Boundary control of non‐well‐posed elliptic equations with nonlinear Neumann boundary conditions

In this paper, we consider a boundary control problem governed by a class of non‐well‐posed elliptic equations with nonlinear Neumann boundary conditions. First, the existence of optimal pairs is proved. Then by considering a well‐posed penalization problem and taking limit in the optimality system for penalization problem, we obtain the necessary optimality conditions for optimal pairs of initial control problem.