Sectorial Approach of the Gradient Observability of the Hyperbolic Semilinear Systems Intern and Boundary Cases

The aim of this paper is to study the notion of the gradient observability on a subregion ω of the evolution domain Ω and also we consider the case where the subregion of interest is a boundary part of the system evolution domain for the class of semilinear hyperbolic systems. We show, under some hypotheses, that the flux reconstruction is guaranteed by means of the sectorial approach combined with fixed point techniques. This leads to several interesting results which are performed through numerical examples and simulations.


Introduction
The regional observability is one of the most important notions of systems theory.It consists to reconstruct the trajectory only in a subregion in the whole domain.This concept has been widely developed see [1] [2].Afterwards, the concept of regional gradient observability for parabolic systems has been developed see [3]- [7] and for hyperbolic systems see [8] [9], it concerns the reconstruction of the gradient conditions initials only in a critical subregion interior to the system domain without the knowledge of the conditions initials.
The aim of this papers is to study the regional gradient observability of an important class of semilinear hyperbolic systems.For the sake of brevity and simplicity, we shall focus our attention on the case where the dynamic of the system is a sectorial operator linear and generating an analytical semigroup The plan of the paper is as follows: Section 2 is devoted to the presentation of problem of regional gradient of semilinear hyperbolic systems, and then we give definitions and propositions of this new concept.Section 3 concerns the sectorial approach.Section 4 concerns the numerical approach which gives algorithm can simulated by a numerical example.

Position of the Problem
Let Ω be an open bounded subset of ( ) and we consider the following hyperbolic semi-linear system and the linear part of the system ( where  is an elliptic and second order operator and  is a nonlinear operator assumed to be locally Lipschitzian, system (1) is augmented with the output function given by ( ) ( ) where ( ) Ω →  if the subregion of interest is a boundary part Γ of the system evolution domain Ω ) is a linear operator, and depends on the number q and the nature of the considered sensors.The observation space is ( ) and the system (1) is equivalent to augmented with the output function the system (4) has a unique solution see [10]- [12] that can be expressed as ( ) ( ) is the semigroup generated by the operator  .
Let's consider a basis of eigenfunctions of the operator  , with the condition of Dirichlet which noted mj Φ and eigenvalues associated are m λ with multiplicity m r .
( ) ( ) if the subregion of interest is a boundary part Γ of the system evolution domain Ω .)* ∇ is the adjoint of ∇ .The initial condition 0 y (initial state 0 y and initial speed 1 y ) and , : , ) which is the restriction operator.The trace operator is defined by ∂Ω is the trace operator of order zero which is linear, continuous, and surjective.* γ (resp.* 0 γ ) denote the adjoint of operator γ (resp.0 γ ).
Finally, we reconstruct the operator as follows resp. from into .
n Im H L ω = • The system (2) together with the output ( 3) is said to be exactly (resp.weakly) ).

Remark 1.
• If the system (2) together with the output ( 3) is exactly G-observable on Γ (resp. in ω ) then it is weakly G-observable on Γ .
Let the gradient ( ) , y y y = be decomposed as follows: where ( ) Given system (1) augmented by the output (3) on ] [ 0,T , is it possible to reconstruct ( ) , y y y = which is the gradient of initial condition of (1) in ω ? (resp.on Γ .)

Sectorial Case
In this section, we study Problem (*) 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 Equation (5) augmented by the output (6) where one suppose that the operator  generates an analytic semigroup ( ) with a is a positive real number and 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 ( ) . .
then the objective is to study the Problem (*) in V endowed with the norm ( ) * .
For , 1 r s > , assume that ( ) and the operator is well defined and satisfies the following conditions: ; 0 0 with : such that lim , 0.
This hypothesis are verified by many important class of semi linear hyperbolic systems.Various examples are given and discussed in ([14]- [16]).
We show that there exists a set of admissible initial gradient states and admissible initial gradient speed, admissible in the sense that allows to obtain system (2) weakly G-observable.
Let's consider given by (8).we assume that ( ) then we have the following result Proposition 1 Suppose that the system (2) is weakly G-observable on Γ , and (10), ( 11) and (12) satisfied, then the following assertion hold: • There exists 1 0 a > and 0 m > such that for all ( ) ( ) , the mapping f is Lipschitzian where ) )  and using (11), we have ) . .
Using Holder's inequality, we obtain  Finally

S y S t y t g t y t g t y
• Let 1 y and 1 x be the solution of the system (5) corresponding respectively to the initial gradient condition, we suppose that we have the same residual part ( 0 ) ( ) , 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 (5) has only one solution in Here, we show that if the 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 ( ) ( ) ( ) Then we have the following result.Proposition 2 Assume that ( ) ( ) ( ) ( ) and 0 such that .

Remark 3
We can consider the regional intern problem in a subregion ω of Ω (see [17]).

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 state and initial gradient speed to be observed on Γ .

Proposition 3
We suppose that the hypothesis of the proposition (3.2) are verified, then for ( ) , the sequence of the initial gradient condition defined in ( ) . , converges to 0 1 y the regional initial gradient condition (the regional initial gradient state 0 1 y and the regional initial gradient speed 1 1 y ) to be observed on Γ .where 0 2 y  is the residual part of the initial gradient condition.Proof.We have,
assumed unknown.For ω ⊂ Ω an open subregion of Ω , consider the restriction operators

y 2 y
is the restriction in Γ and 0  is the residual part of the initial gradient condition 0 y ∇

Figure 3 .
Figure 3.The estimated initial gradient state on Γ .

Figure 4 .
Figure 4.The estimated initial gradient speed on Γ .