Sensors and Regional Gradient Observability of Hyperbolic Systems

This paper presents a method to deal with an extension of regional gradient observability developed for parabolic system [1,2] to hyperbolic one. This concerns the reconstruction of the state gradient only on a subregion of the system domain. Then necessary conditions for sensors structure are established in order to obtain regional gradient observability. An approach is developed which allows the reconstruction of the system state gradient on a given subregion. The obtained results are illustrated by numerical examples and simulations.


Introduction
For a distributed parameter system evolving on a spatial domain , the notion of regional observability concerns the reconstruction of the initial state on a subregion  of .Characterization results and approaches for the reconstruction of regional state are given in [3,4].Similar results were developed for the state gradient of parabolic systems in [2].This led to the so-called regional gradient observability and concerns the possibility to reconstruct the gradient on a subregion   without the knowledge of the system state.The study of gradient observability is motivated by real applications, the case of insulation problems, also there exist systems for which the state is not observable but the state gradient is observable, example is given in [1].
In this paper we present an extension of the above results on regional gradient observability to hyperbolic systems evolving on a spatial domain .That is to say one may be concerned with the observability of the state gradient only in a critical subregion   of  .More precisely let (S) be a linear hyperbolic system with suitable state space and suppose that the initial state 0 y and its gradient 0 y  are unknown and that measurements are given by means of output functions (depending on the number and structure of the sensors).The problem concerns the reconstruction of the state gradient on the subregion  of the system domain  without taking into account the residuel part on \   .
Here, we consider the problem of regional gradient observability of hyperbolic systems and we establish condition that allows the reconstruction of the initial gradient on such a subregion.And the paper is organized as follows.
The second section is devoted to definitions and characterizations of this notion for hyperbolic systems.In the third section we establish a relation between regional gradient observability and sensors structure.The fourth section is focused on regional reconstruction of the initial gradient.In the last section we give a numerical approach, extending the Hilbert Uniqueness Method developed by J.L. Lions [5], and illustrations with efficient simulations., , ,0 ,0 , , 0

Regional Gradient Observability
where A is the second order elliptic linear operator with regular coefficients.
Equation (1) has a unique solution Suppose that measurements on system (1) are given by an output function: where 0 is a linear operator depending on the structure of sensors.
: C H Let us recall that a sensor is defined by a couple , where is the location of the sensor and is the spatial distribution of measurements on .In the case of a pointwise sensor, then the system may be written in the form A has a compact resolvent and generates a strongly continuous semi-group on a subspace of the the associated eigenvalues with multiplicity .Then (3) admits a unique solution .
* which is linear and bounded with its adjoint denoted by K and let  be the operator , ,

H H L y y y y y
while their adjoints are denoted by *  * and  respectively.

Definition 2.1
The system (1) together with the output ( 2) is said to be exactly (resp.approximately) gradient observable if Such a system will be said exactly (resp.approximately) G-observable.
For a positive Lebesgue measure subset  of  , we also consider the operators , , : : 0, ; 1) The system (1) together with the output Equation ( 2) is said to be exactly regionally gradient observable or exactly G-observable on  if The system (1) together with the output equation ( 2) is said to be approximately regionally gradient observ- The notion of regional G-observability on  may be characterized by the following results.

Proposition 2.3
1) The system (1) together with the output Equation ( 2) is exactly G-observable on  if and only if one of the following propositions is holds.
a) For all , there exists , such that 2) The system (1) together with the output Equation ( 2) is approximately G-observable on  if and only if the operator is positive definite.
*  , we have , and by the general result given in [8], this is equivalent to such that .
since the system (1) is exactly G-observable on  , there exists such that .
2) Let such that

  
2) There exist systems which are not G-observable on the whole domain but may be G-observable on some subregion.

Example 2.5
, we consider the two-dimensional system described by the hyperbolic system , , 0 Measurements are given by the output function .
is the sensor support and and we consider the initial state Copyright © 2012 SciRes.
Then the initial state gradient to be observed is We have the result.

Proposition 2.6
The gradient g is not approximately G-observable on the whole domain  , however it is approximately G-observable on the subregion  .

Proof
To prove that g is not approximately G-observable on , we must show that , and then the system is not approximately G-observable once .On the other hand g may be approximately G-observable on  .Indeed, suppose that

Gradient Strategic Sensors
The purpose of this section is to establish a link between regional gradient observability and the sensors structure.
Let us consider the system (1) observed by sensors which may be pointwise or zone.(or a sequence of sensors) is said to be gradient strategic on  if the observed system is G-observable on  , such a sensor will be said G-strategic on  .
We assume that the operator A is of constant coefficients and has a complete set of eigenfunctions in Assume also that is finite, then we have the following result.
If the sequence of sensors is G-strategic on  , then and , where and 1 and is the row vector the elements of which are

Proof
The proof is developed in the case zone sensors.
The sequence of sensors   Suppose that the sequence of sensors i i is Gstrategic on  and there exists , with , and let and  then , , , Integrating on  we obtain Using the fact that , , , , 5), ( 6) and ( 7) we obtain , cos , sin , 0 and  , which contradicts the fact that the sequence of sensors is G-strategic.

Remark 3.3
1) The above proposition implies that the required number of sensors is greater than or equal to the largest multiplicity of eigenvalues.
2) By infinitesimally deforming of the domain, the multiplicity of the eigenvalues can be reduced to one [9,10].Consequently, the regional G-observability on the subregion  may be possible only by one sensor.

Regional Gradient Reconstruction
In this section, we give an approach which allows the reconstruction of the initial state gradient on  of the system (1).This approach extends the Hilbert Uniqueness Method developed for controllability by Lions [6] and don't take into account what must be the residual initial gradient state on the subregion , , ,0 ,0 , , 0 We consider the zone sensor case where the system (1) is observed by the output function ., , is the sensor support, f the function of measure and we consider a semi-norm on F defined by  is the solution of (8).where The reverse system given by and consider the retrograde system which has a unique solution Copyright © 2012 SciRes.
We denote the solution  ,0 Then, the regional gradient observability turns up to solve the equation where ,   which is the gradient of the initial state to be observed on  .

Proof
1) Let us show first that if the system (1) is G-observable, then (10) defines a norm on F .Consider a basis   j w of the eigenfunctions of A , without loss of generality we suppose that the multiplicity of the eigenvalues are simple, then and since the sensor is regionally G-strategic on  , we have and from   on  , ( 10) is a norm.

F completion of 2) Let denote by
F by the norm (10) and * F be its dual.We show that is an isomorphism from and  the corresponding solution for the problem ( 8), multiply the first equation of the system (11) by

and integrate on
, we obtain for the first term, we obtain ˆ., 0 ., 0 , , ˆ., 0 , .,0 , Using Green formula for the second term, we obtain and with the boundary conditions, we obtain , is an isomorphism and consequently the Equation (13) has a unique solution which corresponds to the state gradient to be observed on the subregion  .

Remark 4.2
The previous approach can be established with similar techniques when the output is defined by means of internal or boundary pointwise sensors.

Numerical Approach
In this section we give a numerical approach which leads to explicit formulas for 0 1 , y y   on  .We consider the case where the system (1) is observed by the output equation where

Proof
In the previous section, it has been seen that the regional reconstruction of the initial state gradient on  turns up to solve the Equation (13).For that consider the functional And solving Equation (13) turns up to minimize  with respect to   After development and when , we obtain ., On the other hand, we have , , , , we obtain The minimization of ( 13) is equivalent to solve the two following problems and   t w x be the solution of the system (12) with With these developments, according to (18) and ( 19), we obtain. .
We replace that in the relation ( 16) and (17), we obtain We consider a truncation up to order M IN  , then we obtain the relation ( 14) and (15).
We define a final error Step 2: Choose a low truncation order M .

Remark 5.2
if y and 1 y are regular enough, we have a regular system state, so measurements may be taken with pointwise sensor.In this case we obtain similar formulaes as in the previous proposition given by 6. Simulations

Example
In this section we develop a numerical example that leads to results related to the choice of the subregion, the sensor location and the initial state gradient.On   0,1   , we consider the one dimensional system.

   
Measurements are given by the output function


The previous system is G-observable on 0,1 [7] if and only if We denote that numerically an irrational number does not exist but it can be considered as irrational if truncation number exceeds the desired precision.

Simulating Conjectures
Now we show numerically how the error grows with respect to the subregion area.It means that the larger the region is, the greater the error is.The obtained results are presented in Table 1.       .We note that the reconstr depends on th plitude of initial state gradient.It means that the greater the amplitude is, the greater the error is.The obtained results are presented in Table 2.
uction error e am

Conclusion
ility on a subregion interior to the spa-Gradient Observab tial evolution domain of hyperbolic system is considered.A relation between this notion and the sensors structure is established and numerical approach for its reconstruction is given.This allows the computation of the initial state gradient without the knowledge of the system state.Illustrations by numerical simulations show the efficiency of the approach.Interesting questions remain open, the case where the subregion  is part of the boundary of the system domain.This uestion is under consideration.q

nI
Let  be an open bounded subset of R with a re-


Since the system is exactly G-observable on .

. 1 . 1
Proposition 5.If the sensor is G-strategic on  , then the ini-

AlgorithmStep 1 :
and we have the following algorithm: Data: The region  , the sensor location and D  .
the sensor is located at  .The initial gradient to be reconstructed is given by that the numerical scheme be stable, and in order to obtain a reason-Applying the previous algorithm, using the formulae (20), (21) we respectively obtain the Figures1 and 2 for


Also how both the error decreases with respect to the amplitude 1A of the initial state gradient.For this let take the s region ub