ABSTRACT

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.

1. 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 and its gradient 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.

2. Regional Gradient Observability

Let be an open bounded subset of with a regular boundary. Fix and let denote by and.

Consider the system described by the hyperbolic equation

. (1)

where is the second order elliptic linear operator with regular coefficients.

Equation (1) has a unique solution

[6].

Suppose that measurements on system (1) are given by an output function:

. (2)

where is a linear operator depending on the structure of sensors.

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, and is the Dirac mass concentrated in see [7].

Let and then the system may be written in the form

(3)

with.

has a compact resolvent and generates a strongly continuous semi-group on a subspace of the Hilbert state space given by

is a basis in of eigenfunctions of, orthonormal in and the associated eigenvalues with multiplicity. Then (3) admits a unique solution.

Let us define the observability operator

which is linear and bounded with its adjoint denoted by and let be the operator

where

while their adjoints are denoted by and respectively.

2.1. 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

where

and

while their adjoints, denoted by, and respectively and given by

where

and

We finally introduce the operator

.

2.2. Definition 2.2

1) The system (1) together with the output Equation (2) is said to be exactly regionally gradient observable or exactly G-observable on if

2) The system (1) together with the output equation (2) is said to be approximately regionally gradient observable or approximately G-observable on if .

The notion of regional G-observability on may be characterized by the following results.

2.3. 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

b)

2) The system (1) together with the output Equation (2) is approximately G-observable on if and only if the operator is positive definite.

2.4. Proof

1) a) let us consider the operator and.

Since the system is exactly G-observable on, we have, and by the general result given in [8], this is equivalent to such that

b) Let, then

,

since the system (1) is exactly G-observable on, there exists such that.

Let put where and , then and.

Conversely, let, then , there exist

and such that and .

Since, there exists

such that. Thus, which gives.

2) Let such that .

So which means that

and since (1) is approximately G-observable then, that is is positive definite.

Conversely, let such that, then, there for, that is the system is approximately G-observable on.

2.5. Remark 2.4

1) If a system is exactly (resp. approximately) G-observable on, it is exactly (resp. approximately) G-observable on.

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

2.6. Example 2.5

Let, we consider the two-dimensional system described by the hyperbolic system

The operator, which the eigenvalues are associated to the eigenfunctions.

Measurements are given by the output function

where is the sensor support and

is the function measure.

Let the subregion and we consider the initial state

,

Then the initial state gradient to be observed is

We have the result.

2.7. Proposition 2.6

The gradient is not approximately G-observable on the whole domain, however it is approximately G-observable on the subregion.

2.8. Proof

To prove that is not approximately G-observable on, we must show that. We have

Since

we have

and

This gives, and then the system is not approximately G-observable once.

On the other hand may be approximately G-observable on.

Indeed, suppose that, then

Since for large enough, the set

forms a complete orthonormal set of, we have

but for and, we have

.

witch gives,

.

But for, we have

.

Thus.

3. 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.

3.1. Definition 3.1

A sensor (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 is of constant coefficients and has a complete set of eigenfunctions in denoted by orthonormal in associated to the eigenvalues of multiplicity. Assume also that is finite, then we have the following result.

3.2. Proposition 3.2

If the sequence of sensors is G-strategic on

, then and, where and

(4)

and

is the row vector the elements of which arewith ; for.

3.3. Proof

The proof is developed in the case zone sensors.

The sequence of sensors is G-strategic on if and only if

Suppose that the sequence of sensors is Gstrategic on and there exists, with

then there exists

such that

                              and.                                                  (5)

Let verifying

(6)

Let verifying

(7)

and let

and then

assume that

then

Integrating on we obtain

and

then

but we have

and

Using the fact that

and

then we obtain

and

from (5), (6) and (7) we obtain

Thus

this gives,

and, which contradicts the fact that the sequence of sensors is G-strategic.

3.4. 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.

4. 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. Consider the set

where

for, the system

. (8)

has a unique solution

.

We consider the zone sensor case where the system (1) is observed by the output function

(9)

is the sensor support, the function of measure and we consider a semi-norm on defined by

(10)

where is the solution of (8).

The reverse system given by

(11)

has a unique solution

[5].

We denote the solution by and

by.

Let consider the operator

where,

and consider the retrograde system which has a unique solution

(12)

[5].

We denote the solution by and

by. Then, the regional gradient observability turns up to solve the equation

(13)

where and

.

4.1. Proposition 4.1

If the sensor is G-strategic on, then the equation (13) has a unique solution which is the gradient of the initial state to be observed on.

4.2. Proof

1) Let us show first that if the system (1) is G-observable, then (10) defines a norm on.

Consider a basis of the eigenfunctions of, without loss of generality we suppose that the multiplicity of the eigenvalues are simple, then

on which is equivalent to

The set forms a complete orthogonal set of, then we obtain

and since the sensor is regionally G-strategic on, we have

then.

Consequently and thus.

Conversely, and (constants), since

and from on, (10) is a norm.

2) Let denote by completion of by the norm (10) and be its dual. We show that is an isomorphism from into. Indeed, let 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

Using Green formula for the second term, we obtain

and with the boundary conditions, we obtain

Using Cauchy-Schwartz inequality, we have,

Hence,

which proves that is an isomorphism and consequently the Equation (13) has a unique solution which corresponds to the state gradient to be observed on the subregion.

4.3. 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.

5. Numerical Approach

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

5.1. Proposition 5.1

If the sensor is G-strategic on, then the initial gradients and may be approached by and respectively

(14)

(15)

where is an order of truncation.

5.2. 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

For large enough, we have

On the other hand, we have

and

since, then

                on                                             (16)

and

                on                                              (17)

we obtain

and

The minimization of (13) is equivalent to solve the two following problems

and

which solutions are,

(18)

and

(19)

Now, let be the solution of the system (12) with

Thus

and

then, we obtain

and

With these developments, according to (18) and (19), we obtain.

We replace that in the relation (16) and (17), we obtain

and

We consider a truncation up to order, then we obtain the relation (14) and (15).

We define a final error

.

The good choice of will be such that , and we have the following algorithm:

Algorithm

Step 1: Data: The region, the sensor location and.

Step 2: Choose a low truncation order.

Step 3: Computation of and by the formulae (14) and (15).

Step 4: If then stop, otherwise.

Step 5: and return to step 3.

5.3. Remark 5.2

If and 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

(20)

(21)

6. Simulations

6.1. 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, we consider the one dimensional system.

(22)

Measurements are given by the output function

(23)

The previous system is G-observable on [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.

Let, and the sensor is located at. The initial gradient to be reconstructed is given by

and

The coefficients are chosen such that the numerical scheme be stable, and in order to obtain a reasonable amplitude of and let us take and.

Applying the previous algorithm, using the formulae (20), (21) we respectively obtain the Figures 1 and 2 for and respectively the Figures 3 and 4 for.

The estimated gradient is obtained with error on.

For, the gradient is reconstructed with error on.

6.2. 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.

Also how both the error decreases with respect to the amplitude of the initial state gradient. For this let take the subregion and. We note that the reconstruction error depends on the amplitude 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.

7. Conclusion

Gradient Observability on a subregion interior to the spatial 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 question is under consideration.

REFERENCES