International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 3581431 | https://doi.org/10.1155/2021/3581431

Yacouba Simporé, "Controllability of a Family of Nonlinear Population Dynamics Models", International Journal of Mathematics and Mathematical Sciences, vol. 2021, Article ID 3581431, 17 pages, 2021. https://doi.org/10.1155/2021/3581431

Controllability of a Family of Nonlinear Population Dynamics Models

Academic Editor: Theodore E. Simos
Received16 Jun 2020
Revised24 Aug 2020
Accepted14 Sep 2020
Published11 Jan 2021

Abstract

Considering a nonlinear dynamical system, we study the nonlinear infinite-dimensional system obtained by grafting an operator and an age structure. This system is such that the nonlinearity is at the level of births. We show that there is a time dependent on the constraints on the age and the observability minimal time of the pair ( is the control operator), from which the system is null controllable. We first establish an observability inequality useful for the proof of the null controllability of an auxiliary system. We also apply Schauder’s fixed point in the proof of the null controllability of the nonlinear system..

1. Introduction and Main Results

In this paper, we study the null controllability of an infinite-dimensional nonlinear system describing the dynamics of age-structured population.

Let , and be a four separable Hilbert space. We identify and with their duals. We suppose is dense in and that the following inclusions are checked:where the inclusion of in is continuous.

Now, we consider the operator is defined bywhere is the Hilbert space and , for example, .

And, we suppose thatis measurable. Let be a solution of the following system:where , and is given in

Moreover, the positive function denotes the natural mortality rate of individuals of age , supposed to be independent of the times .

The control function is , depending on and , where is the characteristic function.

We denote by the positive function describing the fertility rate that depends only on the age. The birth law ensures the survival of the species. In several works, one denotes , the number of newborn individuals at time . If we consider an oviparous species, that is to say whose birth process goes through egg laying, we see that is the number of eggs laid at time . Since all the eggs do not reach maturity, we introduce a function giving the proportion of eggs who will arrive there. Let this function. Then, we havethe distribution of newborn individuals at time .

We assume that the fertility rate and the mortality rate satisfy the demographic property:

We define the probability of survival of an individual of age given by

Thus, with hypothesis , the probability of survival at maximal age is zero. Moreover, hypothesis means that individuals of age below are not fertile. This hypothesis, in addition to being a demographic assumption, is also very important in establishing the observability inequality.

For more details about the modelling of such system and the biological significance of the hypotheses, we refer to Webb [1].

Moreover, there exist constants , , and such that, for every , , we have

In the following, we recall that the operator is independent of and and generator of a semigroup on .

Assumptions and are necessary for the existence of the fixed point. Indeed, they make it possible to establish an adequate regularity on the system for the existence of a fixed point.

Theorem 1. Let and ; under assumptions , , and , system (4) admits a unique solution .

Proof. For the proof of the existence, we consider the following auxiliary system:where and .
Using Theorem 2.8 of [2], the auxiliary system (10) admits a unique solution .
Now, we consider the operator defined in on by . Let and be the solutions of (10), where is replaced, respectively, by . We denote by . The function verifiesBy calculating the scalar product of the first equation of (11) by and integrating with respect to the age and the time , we obtainThen,Choosing such that ( is given in assumption ), we obtainTherefore, the operatorWe conclude that, by the Banach fixed point that admits a fixed point, we haveFinally, replacing by in system (10), we obtain that is solution of system (4).
Before we state our main result, let us introduce the notion of null controllability of the pair . We have the following result.

Definition 1. We say that a pair is null controllable in time if, for every , there exists a control such that the solution of the systemsatisfies , for every .
The main result of this paper is as follows.

Theorem 2. Assume that and satisfy conditions above.
Assume that is null controllable in any time , withand . Then, for every and for every , there exists a control such that the solution of (4) satisfies

This result can be seen as a generalization of those obtained by Traoré [3] and Simporé [4] in the case when is an elliptic operator with Neumann or Dirichlet homogeneous boundary conditions and Echarroudi and Maniar [5] when is a degenerate elliptic operator.

Let us now mention some related works in controllability and observability from the literature. In the Network, Information, and Computer Security, Alcaraz and Wolthusen studied strategies for the efficient restoration of controllability following attacks and attacker-defender interactions in power-law networks in [6]. In [7], Alcaraz presents a network infrastructure based on three layers, where the redundant support is primarily concentrated on a fog-based structure to protect a specific subset of cyber-physical control devices.

In the anticancer therapy, Swierniak and Kalmka study a local controllability of a model of combined anticancer therapy with delay in control (see [8]).

For linear and nonlinear population dynamics models (Lotka–Mckendrick model), many results of local or global controllability have been established.

Indeed, Ainseba and Iannelli, in [9], have worked on the controllability of two nonlinear dynamic model problems, the nonlinearity being on mortality and fertility. In the first model, the control corresponds to a supply of individuals on a small age interval. The second one is age and space structured, and the control corresponds to a supply of individuals on a small subdomain of the whole space domain .

In [3], Traoré proved the nonlinear Lotka–McKendrick is null controllable except for a small interval of ages near zero where the controls is localized with respect to the space variable but active for all ages.

In [2], Maity et al. study the null controllability of the linear infinite-dimensional system obtained by grafting an age structure.

To be in line with the studied models, many authors have also worked on constrained controllability problems. Indeed, the authors Maity et al., in [10], solved a problem of null controllability with positivity of the Lotka-Mckendrick system with spatial diffusion where the control is localized in the space variable as well as with respect to the age. The method combines final-state observability estimates with the use of characteristics and the associated semigroup.

In [11], Kalmka worked in constrained approximate controllability IEEE Transactions on Automatic Control.

In this paper, we extend the result of [2] to the nonlinear case. The nonlinearity is at the level of births and can be justified as follows.

If we consider an oviparous species, that is to say whose birth process goes through egg-laying, we see that is the number of eggs laid at time . Since all the eggs do not reach maturity, we introduce a function giving the proportion of eggs which will arrive there. Let be this function. Then, we have,the distribution of newborn individuals at time .

For the establishment of the observability inequality, we use the same technique as in [2, 10]. One of the advantages of this method is that the conditions on the mortality function do not interfere in establishing of the observability inequality which allows us to have a global controllability, which is not the case in [3, 9]. Moreover, we improve the observation domain, and better still we have null controllability for all ages. Indeed, in [3, 5, 9], the control is localized with respect to the space variable but active for all ages, and we do not have necessarily the extinction on the small interval of ages near zero. In this paper, the control is also localized with respect to the age variable, and we obtain the state of the system null for all ages.

The remaining part of this work is organized as follows.

In Section 2, we establish the observability inequality of the adjoint system of an auxiliary linear system. This allows us to prove its approximate null controllability. Section 3 is devoted to the proof of the main result of this paper using a fixed point of Schauder, see for example [12]. To finish, we give some perspectives in the conclusion in Section 4.

2. Approximate Null Controllability of Auxiliary System

2.1. Adjoint System

We suppose that the pair is final-state observable at time , and we consider the following auxiliary system (well chosen so as to obtain the main result of the paper by rotating a fixed point) given by

Applying the scalar product of the first equation of (21) by and integrating with respect to the age and time , we obtain the adjoint system of (21):

For every under assumption and , system (22) admits a unique solution . Moreover, integrating along the characteristic lines, the solution of (22) is given bywhere .

We have the following result.

Theorem 3. Let us assume assumptions , and suppose the pair in null controllable for every time . For , there exists a control such that the solution of system (21) verifies

For the proof of Theorem 1, we need the results of observability.

2.2. Observability Inequality

Theorem 4. Under the assumption of Theorem 3, for every , the solution of system (22) is final-state observable for every . In other words, for every , there exists such that the solution of (22) satisfiesFor the proof, we need the following results.

Proposition 1. Let us assume assumption , and let , , and ; then, there exists a constant such that, for every , the solution of system (22) verifies the following inequality:

Proposition 2. Let us assume assumptions , and let , and ; there exists such that the solution of system (26) verifies the following inequality:

For the proofs of Propositions 1 and 2, we have the following proposition.

Proposition 3. Let us recall that the pair is final-state observable in any time with . Let be a observability cost with as . Let , and be three real numbers such that with . Then, for every , the solution of the problemsatisfies the estimatewhere the constants and depend on and .

Proof. (see [2, 13]).
For , we have ; therefore, system (22) can be written byWe denote by . Then, satisfiesProving inequality (26) also leads to show that there exits a constant such that the solution of (31) satisfiesIndeed, we have

Proof of Proposition 1. We consider the following characteristics trajectory . If , the backward characteristics starting from . If , the trajectory never reaches the observation region (see Figure 1). So, we choose .
Moreover, we estimate on , where . Indeed, all the characteristic starting on with never intersects the observation domain (see Figure 1).
Without loss of the generality, let us assume here .

2.2.1. Case 1: For

We denote by

Then, satisfies

Using Proposition 3 with , we obtainthat is equivalent to

Then, for and , we obtain

Integrating with respect to over , we obtain

Finally,

2.2.2. Case 2: For

We denote by

Then, satisfies

Using Proposition 3 with , we obtainthat is equivalent to

Then, for and , we obtain

Integrating with respect to over , we obtain

Finally,

Combining (40) and (47), we obtain the result.

Proving inequality (27) also leads to show that there exits a constant such that the solution of (31) satisfies

Indeed, we have

Proof of Proposition 1. The proof will be done in two parts.

2.2.3. The First Part: Interval

We consider in this proof the characteristics . For , the characteristics start from .

For and , all the characteristics starting on enter the observation domain (see Figure 2). We have three cases.Case 1: and .Two situations can arise:(i): in this situation, we split the interval as

(ii): in this situation, we keep the interval .Case 2: and .In this case, we keep the interval .Case 3: .

In this case, we prove similarly the observability in and expand to .

Here, we give the proof in the only situation, where

In the remaining part of the proof, we give upper bounds for , where is successively each one of the intervals appearing in decomposition (50) (see Figure 2).

Upper bound on : for , we first set , where .

Then, verifies

By applying Proposition 3 with and , we obtain

Then, we have

Integrating with respect over , we obtain

Asthen

Finally,

Upper bound : for , we consider always system (52), but .

Applying Proposition 3 with and , we obtain

And as before, we obtain

2.2.4. The Second Part: Interval

Here, we suppose and .

And as in [2], we prove the existence of such that

Consequently, combining (58), (60), and (61), we obtain

Lemma 1 (see Figure 3). Let us suppose that with . Then, there exists and such that

Proof. Without loosing the generality, we suppose that .
Suppose that . Then, there exists (well chosen) such that .
We denote by ; then, .
We are in position to prove the observability inequality.

Proof. (proof of theorem 2). Let , as in the previous Lemma. We haveFrom the result of (23) and applying the previous lemma, there exists such thatand thenFinally, with the result of Proposition 1, we obtainWith the result of Proposition 2 and inequality (67), we obtain the result of Theorem 4 (see Figure 3).
The figures below illustrate, respectively, the estimate of the nonlocal term, the estimate of , and the observation time.

Proof. of the Theorem 3. For , we consider the functional define bywhere is the solution of the following system:with .

Lemma 2. The functional is continuous, strictly convex, and coercive. Consequently, reaches its minimum at a point .
Moreover, setting , the associated solution of (69), and , the solution (21) with , has , and there exist and independent of and , such that

Proof. (proof of lemma). It is easy to check that is coercive, continuous, and strictly convex. Then, it admits a unique minimizer . The maximum principle gives that .
By calculating the dot product of the first equation of (69) with by and integrating with respect to the age and the time , we obtainUsing the inequality of Young, we obtain, for any ,Using the observability inequality (25) and choosing , where is given in (25), we obtainThen, the result is obtained.

3. Null Controllability of Nonlinear System

Now, we consider the following system: