Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

We consider an optimal control problem containing a control system described by a partial nonlinear differential equation with the fractional Dirichlet–Laplacian, associated to an integral cost. We investigate the existence of optimal solutions for such a problem. In our study we use Filippov’s approach combined with a lower closure theorem for orientor fields.

A lower closure theorem for autonomous orientor fields

SynopsisGiven a set valued mapping ∑: ℝ → ∑n, we prove a closure property with respect to -convergence for the differential inclusionunder very mild assumptions on ∑.