OPTIMAL DISTRIBUTED CONTROL OF NONLOCAL STEADY DIFFUSION PROBLEMS

We study a control problem constrained by a nonlocal steady diffusion equation that arises in several applications. The control is the right-hand side forcing function and the objective of control is a quadratic matching functional. A recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system. We demonstrate the existence and uniqueness of the optimal state and control when sufficient conditions on certain kernel functions and the volume constraints hold. We also demonstrate the convergence, as the nonlocal interactions vanish, of the optimal nonlocal state to the optimal state of a local PDE-constrained control problem. We define continuous and discontinuous Galerkin finite element discretizations of the optimality system for which we derive a priori error estimates. Numerical examples for 1D problems are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained for the nonlocal diffusion equations and for PDEs.