Differential Equations Seminar

Abstract: We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of symmetric coercive elliptic operators in a bounded domain. These operators can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a 

semi-infinite cylinder in one more spatial dimension. 

Thus, we consider an equivalent optimal control problem with a nonuniformly elliptic operator as the state equation. 

The rapid decay of the optimal state suggests a truncation that is suitable for numerical approximation.

We discretize the proposed truncated state equation using first degree tensor product finite elements on anisotropic meshes.

For the control problem we consider and analyze two approaches: 

one that is semi-discrete based on the

so-called variational approach, where the control is not discretized, and the other one is fully discrete via

the discretization of the control by piecewise constant functions. 

For both approaches, we derive a priori error estimates with respect to the degrees of freedom

on anisotropic meshes.