Mixed Implicit Galerkin – Frank Wolf, Gradient and Gradient Projection Methods for Solving Classical Optimal Control Problem Governed by Variable Coefficients, Linear Hyperbolic, Boundary Value Problem

Abstract

This paper deals with testing a numerical solution for the discrete classical optimal control problem governed by a linear hyperbolic boundary value problem with variable coefficients. When the discrete classical control is fixed, the proof of the existence and uniqueness theorem for the discrete solution of the discrete weak form is achieved. The existence theorem for the discrete classical optimal control and the necessary conditions for optimality of the problem are proved under suitable assumptions. The discrete classical optimal control problem (DCOCP) is solved by using the mixed Galerkin finite element method to find the solution of the discrete weak form (discrete state). Also, it is used to find the solution for the discrete adjoint weak form (discrete adjoint) with the Gradient Projection method (GPM) , the Gradient method (GM), or the Frank Wolfe method (FWM) to the DCOCP. Within each of these three methods, the Armijo step option (ARSO) or the optimal step option (OPSO) is used to improve (to accelerate the step) the solution of the discrete classical control problem. Finally, some illustrative numerical examples for the considered discrete control problem are provided. The results show that the GPM with ARSO method is better than GM or FWM with ARSO methods. On the other hand, the results show that the GPM and GM with OPSO methods are better than the FWM with the OPSO method.

Numerical Solutions for the Optimal Control Governing by Variable Coefficients Nonlinear Hyperbolic Boundary Value Problem Using the Gradient Projection, Gradient and Frank Wolfe Methods

“…These applications are usually governed by partial differential equations (PDEs) or ordinary differential equations (ODEs). Many researchers investigated the numerical solution of optimal control problems (NSOCPs) governed by nonlinear elliptic PDEs [3], semilinear parabolic PDEs [4], one dimensional linear hyperbolic PDEs with constant coefficients(LHPDES) [5], two dimensional linear and nonlinear hyperbolic PDEs with constant coefficients [6][7][8][9], two dimensional linear hyperbolic PDEs but with variable coefficients [10], or by one dimensional nonlinear ODEs [11]. The outcomes of these works have driven us to focus our interest on investigating the NSDCOC governed by the VCNLHBVP.…”