^{1}

^{*}

^{1}

^{*}

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.

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

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 ([

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.

Let

For

where

Let

where

Let

For any

Without loss of generality we note:

The system (3) admits a unique solution

Let denote

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

which is linear and bounded with the adjoint

Consider the operator

where

The initial condition

For

where

Let

where

Definition 1

The System (3)-(2) is said to be exactly (respectively. weakly)

(respectively.

Definition 2

The semilinear system (1) augmented with output (2) is said to be gradient observable in

The study of regional gradient observability leads to solving the following problem:

Problem 1.

Given the semilinear system (1) and output (2) on

Let’s consider

then we have the following results:

Proposition 1.

If the system (3) is weakly

where

where

The solution of the system (4) can be expressed by

where

Using the second decomposition of initial condition we obtain

If the linear part of the system (1) is weakly

where

Finally, solution of problem of

Proposition 2.

If

then

Let

But the operator

condition (10), then

Finally

which is the initial gradient to be observed in

In this section, we study Problem 1 under some supplementary hypothesis on

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

Let’s consider

denotes the real part of spectrum of

and consider

We consider Problem 1 in

We have

where

For

And the operator

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

The assumption is satisfied with

Various examples are given and discussed in ([

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

Let’s consider

where

We assume that

then we have the following result

Proposition 3.

Suppose that system (3) is weakly

· There exist

· There exist

• Proof

• Since

Let us consider

where

Using Holder’s inequality we take

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

It is sufficient to take

Let

but we have

and we deduce that

Finally

Remark 1.

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 measurements are in

Let us consider the mapping

and assume that

Then we have the following result.

Proposition 4.

Assume that

and if the linear system(3)is weakly

is lipschitzian.

Let us consider

or

On the other hand, using the inequality (13), (17) and (18), we have

Let’s consider

In order to have

For

which gives

then

which shows that

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

converges to

We have,

or

Then

We consider

We have

then

which shows that the sequence

On the other hand, we have

Then

Now let’s consider the sequence

Thus we obtain the following algorithm:

1. Algorithm:

2. Given the initial condition

3. Repeat

a)

b)

Until

4.

Else

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.