2019/05/24 by Glusa, Christian, Otarola, Enrique
#49J20 #49M25 #65M12 #65M15 #65M60 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1905.10002
We consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel L2(0,T;L2(Ω)) a priori error estimate. On the basis of the aforementioned solution technique, we propose a fully discrete scheme for our optimal control problem that discretizes the control variable with piecewise constant functions and derive a priori error estimates for it. We illustrate the theory with one- and two-dimensional numerical experiments.