Constraints Optimal Control Governing by Triple Nonlinear Hyperbolic Boundary Value Problem

Abstract

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.

The Classical Continuous Optimal Control for Quaternary Nonlinear Parabolic Boundary Value Problems

“…The classical continuous optimal control problem ) CCOCP) dominated by nonlinear parabolic or elliptic or hyperbolic PDEs are studied in [9][10][11] respectively (resp.). Later, the study of the CCOCPs dominated by the three types of nonlinear PDEs is generalized in [12][13][14] to deal with CCOCPs dominating by coupling NLPDEs of these types resp., and then these studies are generalized also to deal with CCOCPs dominated by triple and NLPDEs of the three types [15][16][17] .…”

The Optimal Classical Continuous Control Quaternary Vector of Quaternary Nonlinear Hyperbolic Boundary Value Problem

“…During the last decade, great attention has been made to studying the subject of OCCCP for a system dominated by nonlinear PDEs (NLPDEs) Ibn Al-Haitham Journal for Pure and Applied Sciences http://jih.uobaghdad.edu.iq/index.php/j/index : Journal homepage of the three types elliptic [5], hyperbolic [6], and parabolic [7]. Latter, the study of this subject expanded to include OCCCP for systems dominated by a couple of NLPDEs of their three types [8][9][10]; through recent years, these studies for these three types expanded to deal with OCCCP for systems dominated by triple NLPDEs [11][12][13]. All these studies encouraged us to investigate the OCCCP dominated by QNLHBVP.…”

The Continuous Classical Boundary Optimal Control of Triple Nonlinear Elliptic Partial Differential Equations with State Constraints

“…All these applications pushed many investigators to a higher level of interest in studying the optimal control problem for nonlinear ordinary differential equations [5], for different types of linear partial differential equations, including the hyperbolic, parabolic and elliptic [6][7][8], or for couple nonlinear partial differential equations of these three types [9][10][11]. While other authors [12,13] studied these three types but included a Neumann boundary control. More recently, optimal control problems were studied for triple partial differential equations of these three types [14][15][16].…”

“…While other authors [12,13] studied these three types but included a Neumann boundary control. More recently, optimal control problems were studied for triple partial differential equations of these three types [14][15][16]. Also, the optimal control problem involving Neumann boundary control for triple partial differential equations of parabolic type was also recently investigated [17].…”

“…) ‖ ‖ ( ) , where . The following assumptions are useful to prove the existence theorem of a unique solution of the weak form(14).Assumption (I): a)( ⃗ ) is coercive , i.e ,( ⃗ ⃗ ⃗ ⃗ ) ‖ ⃗ ⃗ ‖ . ( )/ ‖ ⃗ ‖ .…”