Null Controllability of a Nonlinear Dissipative System and Application to the Detection of the Incomplete Parameter for a Nonlinear Population Dynamics Model

Abstract

We first prove a null controllability result for a nonlinear system derived from a nonlinear population dynamics model. In order to tackle the controllability problem we use an adapted Carleman inequality. Next we consider the nonlinear population dynamics model with a source term called the pollution term. In order to obtain information on the pollution term we use the method of sentinel.

1. Introduction and Main Result

For a given real function , in this paper we consider the following nonlinear system: where is a bounded open subset of , , with a smooth boundary . Let be a positive real and let , be two open subsets of such that and . Here , are the characteristic function of and , is the Laplacian with respect to the spatial variable, and and denote, respectively, the natural fertility and the natural death rate of individuals of age .

The purpose of this paper is to prove a null controllability result for (1) at any time . This means more precisely that we prove here the existence of a control such that the associated solution of (1) verifies a.e. in .

For the sequel we assume that the following assumptions hold:We set and .

Let us now present the main results of this paper.

For this aim we introduce the weightand define the Hilbert spaceendowed with the natural norm. is a real that will be defined in the following.

We have the following.

Theorem 1. Let ; then under the assumptions , , and there exists a control satisfying a.e. in .

The rest of this paper is as follows: the proof of the main result is given in Section 2; in Section 3, we describe the method of sentinel; and Section 4 is devoted to the information provided by the sentinel.

2. Proof of the Null Controllability of the Nonlinear System

2.1. An Adapted Observability Inequality

We recall here that there exists a function such that , for all ; for all and for all , where is an open set such that . See [1].

Now we consider the following system:

Proposition 2. There exist positive constants and and there exists a positive constant which depends on and such that, for , , and for all solution of (4) the following inequality holds: with

For the proof of this proposition see [2].

For we obtain an Observability Inequality.

Lemma 3. There exist and such that the following inequality is true:herein is the solution of system (4).

Proof. If , (5) becomeChoosing and such that , we obtain the result with .

In the following we consider the linear system:where and are in . Each system (9)-(10) is well posed and admits a unique solution. See [3] for the proof of existence and uniqueness of the solution.

Remark 4. For the linear case, that is, can use the Hahn-Banach theorem for proving the existence of a control which verifies , one writesIt is easy to see that is a nonempty subspace of and that It suffices to show that Suppose that Multiplying (9) by and integrating by parts, we haveSince then This implies that for every .

Then on . The unique continuation property of gives Therefore . So .

However, this method cannot allow the study of the nonlinear case.

2.2. The Approached Controllability of the Linear System

Now, for we consider the functional defined by where solves

Proposition 5. The functional is continuous, strictly convex, and coercive. Therefore, reaches its minimum at a point . Moreover, setting the associated solution of (21) and the solution of (10) with has and there exist , independent of , such that

Remark 6. As we will see Proposition 5 solves the problem of existence of the approached sentinel for the linear case.

Proof. It is easy to check that is coercive, continuous, and strictly convex. Then it admits a unique minimizer . The maximum principle gives that .
Multiplying (21) with by and integrating by parts, we obtain So Using the inequality of Young, we obtain for any Now, inequalities (7) and (25) giveChoosing such that we haveWe conclude that

Proposition 7. The function solution of system (21) verifies

Proof. Let be a positive constant. Put .
Then, solvesMultiplying (30) by and integrating by parts, we obtainChoosing , we obtain This gives the desired estimation.

2.3. Study of the Nonlinear Case

In the following we set Let be an operator defined on by where is the solution of the following system:Then we have the following.

Proposition 8. The operator is continuous, bounded, and compact of . Then admits a fixed point.

Proof. The proof will be done in two steps as follows.
Step  1 (boundedness and compactness of ). Let and denote .
Then is the solution of the following system:withUnder assumptions , , and and the results and owing to the estimation on and the function satisfies System (37) is a heat equation where the source term and the initial condition are bounded, respectively, in and .
Then is bounded in and is bounded in . Hence, using the Lions-Aubin lemma we conclude that is bounded and compact in .
Step  2 (continuity of ). Let strongly in . Then we can show that the sequence converges to strongly. For all , and are bounded independently to . Therefore is bounded in . Then we can extract a subsequence such thatThen is a solution of (37). We deduce that the sequence converges to .
Since the operator is continuous, bounded, and compact on onto , Schauder’s fixed-point theorem (see [4, 5]) implies that admits a fixed point.
There exists such that Then solvesFrom the foregoing, we have that and are bounded in .
Then we can extract a subsequence of still denoted by , such that By the same idea Since , then there exists a subsequence still denoted by such that a.e in .
Now since is continuous, thenTherefore, one derives that solves the following system:and we have also, for , a.e. in .

3. Application to the Detection of the Incomplete Parameter for a Nonlinear Population Dynamics Model

For a given positive real function , we consider the following nonlinear population dynamics model:where is a bounded open subset of , , with a smooth boundary . Let be a positive real and let be an open subset such that . Here, is the distribution of individuals of age at time and location , is the characteristic function of , is the maximal live expectancy, is the Laplacian with respect to the spatial variable, and and denote, respectively, the natural fertility and the natural death rate of individuals of age . Thus, the formula denotes the distribution of newborn individuals at time and location . In an oviparus species it denotes the total eggs at time and position . Therefore, the quantity is the distribution of eggs that hatch at time and position .

System (47) describes the evolution of internal controlled age and space structured population under inhospitable boundary conditions in the case that the flux of individuals has the form .

In system (47) is given in .

The data of system (47) are incomplete in the following sense:(i) is unknown and ;(ii) is unknown and ;(iii) is unknown and small enough;(iv) is unknown and small enough.We denote by the observation that is a function in , where is the observation set. We know that (47) has a unique solution that we denote by in some relevant space. The question is as follows: How to calculate the incomplete term independently from the variation around the initial data?The question above is natural and leads to some developments. An answer is given by the least squares method. This method consists in considering the unknowns , as control variables; then the state has to be driven as close as possible to . So one studies an optimal control problem. By this way one looks to the pair but unfortunately there is possibility to find or independently. The sentinel method of Lions [6] is a particular least squares method which is adapted to the identification of parameters in ecosystems with incomplete data. Many papers are devoted to this topic in the literature. The sentinel concept relies on the following three objects: some state equation (e.g., (47)), some observation function , and some control function to be determined. Many papers use the definition of Lions in the theoretical aspect (see, e.g., Bodart [7], Bodart and Fabre [8, 9]), as well as in the numerical one (see Bodart and Demeestere [10], Demeestere [11], and Kernevez [12]).

In [13] the author studies the detection of incomplete parameter for a linear population dynamic model. In this paper we are concerned by the nonlinear case. Moreover our method allows us to easily address the numerical aspect.

We suppose that and .

Proposition 9. The functions and are differentiable at the point .

Proof. Let where .
It is easy to prove that is the solution of the following system:Multiplying (49) by and integrating by parts, we obtain We haveThis implies thenChoosing and using equality (51) we obtain This gives and then Now we consider and solvingFrom the regularity of and the uniform convergence of , we have Multiplying (59) by and integrating by parts, we obtainUsing (60), we obtainChoosing such that we obtain This implies a.e. in as . Then a.e. in .
Therefore is differentiable at . We conclude that is differentiable at .
The proof of the differentiability of the function goes similarly.

Now let be some function in and let be an open and nonempty subset of . For a control function , we define the functionalUsing Proposition 2, one haswhere is the solution of the systemWe say that defines a sentinel for problem (47) if there exists such that is insensitive (at first order) with respect to the missing terms ; this meansfor any and if the norm of is minimal.

The functional assumed to be nonzero is an approached sentinel if, for any , there exists such thatThe existence of sentinel is equivalent to a controllability problem.

Proposition 10. Let us consider the following system:where is the solution of system (47) with . The approached sentinel condition (69) is equivalent to find such that for any . The exact sentinel condition is equivalent to finding such that

Proof. Multiplying (70) by and integrating by parts, we obtainAnd then the approached sentinel condition is equivalent to for everyThat is equivalent to

We have the following result.

Proposition 11. Let , if and ; then under assumptions , , and there exists a unique control satisfying (68).

Proof. Replacing in (21) by we obtain the approached sentinel, and we obtain the sentinel when goes to zero.

4. Information Provided by the Sentinel

The function is the solution of the systemWe have Multiplying system (22) by and integrating by parts, we obtain and thenKnowing , , , and , can be calculated. Thus we obtain an integral equation in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author thanks AIMS South Africa for its partial financial support and for hospitality.