Null controllability for a degenerate population model in divergence form via Carleman estimates

In this paper we consider a degenerate population equation in divergence form depending on time, on age and on space and we prove a related null controllability result via Carleman estimates.


Introduction
We consider the following population model in divergence form describing the dynamics of a single species: (1.1) For example, as k one can consider k(x) = x α ( − x) β , α, β > . Clearly, we say that k is weakly or strongly degenerate only at if De nition 1.1 is satis ed only at , i.e. k ∈ W , ([ , ]), k > in ( , ], k( ) = and, there exists M ∈ ( , ) or M ∈ [ , ) such that xk (x) ≤ M k(x) for a.e. x ∈ [ , ]. Analogously at .
In the last centuries, population models have been widely investigated by many authors from many points of view (see, for example, [5], [9], [14], [20]). From the general theory for the Lotka-McKendrick system, it is known that the asymptotic behavior of the solution depends on the so called net reproduction rate R : if R > , the solution is exponentially growing; if R < , the solution is exponentially decaying; if R = , the solution tends to the steady state solution. Clearly, if R > and the system represents the distribution of a damaging insect population or of a pest population, it is very worrying. For example, in B. Zhong, C. Lv, W. Qin show that the net reproduction rate for the Tirathaba ru vena (which causes a lot of damages for the crop, for example, of fruits and owers) depends on the temperature: it is . if the temperature is • C and it is . is the temperature is • C (see, for example, [25]); in S. S. Win, R. Muhamad, Z. A. M. Ahmad, N. A. Adam show that the net reproduction rate for the Nilaparvata lugens (which caused a lot of damages for the rice crop throughout South and South-East Asia since the early 1970's) was about (see, for example, [24]). For this reason, recently great attention is given to null controllability. For example in [21], where (1.1) models an insect growth, the control corresponds to a removal of individuals by using pesticides.
There are a lot of papers that deal with null controllability for (1.1) when the dispersion coe cient k is a constant or a strictly positive function (see, for example, [3]). If y is independent of a and k degenerates at the boundary or at an interior point of the domain we refer, for example, to [2], [15] and to [17], [18], [19] if µ is singular at the same point of k. To our best knowledge, [1] is the rst paper where y depends on t, a and x and the dispersion coe cient k can degenerate. In particular, the authors assume that k degenerates at the boundary (for example k(x) = x α , being x ∈ ( , ) and α > ). Using Carleman estimates for the adjoint problem, the authors prove null controllability for (1.1) under the condition T ≥ A. However, this assumption is not realistic when A is too large. To overcome this problem in [10], the authors used Carleman estimates and a xed point method via the Leray -Schauder Theorem. However, in [10] the authors consider a dispersion coe cient that can degenerate only at a point of the boundary and they use the xed point technique in which the birth rate β must be in C (Q) specially in the proof of [10,Proposition 4.2]. In the recent paper [13], we studied null controllability for (1.1) in non divergence form and with a di usion coe cient degenerating at a one point of the boundary domain or in an interior point. Observe that, in the case of a boundary degeneracy, we cannot derive the null controllability for (1.1) by the one of the problem in non divergence form or vice versa, see [7]. For this reason here we study the null controllability for (1.1) assuming that k degenerates at the boundary of the domain and T < A completing [1]. We underline that here, contrary to [10] and [13], we assume also that k can degenerate at both points of the boundary domain (see Theorem 4.8) and β is only a continuous function. On the other hand, while in [10] the authors used Carleman estimates, a generalization of the Leray -Schauder xed point Theorem and the multi-valued theory, here we use only Carleman estimates, some results of [13] and a technique based on cut o functions, making the proof slimmer and easier to read. Moreover, the technique that we use to prove Theorem 4.8 can be applied also to the problem in non divergence form considered in [13], generalizing [13,Theorem 4.8]. Finally, in the proof of the last theorem we make precise a calculation of [13,Theorem 4.8] which was not accurate. Observe that in this paper, as in [13], we do not consider the positivity of the solution, even if it is clearly interesting. This problem is related to the minimum time issue, i.e. given T cannot be arbitrarily small, but this study is still a work in progress, see [23] for related results in non degenerate cases.
A nal comment on the notation: by c or C we shall denote universal strictly positive constants, which are allowed to vary from line to line.

Well posedness results
On the rates µ and β we assume: Hypothesis 2.1. The functions µ and β are such that • β ∈ C(Q A, ) and β ≥ in Q A, , • µ ∈ C(Q) and µ ≥ in Q. (2.1) To prove well posedness of (1.1), we introduce, as in [2], the following Hilbert spaces We have, as in [2] or [16], that the operator is self-adjoint, nonpositive and generates a strongly continuous semigroup on the space L ( , ). Now, setting Aa u := ∂u ∂a , we have that generates a strongly continuous semigroup on L (Q A, ) := L ( , A; L ( , )) (see also [4]). Moreover, the operator B(t) de ned as B(t)u := µ(t, a, x)u, where C is a positive constant independent of k, y and f . In addition, if f ≡ , then y ∈ C [ , T]; L (Q A, ) .
For the existence of the solution and the regularity of it we refer, for example, to [11] and [22]. On the other hand, we postpone the proof of (2.2) to the Appendix.

Carleman estimates
From the general theory, it is known that null controllability for a linear parabolic system is, roughly speaking, equivalent to the observability for the associated homogeneous adjoint problem (see, for example, [12]). Thus, the key point is to prove such an inequality. A usual strategy in showing the observability inequality is to prove that certain global Carleman estimates hold true for the adjoint operator. Hence, this section is devoted to obtain global Carleman estimates for the operator which is the adjoint of the given one in both the weakly and the strongly degenerate cases. In particular, we consider the following adjoint system associated to (1.1): (3.1) . . . Carleman inequalities when the degeneracy is at . In this subsection we will consider the case when k( ) = and we assume that µ satis es (2.1). On the other hand, on k we make additional assumptions: We remark that the assumption xa ≤ M a, with M < , is essential in all our results and it is the same made, for example, in [2]. It implies that √ a ∈ L ( , ) and, in particular, if K < , then a ∈ L ( , ). Thus, the case M ≥ is excluded. Summing up, we will con ne our analysis to the case of √ a ∈ L ( , ). This is, however, the interesting case from the viewpoint of null controllability. In fact, if √ a ∉ L ( , ) and y is independent of a, then (1.1) fails to be null controllable on the whole interval [ , ], and regional null controllability is the only property that can be expected, see [8].
In order to prove Theorem 3.1, we de ne, for s > , the function where v is the solution of (3.1) in V; observe that, since v ∈ V, w ∈ V. Clearly, one has that w satis es De ning Lw := w t + wa + (kwx)x and Ls w := e sφ L(e −sφ w), the equation of (3.3) can be recast as follows As usual, we compute the inner product < L + s w, L − s w > L (Q) whose rst expression is given in the following lemma Lemma 3.1. Assume Hypothesis . . The following identity holds Proof. It results, integrating by parts, By [2, Lemma 3.1], we get (3.5) Next, we compute I and I . Integrating by parts, we have On the other hand Adding (3.5) -(3.7), (3.4) follows immediately.
As a consequence of the de nition of φ, one has the next estimate: Now, observe that there exists c > such that Hence, proceeding as in the proof of [2, Lemma 3.5], one can deduce x y k(y) dy. As in [2, Lemma 3.5], one can estimate the last two terms in the following way for s large enough and The same estimate holds also for Using the above estimates in (3.10) the thesis follows immediately for s large enough.
The next lemma holds.

Lemma 3.3. Assume Hypothesis . . The boundary terms in (3.4) become
Proof. Using the de nition of φ, [2, Lemma 3.6], the boundary conditions of w and proceeding as in [13,Lemma 3.2], the boundary terms of As a consequence of Lemmas 3.3 and 3.2, we have Recalling the de nition of w, we have v = e −sφ w and vx = (wx − sφx w)e −sφ . Thus, Theorem 3.1 follows immediately by Proposition 3.1 when µ ≡ . Now, we assume that µ ≢ .
To complete the proof of Theorem 3.1 we consider the function f = f + µv. Hence, there are two strictly positive constants C and s such that, for all s ≥ s , the following inequality holds (3.14) If M ≥ , using the Young's inequality to the function ν := e sφ v, we have Moreover, using Hypothesis 3.1, one has that the function (3. 16) In any case, by (3.14), (3.15) and (3.16), Using this last inequality in (3.13), it follows . . . Carleman inequalities when the degeneracy is at . In this subsection we will consider the case when k( ) = . Again µ satis es (2.1) and on k we make the following assumption: For Hypothesis 3.2 we can make the same considerations made for Hypothesis 3.1.
As in the previous subsection, let us introduce the weight function where Θ is as in (4.7) andp(x) := The following estimate holds: If µ ≢ , we can proceed as in the proof of Theorem 3.1. However, while in that case we use the Hardy-Poincaré inequality proved in [2, Proposition 2.1] which holds only if k( ) = , in this case we have to use the following inequality whose proof we postpone to the Appendix. Assume that k is such that there exists θ ∈ ( , ) such that the function Then, there is a constant C > such that for any function w, locally absolutely continuous on [ , ), continuous at and satisfying w( ) = , and k(x)|w (x)| dx < +∞ , the following inequality holds If hypothesis (HP1) is replaced by Hypothesis (HP1)': Assume that k is such that there exists θ ∈ ( , ) such that the function

Observability and controllability
In this section we will prove, as a consequence of the Carleman estimates established in Section 3, observability inequalities for the associated adjoint problem of (1.1). From now on, we assume that the control set ω is such that ω = (α, ρ) ⊂⊂ ( , ). (4.1) Moreover, on k and β we assume the following assumptions:  Observe that Hypothesis 4.2 is the biological meaningful one. Indeed,ā is the minimal age in which the female of the population become fertile, thus it is natural that beforeā there are no newborns. For other comments on Hypothesis 4.2 we refer to [13].
Under the previous hypotheses, the following observability inequality holds: satis es Here v T (a, x) is such that v T (A, x) = in ( , ). 2. Moreover, as in [13], observe that in (4.4) the presence of the integral δ v T (a, x)dxda is related to the presence of the term β(a, x)v(t, , x) in the equation of (4.3). In fact, estimating such a term using the method of characteristic lines, we obtain the previous integral. Obviously, if v T (a, x) = a.e. in ( , δ) × ( , ), we obtain the classical observability inequality.
Before proving Proposition 4.1 we will give some results that will be very helpful. As a rst step we introduce the following class of functions  for every solution v of (3.1).
The proof of the previous proposition is similar to the one given in [13], but we repeat it in the Appendix for the reader's convenience. Moreover, the following non degenerate inequality proved in [13] is crucial:

Remark 2.
The previous Theorem still holds under the weaker assumption k ∈ W ,∞ ( , ) without any additional assumption.
On the other hand, if we require k ∈ W , ( , ) then we have to add the following hypothesis: there exist two functions g ∈ L ( , ), h ∈ W ,∞ ( , ) and two strictly positive constants g , h such that g(x) ≥ g and In this case, i.e. if k ∈ W , ( , ), the function Ψ in (4.7) becomes where r and c are suitable strictly positive functions. For other comments on Theorem 4.1 we refer to [13].
Proceeding as before one can prove (4.14) Using the previous local Carleman estimates one can prove the next observability inequalities.
Proof. As in [13] and using the method of characteristic lines, one can prove the following implicit formula for v solution of (4. if t ≥ T −ā. Proceeding as in [13,Theorem 4.4], with suitable changes, one has that there exists a positive constant C such that: However, here we make all the calculations in order to make precise some steps in [13,Theorem 4.4], where there is a misprint. Indeed in (4.19) the integration is made on T −ā , T −ā in place of T , T as in [13].
The integration on T , T is correct if T =ā as in the next Corollary or in [13,Corollary 4.1].
Indeed, de ne, for ς > , the function w = e ςt v, where v solves (4.3). Then w satis es Then, integrating over T −ā , T −ā , we have (4.19). Now, take δ ∈ ( , A). By (4.19), we have Consider the term for a strictly positive constant C. Hence, whereΘ is de ned in (4.14) with T := T −ā, T := T, γ = andφ is the function associated toΘ according to (3.2). Analogously, if Hypothesis 3.2 holds, then we obtain whereφ is the function associated toΘ according to (3.18). Thus, by Theorem 4.2 or 4.3 applied toQ := (T −ā, T) × ( , A) × ( , ) and Remark 3, for a strictly positive constant C. By (4.18), (4.24) and proceeding as in [13], we have for a strictly positive constant C. By (4.22) and (4.25), it results Moreover, if v T (a, x) = for all (a, x) ∈ ( ,ā) × ( , ), one has Actually, proceeding as in [13] with suitable changes, we can improve the previous results in the following way: Theorem 4.5. Assume Hypotheses . or . and . . Then, for every δ ∈ (T, A), there exists a strictly positive constant C = C(δ) such that every solution v of (4.3) in V satis es By Theorem 4.5 and using a density argument, one can deduce Proposition 4.1. As a consequence one can prove, as in [13], the following null controllability results: Theorem 4.6. Assume Hypotheses . or . or . and . . Then, given T > and y ∈ L (Q A, ), for every δ ∈ (T, A), there exists a control f δ ∈ L (Q) such that the solution y δ ∈ U of Proof. The proof is similar to the one of [13,Theorem 4.8]. However, here we make all the calculations in order to make precise some steps in [13,Theorem 4.8], where there is a misprint and a term was missing. As a rst step, setT := T −ā ∈ ( , T). By Theorem 2.1, there exists a unique solution u of   Then y δ satis es (1.1) and f δ ∈ L (Q) is such that Indeed y δ (T, a, x) = w δ (T, a, x) = a.e. (a, x) ∈ (δ, A) × ( , ). Now, we prove (4.29). As a rst step, as in [13], we multiply the equation of (4.30) by u. Then, integrating over Q A, , we obtain: Hence, using the fact that u(t, , x) = A β(a, x)u(t, a, x)da, we have Setting F(t) := u(t) L (Q A, ) and multiplying the previous inequality by e − for a strictly positive constant C. Hence, (4.29) follows.
As a consequence of the previous theorem, we obtain the null controllability property if the coe cient k degenerates at and at at the same time. Proof. Fix y ∈ L (Q A, ) and consider the two problems whereᾱ ∈ ( , α) andβ ∈ (β, ). Thus, by Theorem 4.7, there exist two controls h ,δ and h ,δ such that the solutions u ,δ and u ,δ of (P ) and (P ), associated to h ,δ and h ,δ , respectively, satisfy Obviously, the support of f δ is contained in ω and, since k ∈ C (ω), the terms (kϕ u )x, (kξ u )x and (kη u )x are L ( , ) (recall that ϕ (x) = ξ (x) = η (x) = for all x ∈ ( , ) \ ω); thus f δ ∈ L (Q) as required. As in [19], estimate (4.33) follows by the de nition of f δ , (4.36) and (2.2) for u i , i = , , .
Observe that the previous result can be proved also for the problem in non divergence form considered in [13].

A Appendix
A. Proof of (2.