Optimal control of an evolution equation with non-smooth dissipation

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.