Gradient Observability for Semilinear Hyperbolic Systems: Sectorial Approach ()
1. Introduction
The regional observability is one of the most important notions of system theory, and it consists in reconstructing the initials conditions (initial state and initial speed) for hyperbolic systems only in a subregion
of the system evolution domain
. This concept was largely developed (see [1] [2] ) for parabolic systems and for hyperbolic systems (see [3] [4] ). Subsequently, the concept of regional observability was extended to the gradient observability for parabolic systems (see [5] [6] ) and for hyperbolic systems (sees [7] ), which consist in reconstructing directly the gradient of the initial conditions only in a critical subregion interior
without the knowledge of the initial conditions. This concept finds its application in many real world problems.
The aim of this paper is to study the regional gradient observability of an important class of semilinear hyperbolic systems. We will focus our attention on the case where the dynamic of the system is a linear operator and sectorial. This approach was examined for semilinear parabolic systems to reconstruct the initial gradient state ( [8] ) and for semilinear hyperbolic systems to reconstruct the initial state and the initial speed. For observability problem when one is confronted to the question of reconstructing the gradient state and the gradient speed, it is important to take into account the effects of non-linearity. For example, approximate controllability of semilinear system can be obtained when the non-linearity satisfies some conditions (see [9] [10] ), and the used techniques combine a variational approach to controllability problem for linear equation and fixed point method. The techniques are also based on linear infinite dimensional observability theory together with a variety of fixed point theorems.
The plan of the paper is as follows: Section 2 is devoted to the presentation of the problem of regional gradient observability of the considered system. Section 3 concerns the sectorial approach. Numerical approach is developed in the last section.
2. Problem Statement
Let
be an open bounded subset of
.
For
, we denote
,
and we consider the following semilinear hyperbolic system
(1)
where
is a second order elliptic linear operator, symmetric generating a strongly continuous semigroup
and
is a nonlinear operator assumed to be locally Lipshitzian.
Let
denotes the solution of system (1) (see [11] ) and the function of measurements is given by the output function
(2)
where
is a linear operator from
to the space
, and depends on the number and the nature of the considered sensors.
Let
a basis of eigenfunctions of the operator
, with Dirichlet conditions and the associated eigenvalues
of multiplicity
.
For any
the semigroup
is given by
![]()
Without loss of generality we note:
and we associate to the system (1) the linear system
(3)
The system (3) admits a unique solution
(see [12] ).
Let denote
,
for all
,
and
.
The system (1) may be written as
(4)
and the system (3) is equivalent to
(5)
Systems (4) and (5) are augmented with the output function
with (6)
The system (1) can be interpreted in the mild sense as follows
(7)
and the output equation can be expressed by
![]()
Let
be the observation operator defined by
![]()
which is linear and bounded with the adjoint
.
Consider the operator
given by
![]()
where
![]()
is the adjoint of
.
The initial condition
and
its gradient are assumed to be unknown.
For
an open subregion of
and of positive Lebesgue measure, let
be the restriction operator defined by
![]()
where
![]()
. (resp.
) is the adjoint of
(resp.
), and we consider the operator
![]()
Let
be the gradient of the initial condition
, we have
(8)
where
,
and
![]()
Definition 1.
The System (3)-(2) is said to be exactly (respectively. weakly)
-observable in
if ![]()
(respectively. ![]()
Definition 2.
The semilinear system (1) augmented with output (2) is said to be gradient observable in
(
-observable in
) if we can reconstruct the gradient of its state and the gradient of its speed in a subregion
of
at any time
.
The study of regional gradient observability leads to solving the following problem:
Problem 1.
Given the semilinear system (1) and output (2) on
, is it possible to reconstruct
which is the gradient of initial state and the gradient of initial speed of (1) in
?
Let’s consider
and we define, for
, the operator
by
![]()
then we have the following results:
Proposition 1.
If the system (3) is weakly
-observable, then the solution
of the system (6) is a fixed point of the mapping
defined by:
![]()
where
is the pseudo inverse of the operator
and
such that
![]()
where
is the residual part.
Proof
The solution of the system (4) can be expressed by
thus,
so we have
![]()
where
is the output function which allows information about the considered system.
Using the second decomposition of initial condition we obtain
which is equivalent to
.
If the linear part of the system (1) is weakly
-observable in
, then we have
![]()
where
is the pseudo inverse of the operator
.
Finally, solution of problem of
-observability in
is a fixed point of the following function:
define by:
(9)
Proposition 2.
If
is closed in
and if the function (9) has a unique fixed point
such that
(10)
then
is the initial gradient to be observed in
of system (4).
Proof
Let
a fixed point of equation (9), then
![]()
But the operator
is the orthogonal projection of
in
and
satisfy
condition (10), then
.
Finally ![]()
which is the initial gradient to be observed in
of system (4).
3. Sectorial Approach
In this section, we study Problem 1 under some supplementary hypothesis on
and the nonlinear operator
.
With the same notations as in the previous case, we reconsider the semilinear system described by the equations (4) and (6) where one supposed that the operator
generates an analytic semigroup
in the state space
.
Let’s consider
such that
with
is a positive real number and ![]()
denotes the real part of spectrum of
. Then for
, we define the fractional power
as a closed operator with domain
which is a dense Banach space on
endowed with the graph norm
![]()
and consider
.
We consider Problem 1 in
endowed with the norm
(11)
We have
![]()
where
is a constant. For more details, see ( [2] [6] [13] )
For
, assume that
(12)
And the operator
is well defined and satisfies the following conditions:
(13)
Those hypothesis are verified by much important class of semi linear hyperbolic systems. For example the equation governing the flow of neutrons in a nuclear reactor
![]()
which
.
The operators
and
corresponding are
;
The assumption is satisfied with
and
.
Various examples are given and discussed in ( [13] [14] ).
We show that exists a set of admissible initial gradient state and admissible initial gradient speed, admissible in the sense that system (3) be weakly
-observable.
Let’s consider
given by
![]()
where
is the restriction in
and
is the residual part in
of the initial gradient condition
.
We assume that
(14)
then we have the following result.
Proposition 3.
Suppose that system (3) is weakly
-observable in
, and (12), (13) and (14) satisfied, then the following assertion hold:
・ There exist
and
such that for all
the function
has a unique fixed point
in the ball
solution of the system (4).
・ There exist
and
such that
the mapping f is lipschitzian where
![]()
Proof
・ Since
, then there exists
such that
and we have
.
Let us consider
and
in
and
we have
![]()
where
![]()
Using Holder’s inequality we take
and using (13), we have
![]()
On the other hand, we have
![]()
but we have
![]()
and
![]()
and using Holder’s inequality we obtain
![]()
then we have
![]()
and
![]()
or
![]()
where ![]()
Finally
![]()
Let’s consider
and
, then
.
It is sufficient to take
and
, then for all
we have ![]()
Let
and
be the solution of system (4) corresponding respectively to the initial gradient in
, we suppose that we have the same residual part
, then for
we have
![]()
but we have
![]()
and we deduce that
(15)
Finally
is lipschitzian in
.
Remark 1.
The given results show that there exists a set of admissible gradient initial state. If the gradient initial state is taken in
, with a bounded residual part then the system (4) has only one solution in
.
Here we show that if measurements are in
, with
is suitably chosen then the gradient initial state can be obtained as a solution of a fixed point problem.
Let us consider the mapping
(16)
and assume that
.
Then we have the following result.
Proposition 4.
Assume that
(17)
(18)
and if the linear system(3)is weakly
-observable in
and (13) holds, then there exists
and
, such that for all
, the function (16) admit a unique fixed point in
which corresponds to the gradient initial condition
observed in
. Furthermore, the function
![]()
is lipschitzian.
Proof
Let us consider
and
in
, using (11), (13), (15) and (17) we have
![]()
or
, then there exists
such that
and we have
. Then we obtain
.
On the other hand, using the inequality (13), (17) and (18), we have
![]()
Let’s consider ![]()
In order to have
, it suffices to consider
.
For
, we have
![]()
which gives
![]()
then
![]()
which shows that
is Lipschitzian.
4. Numerical Approach
4.1. Numerical Approach
We show the existence of a sequence of the initial gradient state and initial gradient speed which converges respectively to the regional initial gradient states and initial gradient speed to be observed in
.
Proposition 5.
We suppose that the hypothesis of Proposition 4 are verified, then for
, the sequence of the initial gradient condition defined in
by
![]()
converges to
the regional initial gradient condition (the regional initial gradient state
and the regional initial gradient speed
) to be observed in
, where
is the residual part of the initial gradient condition in
.
Proof
We have,
![]()
or
, then there exists
, ![]()
![]()
Then
is a Cauchy sequence on
and its convergence.
We consider
and
with
![]()
We have
, then
![]()
then
![]()
which shows that the sequence
converges to
in
.
On the other hand, we have
![]()
Then
converges to the regional initial gradient
to be observed in
.
4.2. Algorithm
Now let’s consider the sequence
, then we have
and
Thus we obtain the following algorithm:
Algorithm:
1. Given the initial condition
, the region
, The domain
and the function of distribution
and the accuracy threshold
,
.
2. Repeat
a) ![]()
b) ![]()
c) ![]()
Until ![]()
3.
which corresponds to the initial gradient condition to observed
in
.
Else
and go to step 2.
5. Conclusion
The question of the regional gradient observability for semilinear hyperbolic systems was discussed and solved using sectorial approach, which uses sectorial properties of dynamical operators, the fixed point techniques and the properties of the linear part of the considered system. The obtained results are related to the considered subregion and the sensor location. Many questions remain open, such as the case of the regional boundary gradient observability of semilinear systems using Hilbert Uniqueness Method (HUM) and using the sectorial approach. These questions are still under consideration and the results will appear in a separate paper.