Optimal Control for a Degenerate Population Model in Divergence Form With Incomplete Data

In this paper


Introduction
We consider a degenerate population model in its divergence form with incomplete data: Here Q = (0, T )×(0, A)×(0, 1), Q T,A = (0, T )×(0, A), Q A,1 = (0, A)×(0, 1), Q T,1 = (0, T )×(0, 1), and Q ω = (0, T )×(0, A)×ω where the subset ω ⊂ (0, 1) is the region where a control v is acting.The control v can correspond to a supply of individuals or to a removal of individuals on the subdomain ω.In this model, y(t, a, x) is the distribution of certain individuals at the point x ∈ (0, 1), at time t ∈ (0, T ), where T is fixed, and age a ∈ (0, A), A being the life expectancy, β and µ denote the natural rates of fertility and mortality, respectively.The formula A 0 βyda is the proportion of newborns at time t and at location x.In this model, χ ω is the characteristic function of the control domain ω ⊂ (0, 1);y 0 = y 0 (a, x) ∈ L 2 (Q T,1 ) is the initial distribution of individuals; the data f ∈ L 2 (Q) matches to an external supply.The function g belongs to G ⊂ L 2 (Q T,A ).We say that (1) is a system with incomplete data because the information on the boundary are missing.
Then k is a function of the space variable x which designates the dispersion coefficient.We assume that it degenerates at the boundary of the domain.
In recent years, population dynamics models have been widely studied by several authors from many points of view.
The majority of them have investigated the null controllability of the system for example, (Boutaayamou & Echarroudi, 2017), (Fragnelli, 2018), (Fragnelli, 2019), (Fragnelli, 2020).In effect, y can designate the proportion of a pest insect population, for example (He & Ainseba, 2014).Thus it is important to control it.In (He & Ainseba, 2014), the system (1) models insect growth, and the control corresponds to the removal of individuals by using pesticides.Authors (Boutaayamou & Echarroudi, 2017), are concerned with the null controllability of a population dynamics system with an interior degenerate diffusion.To this end, they proved first a new Carleman estimate for the full adjoint system, and afterward, they deduce a suitable observability inequality which will be needed to establish the existence of control acting on a subset of the space which leads the population to extinction in a finite time.(Fragnelli, 2019) and (Fragnelli, 2020) deal with a degenerate system describing the dynamics of a population depending on time, age, and space in divergence form.He assumes that the degeneracy can occur at the boundary or in the interior of the space domain and he focuses on the null controllability problem.To this aim, he proves first Carleman estimates for the associated adjoint problem, then, via cut off functions, he proves the existence of a null control function localized in the interior of the space domain in both papers.In the second one, he considers two cases: either the control region contains the degeneracy point x 0 , or it is a reunion union of two domains each located on one side of x 0 .Whereas in (Fragnelli, 2018), the same previous research is done but on a degenerate population equation in non-divergence form.
According to the authors, the non-trivial solutions of the system (commonly named LotkaCMcKendrick systems) have exponentially rising or falling asymptotic behavior, depending on the size of a certain biological amount (the so called net reproduction rate), see (Anita, 2000) and also (Fragnelli, Martinez, & Vancostenoble, 2005) for related results concerning time-independent steady states.In (Ainseba & Langlais, 2000), authors consider the optimal control problem for a population dynamics system with age dependence, spatial structure, and a nonlocal birth processus appearing as a boundary condition.They examine the controllability at a given time T and prove that the approximate controllability is valid for any fixed finite time T .Accordingly, they established a new result of condition continuation which is unique.
As much as we know, the first null controllability work for an age population dynamics model is due to (Ainseba & Langlais, 2000), where the authors proved that a set of profiles is approximately reachable.Later, in (Ainseba & Anita, 2004), a local exact controllability was proved.In particular, in (Ainseba & Ianneli, 2003), the authors showed that, if the initial data is sufficiently small, it is possible to find a control that drives the population to extinction.In the last one, the null controllability is also studied for a non-linear model of population dynamics in the diffusive form whenever the fertility and the mortality rates respectively depend on the total population.In (Traore, 2006), the authors considered a nonlinear distribution of newborns of the form F( A 0 β(t, a, x)y(t, a, x)da).
But, in all the above articles, the dispersion coefficient k is a scalar or a strictly positive function.To our best knowledge, (Ainseba & Ianneli, 2003) is the first paper where the dispersion coefficient, which depends on the space variable x, can degenerate.In particular, the authors assume that k degenerates at the edges (for example k(x) = x α , being x ∈ (0, 1) and α > 0).The authors apply Carleman estimates on the adjoint problem and prove a zero controllability result for (1) under the condition T ≥ A .But, this hypothesis is incorrect when A becomes large enough.To overcome this problem in (Echarroudi & Maniar, 2014), the authors employed Carleman estimates and the fixed point method of Leray-Schauder.
In (Birgit & Omrane, 2010), B.Jacob and A.Omrane are concerned with the optimal control for linear age-structured population dynamics system with incomplete data.More precisely, the initial population age distribution is supposed to be unknown.They used the notion of no-regret control of J.L.Lions in (Lions, 1992) to such singular population dynamics, following the method by Nakoulima et al. as in (Nakoulima, Omrane, & Velin, 2000).They prove that the problem they are considering has a unique no-regret control that they characterize by a singular optimality system.
In the present paper, we are interested with the no-regret control of a degenerate population dynamics system describing a single species in divergence form with unknown information on the boundary which to our knowledge has not been treated.We consider the minimization of the following cost functional: where z d ∈ L 2 (Q) and N > 0 are given.We deal with solving the optimization problem above: But noticing that we could have obtained: sup We consider the next problem: inf Then we research the control that does not make things worse than a given control v 0 (or to than doing nothing, v 0 = 0 in our case), independently of the perturbations which may be of an infinite number.Lions used the notions of Pareto control (Lions, 1986) and equivalently the no-regret control (Lions, 1992) in application to the control of systems having missing data.The no-regret concept was previously used in statistics by Savage (Savage, 1972).The no-regret control over incomplete data problems is not easy to characterize directly.We will use an approximate control: the low-regret control.
To achieve the no-regret control, we give the singular optimality system for the low-regret control for the incomplete data population dynamics (1)C(2), using a quadratic perturbation used by Nakoulima et al. in (Nakoulima, Omrane, & Velin, 2000) (see also (Nakoulima, Omrane, & Velin, 2003)).Next, we give a singular optimality system that characterizes the no-regret control that is the limit of a standard control problem.
The paper is organized as follows.In Section 2, we give well-posedness and some regularity results.We study the low-regret and no-regret control and their characterizations in sections 3 and 4 respectively.

Well-posedness Result and Preliminaries
In the sequel, we will assume that k satisfies the following hypotheses: In plus, we make the following assumptions about the functions µ and β defined in (1): To show that the problem (1) is well posed, we need to introduce the following Sobolev spaces: √ ku x ∈ L 2 (0, 1) and u(1) = u(0) = 0} and H 2 k = {u ∈ H 1 k (0, 1)| ku x ∈ H 1 (0, 1)}.with their respective norms: Let the unbounded operator A : , closed, symmetric, self-adjoint and negative operator and whose domain is dense in L 2 (0, 1) (Cannarsa, Martinez, & Vancostenoble, 2005).In addition, it generates an analytical semi-group in space L 2 (0, 1).By setting )) the following result on the existence and uniqueness of the solution of the model (1) holds: Theorem 2.1.Assume that k is weakly or strongly degenerated in 0 and/or in 1.For all f ∈ L 2 (Q) and y 0 ∈ L 2 (Q A,1 ), the system (1) admits a unique solution where C is a positive constant independent of k, y 0 and f .In addition The proof is similar to those in ( (Engel & Nagel, 2000), (Fragnelli, 2020), (Lions & Magenes, 1972)) ).One can define also the trace at a = a 0 in L 2 (Q T,1 ).The applications "trace" are continuous for weak and strong topologies.
For more details on the latter lemma, see Oumar in [Sur un problme de dynamique de populations( 2003 (Langlais, 1979) Proposition 2.1.Let y = y(v, .)be solution of system (1), then the application v → y(v, .) We set ȳ = y(v, .)− y(v 0 , .), then ȳ is solution of : If we set z = e −rt ȳ with r > 0, we get that z is solution of: Multiply the first equation of (6) by z then integrate by parts on Q, we get: Proof 2. Let be v 1 , v 2 ∈ L 2 (Q ω ) and let be S = S (v 1 ) − S (v 2 ).Then S satifies the system: If we set z = e −rt S with r > 0, we get that z is solution of: Multiply the first equation of (9) by z then integrate by parts on Q: Then, we can choose r 0 such that: and consequently ).Now let be y(v, 0) , y(0, g) and y(0, 0) the respective solutions of systems: and Remember that according to (7) that the functional S is the solution of system: As a result, as y(v, 0) − y(0, 0) ∈ L 2 (Q), the problem below admits a unique solution.
Also, noting that S Now multiply the first equation of (7) by y(0, g) − y(0, 0) then integrate by parts over Q.We get [y(0, g)(T )S (T ) − y(0, g)(0)S (0)]dxda Finally, we get: On the other hand, noting that:y(v, g) = y(v, 0) + y(0, g) − y(0, 0), we get: and using equality (14), we get: Remark 2. The problem (14) has a meaning if the expression Then, using ( 14), the expression (3) becomes : We can get: The problem ( 16) admits a solution only in the case (18).And the control v is choosen in suitable space U ad subset convex close non empty of L 2 (Q ω ) defined by: As such control is not easy to characterize, we consider the following low-regret control problem: Using the Legendre-Fenchel transformation, problem ( 20) is equivalent to solving:

Existence and Characterization of Low-regret Control
In this section, we propose an existence result for the family of low-regret controls.Then we give the singular optimality system allowing us to characterize it.
Proof 4. The proof uses Propositions 2.1 and 2.2 to show that the functional J γ is continuous, on the one hand, and the strict convexity of J γ , on the other hand, to show the uniqueness of the solution of the problem (21).Therefore, the sequence y(v n , .)weakly converges to y(u γ , .) in L 2 (Q).The sequence S (v n , .)weakly converges to S (u γ , .) in L 2 (Q).By continuity of the trace application, the sequence Thus u γ is a solution to the problem (21).
We now turn to the characterization of low-regret control u γ .

Existence and Characterization of the No-regret Control
In this section, we give an existence result for the no-regret control as a limit of the family of low-regret controls in the neighborhood of the origin.Then we establish the singular optimality system allowing us to characterize it.
Proposition 4.1.The low-regret control u γ converges toward the no-regret control u in L 2 (Q ω ).
We obtain: By combining (45), ( 47), ( 48) and (49), we find that ξ = ξ(u, 0) is solution of Now, using relation (38) and proposition (2.2), we deduce that: If in addition, we exploit the relation (36), we get that: From the uniqueness of the limit, we conclude that: Using the continuity of the trace application: Thus u ∈ U ad .The strict convexity of J γ allows to deduce that u is the unique control, solution of (21).
We conclude that the low-regret control u γ converge in L 2 (Q ω ) toward the unique no-regret control u In what follows, we will try to characterize the unique no-regret control u.
Proposition 4.2.The no-regret control u solution of problem (3) is characterized by the quadruplet {y, ξ, p, q} solution of optimality system : and Nu + q = 0 in Q ω (60) where k Proof 7. The results (56) and (57) have been demonstrated during the proof of the previous proposition.It remains to be demonstrated (58)-( 60).For this, we will proceed by steps: Step 1: show that p is solution of (58).