Mixed Penalty with Gradient, Gradient Projection and Frank Wolfe Methods for Solving Nonlinear Hyperbolic Optimal Control Sate Constraints

Abstract

Abstract This paper is focused on studying the numerical solution (NUSO) for the discrete classical optimal control problem (DISCOPCP) ruled by a nonlinear hyperbolic boundary value problem (NHYBVP) with state constraints (SCONs). When the discrete classical control (DISCC) is given, the existence and uniqueness theorem for the discrete classical solution of the discrete weak form (DISWF) is proved. The proof for the existence theorem of the discrete classical optimal control (DISCOPC) and the necessary and sufficient conditions (NECOs and SUCOs) of the problem are given. Moreover. The DISCOPCP is found numerically from the Galerkin finite element method (GFE) for variable space and implicit finite difference scheme (IFD) for time variable (GFEIFDM) to find the NUSO of the DISWF and then the DISCOPC is found from solving the optimization problem (OPTP) (the minimum of discrete cost functional (DISCF)) by using the mixed Penalty method with the Gradient method (PGMTH), the Gradient projection method (PGPMTH) and the Frank Wolfe method (PFWMTH). Inside these three methods, the Armijo step option (ASO) is used to get a better direction of the optimal search. Finally, illustrative example for the problem is given to exam the accuracy and efficiency of these methods.