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

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.…”

“…To find such solutions, we should discuss the existence and the uniqueness theorem for the NS for the DWF. The proof of the existence theorem for the discrete classical optimal control (DCOC) and the necessary conditions of the problem are studied in a previous article [9] and they are all needed here. On the other hand, the DSCOCP is found numerically by using the GFEM-IFDS to find the NS of the DWF and then the DCOC by solving the optimization problem (the minimum of DCF) by using, separately, each one of the optimization methods; the GM, the GPM and the FWM.…”

“…Existence of the DSCOCP:The following assumptions are useful to study the existence of the DCC. Assumption C: The cost functional is of Caratheodary type, and satisfies : , The proofs of the following theorem and lemmas are shown in a previous article[9]. 4.1 Theorem: The operator is continuous on ( ).…”