Enlarged Asymptotic Compensation in Discrete Distributed Systems

Abstract

Abstract. This work concerns an enlarged analysis of the problem of asymptotic compensation for a class of discrete linear distributed systems. We study the possibility of asymptotic compensation of a disturbance by bringing asymptotically the observation in a given tolerance zone C. Under convenient hypothesis, we show the existence and the unicity of the optimal control ensuring this compensation and we give its characterization.

Domination of Discrete Distributed Systems

“…Characterizations, various properties and results, as well as applications and illustrations of this notion can be found in [6,7,9]. We have the following result.…”

Regional viability and spreadability

“…Example 1: Consider the example given by the following state equation: [2][3][4] and the initial state given by:…”

“…Indeed, z 0 ∈ K and χ σ z 0 ∈ K σ and for all t ∈]0, T [, χ σ z(x, t; z 0 ) ∈ K σ but there exists a time t 1 = such that z(x, t; z 0 ) / ∈ K. Consider now the case where K = {0 L 2 (Ω) } and σ =]2, 4[ then the same state z 0 given by (3)(4)(5) is regionally σ-viable but not globally viable and in the particular case z 0 / ∈ K but χ σ z 0 ∈ K σ and the restriction of the corresponding solution remains in K σ for all t ∈]0, T [. This example shows that there exists some systems for which there exists some states which are regionally viable without being globally viable.…”

“…Thus were introduced concepts of regional controllability/observability and stability/stabilizability. It has been shown that it is possible to control (observe or stabilize) regionally some systems which are not globally controllable (observable or stabilizable ) This [4], [9]. Thus, and more particularly, to describe some environmental phenomena such those of pollution, epidemic or fires of forests..., the concept of spreadability was been introduced by A. El Jai and K. Kassara [10] and developed by Bernoussi and El Jai.…”

Weak and exact domination in distributed systems

“…In this area, the literature is very rich. Characterization results on the notions of weak and exact controllability and observability, as well as strategic actuators and sensors, are established (Afifi and El Jai, 1994;Afifi et al, 2002a;2002b;2008a;Berrahmoune, 1984;Curtain and Pritchard, 1978;Curtain and Zwart, 1995;El Jai and Pritchard, 1987;Jacob and Zwart, 2001;Lions, 1968;1988;Qarai et al, 2008;Uciński and Korbicz, 1990;Zerrik, 1993;Zerrik et al, 2007). But in almost all these works, the problems considered are focused on the possibility to reach a desired state or reconstruct the state of the examined system, i.e., on studying if a system is (or not) controllable or observable, but without comparison between the input or output parameters themselves.…”