Adil KHAZARI,Ali BOUTOULOUT,Imad EL HARRAKI

1.Sidi Moham ed Ben Abdellah University,école Nationale de Commerce et de gestion,Fez,Morocco;

2.TSI Team,MACS Laboratory,Faculty of Sciences,Mou lay Ism ail University,Meknes,Morocco;

3.école nationale supérieure des mines de Rabat,Rabat,Morocco

1 Introduction

Observability represents one of the major concepts of modern control system theory.It was introduced by R.Kalman in 1960.Roughly speaking,observability is concerned with whether without knowing the initial state,one can determine the state of a system given the input and the output.The study of this kind of problem has become fairly common,and is now an established area of research with a long list of publications.

Hyperbolic problem s are one of the problem s which undergoes a detailed investigation,due to the many problem s which rely on this theory.Hyperbolic equations describe various time dependent models of many physical,chemical and biological phenomena so the study of such equations is of substantial contemporary interest.

Modeling any real problem involves approximations.When we model a phenomena,we must make compromises.We attem pt to retain essential factors while keeping within bounds of mathematical tractability.All real problem s are nonlinear,often strongly nonlinear.But in a mathematics point of view we do liniarize them.In this paper we are interested to study the regional observability of semilinear hyperbolic problem s which are linear problems containing a nonlinear term(see[1,2]).

Analysis and control theory are known as as et of technical reasoning and mathematical tools in the service of system s for both a better understanding of its functioning and decision-making.To do this,there must bean accurate description of the system,which requires a fairly detailed know ledge of its various components,their behavior and their interactions.This description may be represented as partial differential equations(PDEs).Exploration and research on the analysis of distributed system sand their control were mainly focused on the global domain of the evolution of system(see[3,4]and[5]).However,many real problem s can’t be formulated in all the domain of development,but only in a sub-region of global area,called regional context(see[2,6]and[7]).

Specifically,the regional observability concept concerns the actual identification of the trajectory based on the information collected on the system which is equivalent to solve an inverse problem(see[8]).The study of this concept becomes more com p lex in the case of infinite dimension spaces.The general approach to reconstruct the initial state is to leave the system in free evolution for a tim e interval[0,T]during which the performances are measured with sensors.For example,the problem of detecting a polluted area in the ocean caused by a sinking oil ship or knowing the region of a pipe leak both problem s are observability problems where we need to reconstruct just the gradient initial conditions without the know ledge of initial conditions(see[9,10]and[11]).

This work focuses on the study of regional observability of a very important class of distributed system s,which is a class of semilinear system s,since they are intermediate between linear system s and the nonlinear ones(see[12]).In fact,m any real problems are modeled by using nonlinear systems,of either the parabolic type(such as the heat equation)or the hyperbolic type(such as the w ave equation).We extend the concept of regional observability of the gradient for linear system s to a class of semilinear hyperbolic system s.This work is discussed in the parabolic case(see[9]).To rebuild the gradient of the initial state in a subregion ω of the evolution domain Ω,we combine the so called Hilbert uniqueness method(see)with a fixed point technique(see[11])for the reconstruction of the flow of initial conditions.This technique was used in[13]for the exact distributed controllability for the semilinear wave equation.The author generalized the theorem s of exact controllability for the linear wave equation with a distributed control to the semilinear case,showing that,given T large enough,for every initial state in a sufficiently small neigh bour hood of the origin in a certain function space,there exists a distributed control,supported on a part of a domain,driving the system to rest.This approach also provides a numerical algorithm tested on a simulation example.

2 Regional observability of the gradient

Let Ω be an open bounded subset of Rn(n=1,2,3).For T＞0,we denote Q=Ω×]0,T[,Σ=?Ω×]0,T[and we consider the follow ing hyperbolic system:

where A is an elliptic and a second order operator,system(1)is augmented with the output function given by

augmented with the output function

Let us consider a basis of eigenfunctions of the operator A,denoted Φmj,with eigenvalues associated are λmwith multiplicity rm.

We can write for any(y1,y2)∈F,

then the output equation can be expressed by

Consider the operatorˉ?given by the formula

w here

For ω ? Ω a nonempty open subregion of Ω with positive Lebesgue measure,letbe the restriction operator defined by

where

De finition1System is said to be gradient observable or G-observable in ω if we can reconstruct the gradient of the initial condition in a subregion ω of Ω.

In what follow s,we say that a system is G-observable in ω.

We consider the following semilinear system:

augmented with the output function

where N is a nonlinear operator as sum ed to be locally Lipschitzian.

System(5)is equivalent to the follow ing system

System(7)is increased by the output function

System(7)has a unique solution that can be expressed in the m ild sense as follow s(see[15]):

Problem(*)Given the semilinear system(5)and(6)on]0,T[,is it possible to reconstructwhich is the gradient of the initial condition of(5)on ω?

3 HUM app roach

In this section,we give an approach that allows the reconstruction of the initial gradient of the state and speed in ω.This approach is an extension of HUM method developed by Lions[15],and does not take into account what m ay be the residual part in the subregion Ωω.

We consider system(5)augmented with the output function(6)and we assume that system(5)is observed by means of internal zone sensor(D,f)with D?Ω is the support of the sensor and f∈L2(D)defined his space repartition.

The problem of regional gradient observability of(5)and(6)is the possibility to reconstruct the component

Let us consider

which can be decomposed as follow s:

and

System(12)has a unique solution

We define the mapping

which in ducesasem i-norm on G,with=(??0,??1),if the linear system(12)is weakly G-observable in ω,the semi-norm define a norm on G we also denote by G the completion of G.

We define the auxiliary system by

The resolution to(15)provides

Consider the operator

where

We decompose as follow s:ˉψ=ψ0+ψ1,w here ψ0is a solution to the problem

and ψ1is a solution to

then

w here

and

with

If the linear part of system(5)is weakly G-observable,then Λ is invertible,and finally

Considering the following system:

We have the following result.

Proposition 1If the linear part(1)of system(5)is weakly G-observable in ω and N satisfies that?c ＞ 0 such as ‖N(x)‖≤ c‖x‖,then(20)admits a unique fixed point corresponding to the gradient of the initial conditions to be observed in the subregion ω.

ProofWe show the proposition in two steps:first the operator M is com pact;second ly:we apply the theorem of the fixed point of Schauder,to prove that the operator Mhas a fixed point that corresponds to the gradients of the initial conditions to observe in the region ω.

?Let p＞ 0 and Bp=B(0,p)×B(0,p),and we have

We have ψ1(·)is a solution to(17),then we obtain

Without loss of generality we assume that rm=1,then we obtain:

or we have

Then

with t? τ=t′.

By the continuity of the gradient we have

where c1is the constant of continuity.

In the other hand,ψ0(·)solution to(16)then

By the continuity of the gradient we have

As well,we obtain that θ(·)is a solution to(13),then

We have ?1is a solution to(12)then

Then we obtain

App lying Gronwall’s theorem for the function θ(·)we have

Therefore,using Gronwall’s theorem for the function ψ1(·)we have

For 0＜t1＜T and 0＜t2＜T,we obtain

Thus,we see that M:Bp→G?is com pact.

?M enforces the ball Bpin itself.If the linear part(1)of system(5)is weakly G-observable in ω,then Λ?1P is bounded,

Then with the Schauder theorem the operator Madmits a unique fixed point and the gradient of the initial condition to be observed in ω is given by□

We have the follow ing algorithm:

Algorithm

Step 1The initial conditionsthe region ω,the domain D,the function of distribution f and the accuracy threshold ε.

Step 2Repeat

?Solving(13)and obtain θk.

Step 3The restriction ofto ω corresponds to the gradientto be reconstructed in ω.

4 Simulation results

Here we present a numerical example illustrating the algorithm based to HUM m ethod.The obtained results are related to the considered subregion and the sensor location.

ExampleConsider the monodimensional semilinear system in Ω=]0,1[,

For the simulations in the zonal case,we assume that in this case the sensor is located in an internal zone D=]0.6,0.7[,T=2,

Using the obtained algorithm,for ω=]0.35,0.65[,we have Figs.1 and 2.

Fig.1 The exact state gradient and estim ated state gradientin ω.

Fig.2 T he exact speed gradientand estimated speed gradient in ω.

The reconstruction is observed with error equals to

For ω = Ω,we have Figs.3 and 4.

The reconstruction is observed with error equals to

Here we study numerically the dependence of the gradient reconstruction error with respect to the subregion area of ω,we have Table 1.

Fig.3 The exact state gradientand estim ated state gradient in Ω.

Fig.4 The exact speed gradientand estimated speed gradientin Ω.

Table 1 The reconstruction error with respect to the subregion area.

5 Conclusions

The regional gradient observability for hyperbolic semilinear systems is considered.The regional gradient observability of linear systems was explored to solve the problem s related to the semilinear ones which constitutes a natural extension.We explored Hilbert Uniqueness reconstruction approach combined with the fixed point techniques.This leads to an algorithm which is numerically implemented.Many questions remain open,such as the case of the regional gradient observability of semilinear system s using sectorial approach[17]and the case of the regional gradient of the boundary observability of hyperbolic system s[18].The questions are still under consideration and the results will appear in a future paper.

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