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

Abstract

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.

قابلية الحل للسيطرة الامثلية الحدودية الرباعية المستمرة التقليدية المسيطر بالنظام الرباعي المكافيء

“…The study of the continuous classical type began with the continuous classical boundary optimal control problems dominated by nonlinear parabolic or elliptic or hyperbolic PDEs. Then, these studies were generalized to deal with systems dominated by coupling nonlinear PDEs of these three types 10,11 , and then were generalized also to deal systems dominated by triple nonlinear PDEs of these three types 12 . In each type of these classical continuous boundary optimal control problems, the problem consists of; an initial or a boundary value problem (the dominating eqs.…”

Classical Continuous Boundary Optimal Control Vector Problem for Triple Nonlinear Parabolic System

“…The classical continuous optimal boundary control problem ) CCOBCP) dominated by nonlinear parabolic or elliptic or hyperbolic PDEs is studied in [8][9][10] respectively (resp.). Later, the study of the CCOBCPs dominated by the three types of PDEs is generalized in [11][12][13] to deal with CCOBCPs dominating by couple nonlinear PDEs (CNLPDES) of these types resp., and then the studies of the second and the third types are generalized also to deal with continuous classical optimal control problems (CCOCPs) dominated by triple and NLPDEs of the elliptic and the hyperbolic types [14,15]. All of the above-mentioned studies encouraged us to think about generalizing the study of the CCOCP dominated by CNLPDEs of parabolic type to a CCOCP dominated by TNLPBVP.…”

Quaternary Boundary Optimal Control Problem Dominating by Quaternary Nonlinear Parabolic System

“…The Continuous Classical boundary optimal control problem (CCBOCP) is dominated by a couple of nonlinear parabolic, elliptic or hyperbolic PDEqs that were studied in [8][9][10]. Later, these studies for these three types were generalized to deal with CCBOCP dominated by triple nonlinear PDEqs (TNLPDEqs) so as [11][12][13]. All of the above-mentioned studies and many others encouraged us to think about generalizing the study of the CCBOCP dominated by TNLPDEqs of parabolic type to a CCBOQCP dominated by QNLPBVP.…”