ABSTRACT
This chapter discusses a bounded domain with a specific boundary in a m-dimensional Euclidean space. It considers a semilinear heat equation with a specific control appearing in a nonlinear boundary condition. The optimal control problem is regarded as that of minimizing a cost functional among all controls satisfying a constraint whose corresponding solutions fulfill a target condition. The problems of specific constraints are studied using a theory of nonlinear programming problems in metric spaces developed for Banach spaces. The chapter discusses related treatments of other boundary control problems (both linear and nonlinear). It also discusses the Neumann function, boundary value problems as integral equations, existence theory, directional derivatives, the maximum principle, and the Dirichlet boundary condition.
