Boundary Control of Systems Governed by Semilinear Elliptic Equation for Infinite Order Operator with Finite Dimension

Abstract

This paper studies a boundary optimal control problem governed by a semilinear elliptic equation involving an elliptic operator of infinite-order with a finite dimension. The state equation is defined on a bounded domain with a nonlinear boundary condition where the control variable acts on the boundary. The analysis is carried out within the framework of Sobolev spaces of infinite order. We first establish the existence and uniqueness of the solution to the state equation and define the associated control-to-state mapping. Under suitable assumptions on nonlinear boundary term, we give the differentiability properties of this mapping. Furthermore, we derive the first-order necessary optimality conditions for the optimal control problem through the associated adjoint system and a variational inequality. Finally, we analyze the second order derivatives of the reduced cost functional.