Optimal Venttsel Boundary Control of Parabolic Equations

TL;DR

This paper derives necessary and sufficient optimality conditions for controlling parabolic equations through Venttsel boundary conditions, advancing the mathematical understanding of boundary control problems.

Contribution

It provides the first derivation of first-order optimality conditions for Venttsel boundary control of parabolic equations, including both constrained and unconstrained cases.

Findings

First-order necessary conditions established for optimal Venttsel boundary control.

Sufficient conditions identified for unconstrained problems.

Framework applicable to both constrained and unconstrained control scenarios.

Abstract

In this paper we study the optimality condition for the Venttsel boundary control of a parabolic equation, that is, the state of the dynamic system is governed by a parabolic equation together with an initial condition while the control is applied to the system via the Venttsel boundary condition. A first order necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained problem.