Ukraine

^*Corresponding author: Olha Kupenko
^*Corresponding author: Olha Kupenko

Received: March 2022

Revised: July 2022

Early access: September 2022

Published: December 2023

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Abstract

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 problem is to minimize the discrepancy between a given distribution $ u_d\in L^2(\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 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.

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Mathematics Subject Classification: Primary: 49J20, 49K20; Secondary: 35Q93.

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