On Optimal Control of the Heat Flux at the Left-Hand Side in a Heat Conductivity System

Abstract

We deal with an optimal boundary control problem in a 1-d heat equation with Neumann boundary conditions. We search for a Neumann boundary function which is the minimum element of a quadratic cost functional involving the $H^1$-norm of boundary controls. We prove that the cost functional has a unique minimum element and is Frechet differentiable. We give a necessary condition for the optimal solution and construct a minimizing sequence using the gradient of the cost functional.