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The aim of this paper is to study the notion of the gradient observability on a subregion w of the evolution domain W 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.

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 [

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.

Let

and the linear part of the system (1) is

where

where

Let

For

and the system (1) is equivalent to

augmented with the output function

with

Let’s consider a basis of eigenfunctions of the operator

We can write for any

The system (5) has a unique solution that can be expressed as follows see [

then the output Equation (6) can be expressed by

Let

which is linear and bounded with the adjoint

Consider the operator

where

(resp. if the subregion of interest is a boundary part of the system evolution domain.)

The initial condition

with

(resp. For

with

The trace operator is defined by

with

and

Finally, we reconstruct the operator as follows

Definition 1

・ The system (2) together with the output (3) is said to be exactly (resp. weakly) G-observable in

・ The system (2) together with the output (3) is said to be exactly (resp. weakly) G-observable in

Remark 1.

・ If the system (2) together with the output (3) is exactly G-observable on

・ For

Definition 2 The semilinear system (1) augmented by the output function (3) is said to be gradient observable or G-observable on

Let the gradient

where

Problem (*)

Given system (1) augmented by the output (3) on

In this section, we study Problem (*) under some supplementary hypothesis on

With the same notations as in the previous case, we reconsider the semilinear system described by the Equ- ation (5) augmented by the output (6) where one suppose that the operator

Let’s consider

denotes the real part of spectrum of

Let us consider

we have

where c is a constant. For more details, see ( [

For

and the operator

This hypothesis are verified by many important class of semi linear hyperbolic systems. Various examples are given and discussed in ( [

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

where

then we have the following result

Proposition 1 Suppose that the system (2) is weakly G-observable on

・ There exists

fixed point

・ There exist

Proof.

・ Since

and we have

Let us consider

where

Using Holder’s inequality we take

and using (11), we have

On the other hand, we have

but we have

and

Using Holder’s inequality, we obtain

then we have

and

where

Finally

Let’s consider

and

It is sufficient to take

・ Let

but we have

and we deduce that

Finally, f is Lipschitzian in

Remark 2 The given results show that there exists a set of admissible gradient initial state. If the gradient initial state is taken in

Here, we show that if the measurements are in

Let us consider the mapping

and assume that

Then we have the following result.

Proposition 2 Assume that

and if the linear system (2) is weakly G-observable on Γ and (11) holds, then there exists a_{2} > 0 and

Proof. Let us consider

Or

and we have

Then we obtain

On the other hand, using the inequalities (11), (15) and (16), we have

Let’s consider

In order to have

For

which gives

Then

which shows that h is Lipschitzian.

Remark 3 We can consider the regional intern problem in a subregion

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

Proposition 3 We suppose that the hypothesis of the proposition (3.2) are verified, then for

converges to

Proof. We have,

or

Then

We consider

we have

So

then

which show that the sequence

hence

Algorithm

Let’s consider

Thus, we obtain the following algorithm:

In this part, we give a numerical illustrating example and the simulations are related to the choice of the subregion, the sensor location.

Consider the one dimensional semilinear hyperbolic system

augmented with the output function described by a pointwise sensor located in

where

Using the previous algorithm, we obtain the following figures.

・

・

Consider the two dimensional system described in

where

The system (20) augmented by output function described by a pointwise sensor located in b.

with

・

・

・

・ The initials gradient conditions

to be observed on

Using the previous algorithm, we obtain the following results:

・

・

The question of the regional internal and boundary gradient observability for semilinear hyperbolic systems was discussed and solved using the sectorial approach, which used sectorial properties of dynamical operators, the fixed point techniques and the properties of the linear part of the considered system. Many questions remain open, such as the case of the regional gradient observability of semilinear systems using Hilbert Uniqueness Method (HUM) and the constrained observability of semilinear hyperbolic system.

Adil Khazari,Ali Boutoulout, (2016) Sectorial Approach of the Gradient Observability of the Hyperbolic Semilinear Systems Intern and Boundary Cases. Applied Mathematics,07,1326-1339. doi: 10.4236/am.2016.712117