Simultaneous distributed and Neumann boundary optimal control problems for elliptic hemivariational inequalities

Abstract

In this paper, we study boundary optimal control problems on the heat flux and simultaneous distributed-boundary optimal control problems on the internal energy and the heat flux for a system governed by a class of elliptic boundary hemivariational inequalities with a parameter. The system was originated by a steady-state heat conduction problem with non-monotone multivalued subdifferential boundary condition on a portion of the boundary of the domain described by the Clarke generalized gradient of a locally Lipschitz function. We prove an existence result for the boundary optimal control problem and simultaneous distributed-boundary optimal control problems. We show an asymptotic behavior result for the optimal controls and the system states for both optimal control problems, when the parameter, like a heat transfer coefficient, tends to infinity on a portion of the boundary.

Numerical analysis of a family of simultaneous distributed-boundary mixed elliptic optimal control problems and their asymptotic behaviour through a commutative diagram and error estimates

“…is locally Lipschitz for a.e. x ∈ Γ 1 and not necessary differentiable following [3,13,17,20]. Therefore, the variational formulation, for the system state, will be given by an elliptic hemivariational inequality, and the corresponding control variable can be the energy g, or the heat flux q or the vectorial control (g, q).…”