Abstract
This paper is concerned with studying the numerical solution for the discrete classical optimal control problem (NSDCOCP) governed by a variable coefficients nonlinear hyperbolic boundary value problem (VCNLHBVP). The DSCOCP is solved by using the Galerkin finite element method (GFEM) for the space variable and implicit finite difference scheme (GFEM-IFDS) for the time variable to get the NS for the discrete weak form (DWF) and for the discrete adjoint weak form (DSAWF) While, the gradient projection method (GRPM), also called the gradient method (GRM), or the Frank Wolfe method (FRM) are used to minimize the discrete cost function (DCF) to find the DSCOC. Within these three methods, the Armijo step option (ARMSO) or the optimal step option (OPSO) are used to improve the discrete classical control (DSCC). Finally, some illustrative examples for the problem are given to show the accuracy and efficiency of the methods.
Highlights
Many researchers investigated the numerical solution of optimal control problems (NSOCPs) governed by nonlinear elliptic partial differential equations (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 ordinary differential equations (ODEs) [11]
The continuous classical optimal control problem (COCOCP) described by the VCNLHBVP is discretized by applying the Galerkin finite element method (GFEM) for the space variable and implicit finite difference scheme (GFEM-IFDS) for the time variable to get the DSCOCP
Main results (Solution methods): This section is devoted to present our method which is used to solve the DCCOC governed by the VCNLHBVP, the discrete weak form (DWF) are solved by using the mixed GFEM-IFDS, while the minimum values for the discrete cost function (DCF) and the discrete classical optimal control (DCOC) are found by using,separately, each one of the GM, FWM, or GPM
Summary
Optimization problems have wide applications in medicine, sciences and many other fields [1, 2]. The continuous classical optimal control problem (COCOCP) described by the VCNLHBVP is discretized by applying the Galerkin finite element method (GFEM) for the space variable and implicit finite difference scheme (GFEM-IFDS) for the time variable to get the DSCOCP ((the discrete weak form (DWF) for the VCNLHBVP and the discrete cost function (DGF)). To find such solutions, we should discuss the existence and the uniqueness theorem for the NS for the DWF. The proofs of the following theorem and lemmas are shown in a previous article [9]
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