Gradient Observability for Semilinear Hyperbolic Systems : Sectorial Approach

The aim of this work is to study the notion of the gradient observability on a subregion ω of the evolution domain Ω for a class of semilinear hyperbolic systems. We show, under some hypothesis, that the gradient reconstruction is achieved following sectorial approach combined with fixed point techniques. The obtained results lead to an algorithm which can be implemented numerically.


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 ω of the system evolution domain Ω .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 ω 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.

Problem Statement
Let Ω be an open bounded subset of ( ) and we consider the following semilinear hyperbolic system 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 where C is a linear operator from  to the space q IR , and depends on the number and the nature of the considered sensors.Let mj Φ a basis of eigenfunctions of the operator  , with Dirichlet conditions and the associated eigenva- lues m λ of multiplicity m r .

∑∑ ∑∑
Without loss of generality we note: ( ) ( ) and we associate to the system (1) the linear system The system (3) admits a unique solution (see [12]).
Let denote , y y y t The system (1) may be written as and the system (3) is equivalent to Systems ( 4) and ( 5) are augmented with the output function ( ) ( ) The system (1) can be interpreted in the mild sense as follows and the output equation can be expressed by Let K be the observation operator defined by : 0, ; 0, ; .
which is linear and bounded with the adjoint * K .Consider the operator ∇ given by ( ) ( The initial condition 0 y and 0 y ∇ its gradient are assumed to be unknown.
For ω ⊂ Ω an open subregion of Ω and of positive Lebesgue measure, let ω χ be the restriction operator defined by : ) is the adjoint of ω χ (resp.ω χ ), and we consider the operator be the gradient of the initial condition ( ) where , y y y y t The System (3)-( 2) is said to be exactly (respectively.weakly) The semilinear system (1) augmented with output ( 2) is said to be gradient observable in ω ( G -observable in ω ) if we can reconstruct the gradient of its state and the gradient of its speed in a subregion ω of Ω at any time t .
The study of regional gradient observability leads to solving the following problem: Problem 1.Given the semilinear system (1) and output (2) on [0, ] T , is it possible to reconstruct ( ) which is the gradient of initial state and the gradient of initial speed of (1) in ω ?Let's consider ( ) and we define, for ] [ d , 0, and then we have the following results: Proposition 1.
If the system (3) is weakly G -observable, then the solution ( ) .y of the system ( 6) is a fixed point of the mapping ( ) The solution of the system (4) can be expressed by ( ) ( ) ( ) ( )

Cy t CS t y CL t y
where z is the output function which allows information about the considered system.Using the second decomposition of initial condition we obtain If the linear part of the system (1) is weakly G -observable in ω , then we have Finally, solution of problem of G -observability in ω is a fixed point of the following function: ( ) Proposition 2.
and if the function ( 9) has a unique fixed point ( ) then ( ) is the initial gradient to be observed in ω of system (4).

Let ( )
. y a fixed point of equation ( 9), then which is the initial gradient to be observed in ω of system (4).

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 ( ) with a is a positive real number and ( ) ( ) Re σ  denotes the real part of spectrum of 1  .Then for 0 1 α ≤ < , we define the fractional power ( ) as a closed operator with domain ( ) which is a dense Banach space on E endowed with the graph norm ( ) .
We consider Problem 1 in V endowed with the norm ( ) We have where c is a constant.For more details, see ( For , 1 r s > , assume that ( ) And the operator ( ) is well defined and satisfies the following conditions: 0, ; 0, ; 1 2 , 0,0 , , 0, ; 0 0 with : 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 The assumption is satisfied with 2, 1 r s = = and ) ( ) Various examples are given and discussed in ( [13] [14]).We show that a set of admissible initial gradient state and admissible initial gradient speed, admissible in the sense that system (3) be weakly G -observable.
then we have the following result.Proposition 3.
Suppose that system (3) is weakly G -observable in ω , and ( 12), ( 13) and ( 14) satisfied, then the following assertion hold: • There exist 1 0 a > and 0 m > such that for all ( ) ( ) the mapping f is lipschitzian where Let us consider y and x in ( ) ( ) Using Holder's inequality we take and using ( 13), we have ) On the other hand, we have and using Holder's inequality we obtain It is sufficient to take x be the solution of system (4) corresponding respectively to the initial gradient in ω , we suppose that we have the same residual part ( ) , then for ( ) ) ( ) ( ) . sup , and we deduce that Finally f is lipschitzian in ( ) The given results show that there exists a set of admissible gradient initial state.If the gradient initial state is taken in ( ) 0, B m , with a bounded residual part then the system (4) has only one solution in ( ) Here we show that if measurements are in ( ) 0, B ρ , with ρ is suitably chosen then the gradient initial state can be obtained as a solution of a fixed point problem.
Let us consider the mapping and assume that ( ) ( ) ( ) and 0 such that .

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 ω .
the function (16) admit a unique fixed point in ( ) 0, B m which corresponds to the gradient initial condition 1 0 y observed in ω .Furthermore, the function

Proposition 5 . 1 y 1 y and the regional initial gradient speed 1 1 y 2 ŷ
We suppose that the hypothesis of Proposition 4 are verified, then for the regional initial gradient condition (the regional initial gradient state 0 ) to be observed in ω , where 0 is the residual part of the initial gradient condition in is a Cauchy sequence on V and its convergence.
Then we have the following result.