The Discrete Classical Optimal Control Problem of a Nonlinear Hyperbolic Partial Differential Equation (Dcocp)

Abstract

In this paper, we consider a continuous classical optimal control for systems of nonlinear hyperbolic partial differential equations, with several equality and inequality state constraints. First, the considered continuous classical optimal control problem is discretized into a discrete classical optimal control problem by using the Galerkin finite element method in space and the implicit finite difference scheme in time. The classical continuous controls are approximated by picewise constants. Second the existence of a unique solution of the discrete state equations for fixed discrete classical control is studied. Third, we develop the existence theory for optimality of the discrete classical problem, and the discrete adjoint equations are developed corresponding to the discrete state equations. Finally the necessary conditions and a picewise minimum principle are developed for optimality of the discrete classical problem.

Constraints Optimal Control Governing by Triple Nonlinear Hyperbolic Boundary Value Problem

“…us, (66) and (67) are holded for every υ i ∈ Υ , since C(E) is dense in Υ; hence, we get the wkf (10), (12), and (14).…”

“…Such OCP problems are studied at the beginning for the systems which are controlled by nonlinear ordinary deqs (nodeqs) [7] or by linear deqs (lpdeqs) [8]. Later great interests have been made to study this subject but for systems which are controlled by pdeqs of elliptic type (ET) [9], or of hyperbolic type (HT) [10], or of parabolic type (PT) [11], or by couple of npdeqs of ET [12], or of PT [13], or of HT [14].…”

The Classical Continuous Optimal Control for Quaternary Nonlinear Parabolic Boundary Value Problems with State Vector Constraints

“…OCPs are typically ruled by nonlinear ODEs (NLODEs) [5] or nonlinear PDEs (NLPDEs) [6]. During the last decade, great attention has been made to studying OCPs for system ruling by NLPDEs of elliptic, hyperbolic, and parabolic types [7][8][9]. Later, the study of this subject is expanded to include classical continuous optimal control problem (CCOCP) for systems ruling by couple NLPDEs and then recently by triple NLPDEs for the above three Ibn Al-Haitham Journal for Pure and Applied Sciences http://jih.uobaghdad.edu.iq/index.php/j/index : Journal homepage indicated types of NLPDEs [10 -15].…”

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

“…The optimal control problems play an important role in many fields in the real life problems, for examples in robotics [1], in an electric power [2], in civil engineering [3], in Aeronautics and Astronautics [4], in medicine [5], in economic [6], in heat conduction [7], in biology [8] and many others fields. This importance of optimal control problems encouraged many researchers interested to study the optimal control problems of systems are governed either by nonlinear ordinary differential equations as in [9] and [10] or by linear partial differential equations as in [11] or are governed by nonlinear partial differential equations either of a hyperbolic type as in [12] or of a parabolic type as in [13] or by an elliptic type as in [14], or optimal control problem are governed either by a couple of nonlinear partial differential equations of a hyperbolic type as in [15] or of a parabolic type as in [16] or by an elliptic type as in [17], or of an elliptic type but involve a boundary control as in [18]. While the optimal control problem which, is considered in this work is an optimal boundary (Neumann boundary conditions NBCs) control problem governed by a couple of nonlinear partial differential equations of elliptic type.…”