Necessary Optimality Conditions for Some Control Problems of Elliptic Equations with Venttsel Boundary Conditions

TL;DR

This paper establishes necessary optimality conditions for control problems governed by semi-linear elliptic equations with Venttsel boundary conditions, focusing on boundary control and state constraints.

Contribution

It introduces the first derivation of necessary optimality conditions for elliptic control problems with Venttsel boundary conditions involving boundary controls.

Findings

Derived necessary optimality conditions for the control problem

Addressed boundary control with state constraints

Extended optimality theory to Venttsel boundary conditions

Abstract

In this paper we derive a necessary optimality condition for a local optimal solution of some control problems. These optimal control problems are governed by a semi-linear Vettsel boundary value problem of a linear elliptic equation. The control is applied to the state equation via the boundary and a functional of the control together with the solution of the state equation under such a control will be minimized. A constrain on the solution of the state equation is also considered.