OPTIMAL NEUMANN BOUNDARY CONTROL OF THE HEAT EQUATION WITH H1 CONTROLS: EXISTENCE, OPTIMALITY CONDITIONS AND STRONG CONVEXITY

Abstract

This paper investigates an optimal Neumann boundary control problem for the one-dimensional heat equation. The control input is the heat flux applied at the left boundary of the spatial domain, while the objective is to drive the temperature at the right boundary toward a prescribed target trajectory over a finite time interval. The admissible control set is taken as a bounded, closed, and convex subset of the Hilbert space H¹(0,T), which yields enhanced regularity compared with the classical L²(0,T),  framework. The existence and uniqueness of the optimal control are established via the direct method of the calculus of variations, together with the strong convexity of the cost functional. The Fréchet differentiability of the cost functional is proved via a Lagrange multiplier approach, and an explicit expression for the gradient is derived through the associated adjoint problem. A necessary optimality condition is obtained in the form of a variational inequality. Furthermore, the strong convexity of the cost functional guarantees the uniqueness of the optimal control and implies that the optimality condition is also sufficient.