joint work with Harbir Antil, Mahamadii Warma
We consider external optimal control with fractional parabolic PDEs as constraints. The need stems from the fact that the classical PDE models only allow placing the source either on the boundary or in the interior whereas the nonlocal behavior of the fractional operator allows placing the control in the exterior. We introduce the notions of weak and very-weak solutions to the parabolic Dirichlet problem and then approximate its solutions by the parabolic Robin solutions. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models.
joint work with Stefan Ulbrich
We present an optimization approach for nonsmooth, nonconvex and constraint optimization problems. The approach has been developed in the context of shape optimization governed by a contact problem with Coulomb friction including state gradient constraints. For the solution operator of the weak formulation of the underlying variational inequality, we prove the local Lipschitz continuity with respect to the geometry. The shape optimization problem is solved with a bundle trust-region algorithm, whereas the handling of the nonlinear constraints is motivated by a constraint linearization method.