Regional Boundary Observability with Constraints of the Gradient

The aim of this paper is to explore the concept of observability with constraints of the gradient for distributed parabolic system evolving in spatial domain Ω, and which the state gradient is to be observed only on a part of the boundary of the system evolution domain. It consists in the reconstruction of the initial state gradient which must be between two prescribed functions in a subregion Γ of. Two necessary conditions are given. The first is formulated in terms of the subdifferential associated with a minimized functional, and the second uses the Lagrangian multiplier method. Numerical illustrations are given to show the efficiency of the second approach and lead to open questions.


Introduction
For a distributed parameter system evolving on a special domain Ω, the observability concept has been widely developed and survey of these developments can be found [1][2][3].Later, the regional observability notion was introduced, and interesting results have been obtained [4,5], in particular, the possibility to observe a state only on a subregion  interior to Ω.These results have been extended to the case where  is a part of the boundary of Ω [6].Then the concepts of regional gradient observability and regional observability with constraints were introduced and developed by [7][8][9][10][11] in the case where the subregion is interior to Ω and the case where the subregion is a part of .Here we are interested to approach the initial state gradient and the reconstructed state between two prescribed functions given only on a boundary subregion of system evolution domain.There are many reasons motivating this problem: first the mathematical model of system is obtained from measurements or from approximation techniques and is very often affected by perturbations.Consequently, the solution of such a system is approximately known, and second, in various real problems the target required to be between two bounds.This is the case, for example of a biological reactor "Figure 1" in which the concentration regulation of a substrate at the bottom of the reactor is observed between two levels.

  
The paper is organized as follows: first we provide results on regional observability for distributed parameter system of parabolic type and we give definitions related to regional boundary observability with constraints of the gradient of parabolic systems.The next section is focused on the reconstruction of the initial state gradient by using an approach based on sub-differential tools.The same objective is achieved in Section 4 by applying the multiplier Lagrangian approach which gives a practice algorithm.The last section is devoted to compute the obtained algorithm with numerical example and simulations.

Problem Statement
Let with the measurements given by the output function where 0 is linear and depends on the considered sensors structure.

 
The observation space is .A is a second order differential linear and elliptic operator which generates a strongly continuous semigroup in the Hilbert space .A denotes the adjoint operators of A. The initial state 0 and its gradient 0 are assumeed to be unknown.The system (1) is autonomous and (2) allows writing We define the operator which is linear bounded with the adjoint K * given by * * 0 : is the extension of the trace operator of order zero which is linear and surjective.*  , 0 denotes the adjoint operators of  and : We recall the following definitions Definition 2.1 1.The system (1) together with the output ( 2) is said to be exactly (respectively weakly) gradient observable on 2. The sensor (D, f) (or a sequence of sensors) is said to be gradient strategic on Γ if the observed system is weakly gradient observable on Γ.
For more details, we refer the reader to [11].
a.e. on 1, 2, , ).The system (1) together with the output (2) is said to be exactly 2).The system (1) together with the output (2) is said to be weakly ).If the system (1) together with the output (2) is exactly gradient observable on Γ then it is exactly • -gradient observable on Γ.   3).If the system (1) together with the output (2) is exactly (resp.weakly) • -gradient observable on Γ 1 then it is exactly (resp.weakly) There exist systems which are not weakly gradient observable on a subregion Γ but which are weakly -gradient observable on Γ.  Example 2. 4 Consider the two-dimensional system described by the diffusion equation , 0 on , the time interval is ]0, T[ and let Γ be the boundary subregion given by Thus, the output function is given by , , The operator where Then we have the result: Proposition 2.5 The system (3) together with the output (4) is not weakly gradient observable on Γ but it is weakly -gradient observable on Γ. Proof Let g 1 be the function defined in Ω by x  be the gradient to be observed on Γ and show that g 1 is not weakly gradient observable on Γ.
we have   .Consequently, the gradient g 1 is not weakly gradient observable on Γ.Then the system (3) together with the output ( 4) is not weakly gradient observable on Γ. but we can show that it is weakly which show that the gradient g 2 is weakly gradient observable on Γ.
 , then the system (3) together with the output ( 4) is weakly Proposition 2. 6 The system (1) together with the output (2) is exactly which shows that the system (1) together with the output ( 2) is exactly -Assume that the system (1) together with the output (2) is exactly then there exists and and which shows that Proposition 2. 7 The system (1) together with the output (2) is weakly -gradient observable on Γ if and only if , where which implies that the system (1) together with the output ( 2) is weakly • -gradient observable on Γ.
-Suppose that the system (1) together with the output (2) is weakly

Subdifferential Approach
This section is focused on the characterization of the initial state of the system (1) together with the output (2) in the nonempty subregion Γ with constraints on the gradient by using an approach based on subdifferential tools [12].So we consider the optimization problem where proper, lower semi-continuous (l.s.c.) and co x.
  ed, convex and nonempty, then  the polar fun tion f * 0 0 of f be given by denotes the subdifferential of f at y 0 , then we have -For D a nonem The solution of this problem may be ch the following result.

Proof
We have that y * is a solution of (7) if and only if Also, according to the hypothesis of th n of (7) if and only if it follows that y * is a solutio In this section we propose to solve the problem (6) sing the Lagrangian multiplier method [13].Also we describe of

Lagrangian Multiplier Approach
u a numerical algorithm which allows the computation the initial state gradient on the boundary subregion Γ and finally we illustrate the obtained results by numerical simulation which tests the efficiency of the numerical scheme.
From the definition of the exact • -gradi- ent observability on Γ all state we will consider are of the form we have the following result: Proposition 4.1 If the system (1) together with the output (2) is ex tly ob radient in Γ of the solution of the problem ( 6) is and the g given by where *  is the solution of while  

Proof
The system (1) together with e outpu

 ,  
able on Γ n   and lem (8) has a so e problem ( 8) is t to the saddle point problem the prob eq lution.Th uivalen where ociate with problem (12) the Lagrangian funcefined by the formula , , 0 y , • -gradient observable on Γ and, there- fore, there exist 0  and O Let   be a saddle point of L and prove is the restric gradient on Γ of the solution of (6).
, , , The first inequality above gives The second inequality means that From ( 13) we have we assume that the system is obse KK KK is invertible, and If the system (1) together with the output ( 2) is observable in Ω, exactly gradient observable Γ and the function coercive, then for ρ convenably chosen, the system (11) has a unique solution   * * , y It follows that y is a fixed point of the function ,

Numerical Approach
In this section we describe a numerical scheme which allows the calculation of the initial state gradient between    • and    • on the subregion Γ.
We have seen in the previous section that in order to reconstruct the initial sta e between   we obta he following algorithm then in t (Table 1).

Simulation Results
In this section we give a numerical example which illustrates the efficiency of the previous approach.The results are related to the choice of the subregion, the initial conditions and Let us consider a t the sensor location.wo-dimensional system defined in     0,1 0,1    and escribed by the following parabolic uation d eq , , , , , y x x t x t t t easurements are given by a pointwise sensor , , , , g x x g x x g x x  the initial gradient to be observed on Γ with g 1 and g 2 are given by   • -gradient strategic on Γ.The estimated initial gradient is obtained with reconstruction error  shows that the estimated initial gradient is not between

   
• and   • on the subregion Γ, which implies that the sensor located in

Conclusions
The problem of servability on Γ of parabolic system is considered.The initial state gradient is characterized by two approaches based on regional observability tools in connection with Lagrangian and subdifferential techniques.
Moreover, we have explored à useful numerical algorithm which allows the computation of initial state gradient and which is illustrated by numerical example and simulations.Various  question is under consideration and will be the subject of the future paper.

1 Figure 1 .
Figure 1.Regulation of the concentration flux of the substratum at a bottom of the reactor.
, y   is a saddle point of L then t e system (11) is equivalent to with b is the location of the sensor and 2
pointwise measurement, and one can obtain similar results with zone (internal or boundary) measurement.
questions are still open.The characterization of gradient observability by a rank condition as stated for usual gradil eat interest.This ent observability or regiona gradient observability of distributed parameter systems is of gr     ,
Ω be an open bounded subset of IR n (n = 2, 3) with