The Nonlocal Kelvin Principle and the Dual Approach to Nonlocal Control in the Conduction Coefficients

We explore the dual approach to nonlocal optimal control in the coefficients, specifically for a classical min-max problem which in this study is associated with a nonlocal scalar diffusion equation. We reformulate the optimal control problem utilizing a dual variational principle, which is expressed in terms of nonlocal two-point fluxes. We introduce the proper functional space framework to deal with this formulation and establish its well-posedness. The key ingredient is the inf-sup (Ladyzhenskaya–Babuška–Brezzi) condition, which holds uniformly with respect to small nonlocal horizons. As a by-product of this fact, we are able to prove convergence of nonlocal optimal control problems toward their local counterparts in a straightforward fashion.