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

The focus of this work lies on proving the existence theorem of a unique state vector solution (Stvs) of the triple nonlinear hyperbolic boundary value problem (TNHBVP) when the classical continuous control vector (CCCVE) is fixed by using the Galerkin method (Galm), proving the existence theorem of a unique constraints classical continuous optimal control vector (CCCOCVE) with vector state constraints (equality EQVC and inequality INEQVC). Also, it consists of studying for the existence and uniqueness adjoint vector solution (Advs) of the triple adjoint vector equations (TAEqs) associated with the considered triple state equations (Tsteqs). The Fréchet Derivative (Frde.) of the Hamiltonian (HAM) is found. At the end, the theorems for the necessary conditions and the sufficient conditions of optimality (Necoop and Sucoop) are achieved.

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Boundary Optimal Control for Triple Nonlinear Hyperbolic Boundary Value Problem with State Constraints

Iraqi Journal of Science ◽

10.24996/ijs.2021.62.6.27 ◽

2021 ◽

pp. 2009-2021

Author(s):

Lamyaa H Ali ◽

Jamil A. Al-Hawasy

Keyword(s):

Optimal Control ◽

Boundary Value Problem ◽

State Vector ◽

Boundary Value ◽

Sufficient Conditions ◽

The State ◽

Control Vector ◽

Vector Solution ◽

Solvability Theorem ◽

Hyperbolic Boundary

The paper is concerned with the state and proof of the solvability theorem of unique state vector solution (SVS) of triple nonlinear hyperbolic boundary value problem (TNLHBVP), via utilizing the Galerkin method (GAM) with the Aubin theorem (AUTH), when the boundary control vector (BCV) is known. Solvability theorem of a boundary optimal control vector (BOCV) with equality and inequality state vector constraints (EINESVC) is proved. We studied the solvability theorem of a unique solution for the adjoint triple boundary value problem (ATHBVP) associated with TNLHBVP. The directional derivation (DRD) of the "Hamiltonian"(DRDH) is deduced. Finally, the necessary theorem (necessary conditions "NCOs") and the sufficient theorem (sufficient conditions" SCOs"), together denoted as NSCOs, for the optimality (OP) of the state constrained problem (SCP) are stated and proved.

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

Iraqi Journal of Science ◽

10.24996/ijs.2020.si.1.21 ◽

2020 ◽

pp. 161-169

Author(s):

Jamil A. Ali Al-Hawasy ◽

Eman H. Mukhalef Al-Rawdanee

Keyword(s):

Optimal Control ◽

Boundary Value Problem ◽

Numerical Solutions ◽

Boundary Value ◽

Weak Form ◽

Variable Coefficients ◽

Gradient Projection ◽

Implicit Finite Difference Scheme ◽

Classical Control ◽

Hyperbolic Boundary

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.

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