ON EXACT AND APPROXIMATE BOUNDARY CONTROLLABILITIES FOR THE HEAT-EQUATION - A NUMERICAL APPROACH

The present article is concerned with the numerical implementation of the Hilbert uniqueness method for solving exact and approximate bounda ry controllability problems for the heat equation. Using convex dualit y, we reduce the solution of the boundary control problems to the solu tion of identification problems for the initial data of an adjoint hea t equation. To solve these identification problems, we use a combinati on of finite difference methods for the time discretization, finite el ement methods for the space discretization, and of conjugate gradient and operator splitting methods for the iterative solution of the discr ete control problems. We apply then the above methodology to the solut ion of exact and approximate boundary controllability test problems in two space dimensions. The numerical results validate the methods disc ussed in this article and clearly show the computational advantage of using second-order accurate time discretization methods to approximate the control problems.