Boundary Optimal Control of Infinite Order Linear Elliptic Systems Under Pointwise Control Constraints

This paper presents a rigorous analysis of an optimal boundary control problem governed by a linear elliptic equation of infinite order subject to pointwise control constraints. Such problems arise naturally in various applications but remain insufficiently studied due to the analytical difficulties associated with infinite-order operators and control constraints. The main objective of this work is to establish the Well-posedness of the state equation and derive optimality conditions for the associated control problem. Under assumptions on the system coefficient and admissible control set, we prove the existence and uniqueness of the weak solution to the state equation. Under pointwise control constraints on the boundary, we demonstrate the existence of an optimal control using convexity and compactness arguments that are adapted to the infinite order setting. By deriving the associated adjoint system, we formulate first order necessary optimality conditions in the form of a variational inequality involving the boundary adjoint variable. Furthermore, we discuss optimality conditions under coercivity assumptions on the infinite order operator. The results presented in this paper extend several known results for finite order elliptic systems to the infinite order framework, thereby filling an important gap in the existing literature on boundary optimal control.