Optimal control of an evolution equation with non-smooth dissipation

TL;DR

This paper develops a method for optimal control of evolution equations with non-smooth dissipation by regularizing the dissipation, deriving optimality conditions for the smooth approximation, and then passing to the limit to handle the original non-smooth problem.

Contribution

It introduces a regularization approach to derive optimality conditions for non-smooth dissipation systems, addressing the challenge of non-smooth solution mappings.

Findings

Derived optimality conditions for smooth approximations

Established necessary conditions for the original non-smooth problem

Analyzed the impact of regularization on adjoint regularity

Abstract

In the present work we study the optimal control of an evolution equation with non-smooth dissipation. The solution mapping of this system is non-smooth and hence the analysis is quite challenging. Our approach is to regularize the dissipation via approximation by a smooth function. We derive optimality conditions for the corresponding smooth optimal control problem. Then we drive the regularization parameter to zero and obtain necessary optimality conditions for the original non-smooth problem. However, in this process we lose regularity of the adjoint variables.