Barycentric Interpolation Collocation Methods Based on the Crank-Nicolson Scheme for Nonlinear Parabolic Optimal Control Problems

Abstract

In this paper, two numerical schemes for nonlinear optimal control problems governed by parabolic equations are presented to deal with the challenge arising from the mutually coupled state and co-state variables in numerical simulation. Using Lagrangian multipliers, the continuous optimality system is derived, which consists of state and co-state equations, coupled with an optimality condition. To achieve higher spatial accuracy, meshless and high-precision barycentric interpolation collocation methods are applied. Fully discrete collocation approximation schemes are presented, utilizing the Crank-Nicolson scheme in time and the Newton linearization for the nonlinear term. To avoid solving the large coupled scheme directly, a classical iterative method is adopted. Furthermore, we provide consistency analyses of semi-discretized schemes in space, as well as nonlinear fully discretized schemes based on approximation properties of collocation methods. Finally, several numerical experiments are conducted to validate the efficiency of our proposed methods. Comparisons with a classical finite difference method indicate that the proposed collocation schemes offer superior accuracy while requiring fewer nodes.