The Ritz–Galerkin Procedure for Parabolic Control Problems

Abstract

A systematic numerical approximation approach to the problem of minimizing an integral cost functional subject to a parabolic partial differential equation constraint is discussed. This problem is viewed as a standard variational minimization problem subject to nonholonomic constraints and is treated using Lagrange multipliers. A practical computational procedure for obtaining approximate solutions to this problem is developed. Error estimates for the control, state, costate, and cost functional are established under appropriate smoothness and boundedness conditions. Also, explicit order bounds are obtained for these estimates over generalized spline interpolating spaces.