Sensors and Regional Gradient Observability of Hyperbolic Systems ()
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 are
with 
; 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.
Figure 1. Iheb2.eps: initial state gradient Ñy0 (continuous line) and estimate initial state gradient bar (Ñ)y0 (dashed line).
Figure 2. Nouha2.eps: initial speed gradient Ñy1 (continuous line) and estimate initial speed gradient bar (Ñ)y1 (dashed line).
Figure 3. Iheb1.eps: initial state gradient Ñy0 (continuous line) and estimate initial state gradient bar (Ñ)y0 (dashed line).
Figure 4. Nouha1.eps: initial speed gradient Ñy1 (continuous line) and estimate initial speed gradient bar (Ñ)y1 (dashed line).
Table 1. Evolution error with respect to the area of the subregion.
Table 2. Evolution error with respect to the initial state gradient amplitude.
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.