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

Lamyaa H Ali,Jamil A Al-Hawasy

doi:10.24996/ijs.2021.62.6.27

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.

Highlights

The problems of optimal control (OCPs) have a major significant and vital role in numerous fields, such as biology [1], electric power [2], robotics [3], economic [4], and many other different fields. This significance has motivated many investigators to be concerned with studding the OCPs for mathematical modules dominated by the three types of nonlinear PDEs; elliptic [5], hyperbolic [6] and parabolic [7], whilst many others [8,9,10] are concerned with studying the boundary OCPs (BOCPs)
Iraqi Journal of Science, 2021, Vol 62, No 6, pp: 2009-2021 triple PDEs (TNBVPs) of elliptic and parabolic types [14,15]. All these investigations took our attention to think about generalizing the work in [12] for the BOCP dominated by couple nonlinear BVPs (CNBVPs) into BOCP dominated by NTBVPs of a hyperbolic type (NTHBVPs). This includes the investigation of the solvability theorem for the state vector solution (SVS), the solvability theorem of a boundary optimal control vector (BOCV) with the EINESVC, the derivation for the DRDH, and the demonstration theorems for both the NCOs and the SCOs of optimality
The solvability theorem for the SVS of the triple nonlinear hyperbolic boundary value problem (TNLHBVP) when the boundary control vector (BCV) is given, utilizing the Galerkin method (GAM) with the Aubin theorem (AUTH), is proved successfully

Summary

Introduction

The problems of optimal control (OCPs) have a major significant and vital role in numerous fields, such as biology [1], electric power [2], robotics [3], economic [4], and many other different fields This significance has motivated many investigators to be concerned with studding the OCPs for mathematical modules dominated by the three types of nonlinear PDEs; elliptic [5], hyperbolic [6] and parabolic [7], whilst many others [8,9,10] are concerned with studying the boundary OCPs (BOCPs). Iraqi Journal of Science, 2021, Vol 62, No 6, pp: 2009-2021 triple PDEs (TNBVPs) of elliptic and parabolic types [14,15] All these investigations took our attention to think about generalizing the work in [12] for the BOCP dominated by CNBVPs into BOCP dominated by NTBVPs of a hyperbolic type (NTHBVPs). The DRDH is derived and, the theorems of both the NCOs and SCOs of optimality of the SCP are demonstrated

Description of the problem

Solvability of the SVS

Solvability of BOCV

Conclusions