Mixed Methods for Solving Classical Optimal Control Governing by Nonlinear Hyperbolic Boundary Value Problem

Abstract

This paper is concerned with studying the numerical solution for the discrete classical optimal control problem governing by a nonlinear hyperbolic boundary value problem. When the discrete classical control is given, the existence theorem for a unique discrete solution of the discrete weak form is proved. The existence theorem for the discrete classical optimal control and the necessary theorem “conditions” for optimality of the problem are proved under a suitable assumption. The discrete classical optimal control problem is solved by mixing the Galerkin finite element method for space variable with the implicitfinite difference method for the time variable to find the discrete state of discrete weak form (and the discrete adjoint solution of discrete adjoint weak form), while the Gradient Projection method or of the Gradient method or of the Frank Wolfe method are used to find the discrete classical optimal control. Inside these three methods the Armijo step option or the optimal step option are used to improve the (solution) discrete classical control. Finally, an illustrative example for the problem is given to show the accuracy and efficiency of the methods.