On convergence of solutions to nonlocal optimal control problems with quasi-minimization constraints
Abstract
This paper studies a new class of nonlocal optimal control problems where the constraints involve approximations of minimizers of quasiconvex energies. These problems are parameterized by a fractional parameter $s \in (0, 1)$ and a horizon parameter $\delta>0$, and the energy density depends on a nonlocal fractional gradient. Here, the constraint consists of finding quasi-minimizers with respect to the quasiconvex energy. Despite the fact that the energies of interest may not have unique minimizers, we may prove the existence of solutions to this class of control problems. The constraint in Cueto-Siktar 2026 was finding global minimizers of the energy, and this problem's main limitation was an inability to prove convergence of solutions for the nonlocal control problems to those of a corresponding local, PDE-constrained optimal control problem. While this issue arises from the lack of uniqueness of minimizers for the constraining energy, we get stronger convergence results with our new choice of constraints. Namely, we obtain convergence of minimizers for nonlocal optimal control problems that have a general cost functional depending on the nonlocal gradient.