The Continuous Classical Optimal Control Problems for Triple Nonlinear Elliptic Boundary Value Problem

Doaa Kateb Jasim,Jamil A Al-Hawasy

doi:10.30526/33.3.2477

Abstract

In this research, our aim is to study the optimal control problem (OCP) for triple nonlinear elliptic boundary value problem (TNLEBVP). The Mint-Browder theorem is used to prove the existence and uniqueness theorem of the solution of the state vector for fixed control vector. The existence theorem for the triple continuous classical optimal control vector (TCCOCV) related to the TNLEBVP is also proved. After studying the existence of a unique solution for the triple adjoint equations (TAEqs) related to the triple of the state equations, we derive The Fréchet derivative (FD) of the cost function using Hamiltonian function. Then the theorems of necessity conditions and the sufficient condition for optimality of the constraints problem are proved

Highlights

The optimal control problem (OCP) is one of the most important subject in mathematics, but in all branches of science, for instance, in engineering such as robotics [1]
In the past few decades, there were many studies and papers published in OCPs for systems that related to nonlinear ordinary differential equations [5]. or systems related to nonlinear partial differential equation (NLPDEqs) either of: a hyperbolic type [6]
While other papers deals with the optimal control problems that are related to triple linear partial differential equation of : an elliptic type [15]

Summary

Introduction

The OCP is one of the most important subject in mathematics, but in all branches of science, for instance, in engineering such as robotics [1]. Systems related to nonlinear partial differential equation (NLPDEqs) either of: a hyperbolic type [6]. Of hyperbolic but include a boundary control [10]. Of a parabolic type [11].Or of a parabolic type but includes a boundary control [12]. Of an elliptic type that includes a Numann boundary control [14]. While other papers deals with the optimal control problems that are related to triple linear partial differential equation of : an elliptic type [15]. The TCCOC problem is to minimize the cost function (5) subject to the state constraints of (6) and (7), i.e. to find v⃗ such that v⃗ ∈ U⃗ and Y v⃗ min Y u⃗. Let W⃗ WWWHΛHΛHΛ , y ‖w‖ and ‖w⃗‖ are denoted by the norm in H Λ and

Weak Formulation of the TNLEBVP

Conclusion