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Boundary Controllability of Co-operative Neumann Parabolic Systems with Distributed or Boundary Observation

ABSTRACT

In this communication,  acontrollability problems for co-operative parabolic linear system involving Laplace operator with boundary Neumann control and  distributed or  boundary observations are considered.

Keywords

Optimal control problem, Controllability, solutions of parabolic system, co-operative system.

INTRODUCTION

Controllability is a mathematical problem, which consists in determining the targets to which one can drive the state of some dynamical system, by means of a control parameter present in the equation. Many physical systems such as quantum systems, fluid mechanical systems, wave propagation, diffusion phenomena, etc. are represented by an infinite number of degrees of freedom, and their evolution follows some partial differential equation. Finding active controls in order to properly influence the dynamics of these systems generate highly involved problems. The control theory for PDEs, and among this theory, controllability problems, is a mathematical description of such situations. Any dynamical system represented by a PDE, and on which an external influence can be described, can be the object of a study from this point of view. In 1978, D.L. Russell [1] made a rather complete survey of the most relevant results that were available in the literature at that time. In that paper, the author described a number of different tools that were developed to address controllability problems, often inspired and related to other subjects concerning partial differential equations: multipliers, moment problems, nonharmonic Fourier series, etc.

Various types of controllability of linear abstract dynamical systems defined in a Banach or Hilbert spaces have been recently extensively explored by several authors (see e.g.[2]-[18]). More recently, J.-L. Lions introduced the so called Hilbert Uniqueness Method (H.U.M.; see [19]).

In this work, we will focus our attention on some special aspects of controllability problems for parabolic system involving Laplace operator with pointwise observation. In order to explain the results we have in mind, it is convenient to consider the abstract form:

Let  and  be two real Hilbert spaces such that  is a dense subspace of  Identifying the dual of  with  we may consider  where the embedding is dense in the following space. Let ( ) be a family of continuous operators associated with a bilinear forms  defined on  which are satisfied Gårding’s inequality

                                       (1)

Then, from [20] and [21], for given  and  be a bounded linear operator the following abstract systems:

                                                                       (2)

 have a unique solution, we denote it by

We also given an observation equation

                         being a Hilbert space

Definition 1 The system whose state is defined by (2)is said to be controllable if the observation  generates a dense (affine) subspace of the space of observation.

 

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About the Author

Mohammed Shehata

Mohammed Shehata

Department of Mathematics, Faculty of Science, Jazan University, Kingdom of Saudi Arabia.

mashehata_math@yahoo.com


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