NULL CONTROLLABILITY OF POPULATION DYNAMICS WITH INTERIOR DEGENERACY. 1. Introduction

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1 Electronic Journal of Differential Euations, Vol. 217 (217, No. 131, pp ISSN: URL: or NULL CONTROLLABILITY OF POPULATION DYNAMICS WITH INTERIOR DEGENERACY IDRISS BOUTAAYAMOU, YOUNES ECHARROUDI Communicated by Jerome A. Goldstein Abstract. In this article, we study the null controllability of population model with an interior degenerate diffusion. To this end, we proved first a new Carleman estimate for the full adjoint system and then we deduce a suitable observability ineuality which will be needed to establish the existence of a control acting on a subset of the space which lead the population to extinction in a finite time. Consider the system 1. Introduction y t + y a ((xy x x + µ(t, a, xy = ϑχ in, y(t, a, 1 = y(t, a, = on (, T (, A, y(t,, x = y(, a, x = y (a, x in A, A β(t, a, xy(t, a, xda in T, (1.1 where = (, T (, A (, 1, A = (, A (, 1, T = (, T (, 1 and we will denote = (, T (, A, where = (x 1, x 2 (, 1 is the region where the control ϑ is acting. This control corresponds to an external supply or to removal of individuals on the subdomain. Since (1.1 models the dispersion of gene of a given population, x represents the gene type and y(t, a, x is the distribution of individuals of age a at time t and of gene type x. The parameters β(t, a, x and µ(t, a, x are respectively the natural fertility and mortality rates, A is the maximal age of life and is the gene dispersion coefficient. y L 2 ( A is the initial distribution of population. Finally, A β(t, a, xy(t, a, xda is the distribution of the newborns of the population that are of gene type x at time t. As usual, we will suppose that no individual reaches the maximal age A. We note that in the most wors concerned with the diffusion population dynamics models, x is viewed as the space variable. 21 Mathematics Subject Classification. 35K65, 92D25, 93B5, 93B7. Key words and phrases. Population dynamics model; interior degeneracy; Carleman estimate; observability ineuality; null controllability. c 217 Texas State University. Submitted January 9, 217. Published May 12,

2 2 I. BOUTAAYAMOU, Y. ECHARROUDI EJDE-217/131 The population models in their different aspects attracted many authors that investigated them from many sides (see for example [18, 2, 21, 25, 28, 29]. Among those uestions, we find the null controllability problem or in general the controllability problems for age and space structured population dynamics models which were studied in a intensive literature basing, in general, on the references interested on the controllability of heat euation (see for instance [12, 13, 1, 15, 16, 19] for a different controllability problems of heat euation. In this context, we can cite the pioneering items of Barbu and al. [7], Ainseba and Anita [1, 2, 3, ]. In [7], the authors proved the null controllability for a population dynamics model without diffusion both in the cases of migration and birth control for T A showing directly an appropriate observability ineuality for the associated adjoint system and they concluded that in the case of the migration control, only a classes of age was controlled in contrary with the birth control which allows to steer all population to extinction. In [1, 2, 3, ], the diffusion was taen into account in a age-space structured model and the null controllability of (1.1 for classes of age was established in the case where = 1 and for any dimension n by means of a weighted estimates called Carleman estimates and exploiting the results gotten for heat euation in [26]. Ainseba et al. [5] studied a more general case allowing the dispersion coefficient to depend on the variable x and satisfies ( = (i.e, the coefficient of dispersion degenerates at. The authors tried to obtain (1.2 in such a situation with β L basing on the wor done in [6] for the degenerate heat euation to establish a new Carleman estimate for the full adjoint system (2.3 and afterwards his observability ineuality. However, the null controllability property of this paper was showed under the condition T A (as in [7] and this constitutes a restrictiveness on the optimality of the control time T since it means, for example, that for a pest population whose the maximal age A may eual to a many days (may be many months or years we need much time to bring the population to the zero euilibrium. In the same trend and to overcome the condition T A, Maniar et al. [17] suggested the fixed point techniue in which the birth rate β must be in C 2 ( specially in the proof of [17, Proposition.2]. Such a techniue consists briefly to demonstrate in a first time the null controllability for an intermediate system with a fertility function b L 2 ( T instead of A β(t, a, xy(t, a, xda and to achieve the tas via a Leray-Schauder Theorem. Thereby, the main goal of this article is to sutdy the null controllability property with a minimum of regularity of β (see (2.2 and a positive small control time T taing into account that depends on the gene type and degenerates at a point x, i.e (x =, e.g (x = x x α. To be more accurate, for a fixed T (, δ with δ (, A small enough, we investigate the existence of a suitable control ϑ L 2 ( which depends on y and δ and such that the associated solution y of (1.1 satisfies y(t, a, x =, a.e. in (δ, A (, 1. (1.2 If (x = in a point x, we say that (1.1 is a population dynamics model with interior degeneracy. Genetically speaing, the meaning of (x = is that the gene of type x (, 1 can not be transmitted from the studied population to its offspring. This objective will be attained via the classical procedure following the strategy in [22]. On other words, we will establish an appropriate observability ineuality for the full adjoint system of (1.1 which is an outcome of a suitable Carleman estimate. We highlight that such a result can be shown if we

3 EJDE-217/131 NULL CONTROLLABILITY 3 replace the homogeneous Dirichlet boundary conditions by the ones of Neumann, i.e., y x (t, a, = y x (t, a, 1 =, (t, a (, T (, A using the same way done in [1]. Another interesting null controllability problem of (1.1 can be elaborate using the wor of Fragnelli et al. [23] arising in the case when the potential term admits an interior singularity belonging to gene type domain. The remainder of this paper is organized as follows: in Section 2, we will provide the well-posedness of (1.1 and give the proof of the Carleman estimate of its adjoint system. The Section 3 will be devoted to the observability ineuality and hence we obtain the null controllability result (1.2. The last section will tae the form of an appendix where we will bring out a Caccioppoli s ineuality which plays an important role to show the desired Carleman estimate. 2. Well-posedness and Carleman estimate results 2.1. Well-posedness. For this section and for the seuel, we assume that the dispersion coefficient satisfies x (, 1, C([, 1] C 1 ([, 1]\{x }, > in [, 1]\{x } and (x =, γ [, 1 : (x x (x γ(x, x [, 1]\{x }. (2.1 It is well-nown in the literature of degenerate problems that there exist two inds of degeneracy namely the wealy degenerate and the strong degenerate problems, in our study we will restrict ourselves to the first one and this fact explains the choice of γ [, 1 which in fact are associated to the Dirichlet boundary conditions (see [22, Hypothesis 1.1]. On the other hand, the last hypothesis on means in the case of (x = x x α that α < 1. The investigation of (1.2 needs also the following assumptions on the natural rates β and µ: µ, β L (, β(t, a, x, µ(t, a, x, a.e. in, β(,, a.e. in (, T (, 1. (2.2 The last assumption in (2.2 is natural since the newborns are not fertile. Also, it is worth mentioning to point out that, as in [5] we do not need to reuire that µ satisfies an hypotheses lie A µ(t s, A s, xds = +, (t, x [, T ] [, 1] since it does not play any role on the well-posedness result and the computations concerning the proofs of our controllability result as well. However, we will suppose that no individual can reach the maximal age A as mentioned in the introduction. In the same context, we emphasize that in [17], the L regularity of β is sufficient to prove the well posedness of the studied model which is exactly our case. To this end, we introduce the following weighted Sobolev spaces: H 1 (, 1 := { u L 2 (, 1 : u is abs. cont. on [, 1] : ux L 2 (, 1, u(1 = u( = }, H 2 (, 1 := { u H 1 (, 1 : (xu x H 1 (, 1 }, endowed respectively with the norms u 2 H 1 (,1 := u 2 L 2 (,1 + u x 2 L 2 (,1, u H1 (, 1, u 2 H := u 2 2 H 1(,1 + ((xu x x 2 L 2 (,1, u H2 (, 1.

4 I. BOUTAAYAMOU, Y. ECHARROUDI EJDE-217/131 We recall from [2, Theorem 2.2] that the operator (P, D(P defined by P u := ((xu x x, u D(P = H 2 (, 1, is closed self-adjoint and negative with dense domain in L 2 (, 1. Conseuently, from [32, Theorem 5] the operator A := a +P generates a C -semigroup on the space L 2 ((, A (, 1. Then, the following wellposedness result holds. Theorem 2.1. Under assumptions (2.1 and (2.2, for all ϑ L 2 ( and y in L 2 ( A, system (1.1 admits a uniue solution y. This solution belongs to E :=C([, T ], L 2 ((, A (, 1 C([, A], L 2 ((, T (, 1 L 2 ((, T (, A, H 1 (, 1. Moreover, the solution of (1.1 satisfies sup t [,T ] ( C y(t 2 L 2 ( A + sup a [,A] ϑ 2 + y 2 dx dx. A y(a 2 L 2 ( T + 1 A T ( (xy x 2 dt da dx 2.2. Carleman estimates. As we said in the introduction, we will show the main ey of this paper namely the Carleman type ineuality. In general, it is wellnown that to prove a controllability result of a studied model through this a priori estimate, we must show this last for the associated adjoint system. In our case, this adjoint system taes the form w t + w a + ((xw x x µ(t, a, xw = β(t, a, xw(t,, x, w(t, a, 1 = w(t, a, =, w(t, a, x = w T (a, x, w(t, A, x =, (2.3 where T > and assume that w T L 2 ( A. Of course, assumptions (2.1 and (2.2 on, µ and β are perpetuated. To attaint our goal, we will introduce the weight functions ϕ(t, a, x := Θ(t, aψ(x, Θ(t, a := ψ(x := c 1 ( x 1 (t(t t a, r x x (r dr c 2. (2. For the moment, we will assume that c 2 > max{ (1 x2 (1(2 γ, x 2 ((2 γ } and c 1 >. A more precise restriction on c 1 will be given later. On the other hand, using the relation satisfied by c 2 and with the aid of [22, Lemma 2.1] one can prove that ψ(x < for all x [, 1]. Observe also that Θ(a, t + as t T, + and a +. To demonstrate our Carleman estimate, we reuire that fulfills, besides (2.1 the following hypothesis θ (, γ] such that x (x is non-increasing on the left of x x θ x = x and nondecreasing on the right of x = x, (2.5

5 EJDE-217/131 NULL CONTROLLABILITY 5 where γ is defined by (2.1. The first Carleman estimate result is the following result. Proposition 2.2. Consider the following two systems with h L 2 (, and w t + w a + ((xw x x = h, w(a, t, 1 = w(a, t, =, w(t, a, x = w T (a, x, w(t, A, x =, w t + w a + ((xw x x µ(t, a, xw = h, w(t, a, 1 = w(t, a, =, w(t, a, x = w T (a, x, w(t, A, x =. (2.6 (2.7 Then, there exist two positive constants C and s, such that every solution of (2.6 and (2.7 satisfy, for all s s, the ineuality s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx + s Θ(xw (x xe 2 2sϕ dt da dx ( A T (2.8 C h 2 e 2sϕ dt da dx + s [Θe 2sϕ (x x wx] 2 x=1 x= dt da. Proof. Firstly, we prove (2.8 for system (2.6 and replacing h by h + µw we will get the same ineuality for (2.7. So, let w be the solution of (2.6 and put Then, ν satisfies the system where ν(t, a, x := e sϕ(t,a,x w(t, a, x. L + s ν + L s ν = e sϕ(t,a,x h, ν(t, a, 1 = ν(t, a, =, ν(t, a, x = ν(, a, x =, ν(t, A, x = ν(t,, x =, L + s ν := ((xν x x s(ϕ a + ϕ t ν + s 2 ϕ 2 x(xν, Passing to the norm in (2.9, one has L s ν := ν t + ν a 2s(xϕ x ν x s((xϕ x x ν. L + s ν 2 L 2 ( + L s ν 2 L 2 ( + 2 L+ s ν, L s ν = e sϕ(a,t,x h 2 L 2 (, (2.9 where, denotes here the inner product in L 2 (. Then, the proof of step one is based on the calculus of the inner product L + s ν, L s ν whose a first expression is given in the following lemma. Lemma 2.3. The following identity holds L + s ν, L s ν = S 1 + S 2,

6 6 I. BOUTAAYAMOU, Y. ECHARROUDI EJDE-217/131 with S 1 = s ((xν x 2 ϕ xx dt da dx s 3 ((xϕ x x (xϕ 2 xν 2 dt da dx + s 2 (ϕ a + ϕ t ((xϕ x x ν 2 dt da dx + s (xν x (((xϕ x xx ν + ((xϕ x x ν x, dt da dx + s 3 ( 2 ϕ 3 x x ν 2 dt da dx s 2 ((x(ϕ a + ϕ t ϕ x x ν 2 dt da dx + s 2 + s 2 (ϕ at + ϕ tt ν 2 dt da dx s2 (ϕ 2 2 x t (xν 2 dt da dx (ϕ at + ϕ aa ν 2 dt da dx s2 (ϕ 2 2 x a (xν 2 dt da dx, (2.1 and S 2 A T = + s 2 A s A T [(xν x ν a ] 1 dt da + T A T [(xν x ν t ] 1 dt da [(xϕ x (ϕ a + ϕ t ν 2 ] 1 dt da s 3 A [(xνν x ((xϕ x x ] 1 dt da s T A T [ 2 (xϕ 3 xν 2 ] 1 dt da [((xν x 2 ϕ x ] 1 dt da. (2.11 For the proof of Lemma 2.3, see [17, Lemma 3.2]. The previous expressions of S 1 and S 2 can be simplified using the functions ϕ and ψ given in (2. and also the homogeneous Dirichlet boundary conditions satisfied by ν. Hence, one has S 1 = s (Θ aa + Θ tt ψν 2 dx dt da + s Θ ta ψν 2 dt da dx 2 + sc 1 Θ(2(x (x x (xνx 2 dt da dx 2s 2 Θc 2 (x x 2 1 (Θ a + Θ t ν 2 dt da dx (x ( x + s 3 Θ 3 c 3 x 1 (x 2 (2(x (x x (x ν 2 dt da dx, (2.12 and S 2 = sc 1 A T [Θe 2sϕ (x x ν 2 x] x=1 x= dt da.

7 EJDE-217/131 NULL CONTROLLABILITY 7 Accordingly, L + s ν, L s ν = s (Θ aa + Θ tt ψν 2 dx dt da + s Θ ta ψν 2 dt da dx 2 + sc 1 Θ(2(x (x x (xνx 2 dt da dx 2s 2 Θc 2 (x x 2 1 (Θ a + Θ t ν 2 dt da dx (x + s 3 Θ 3 c 3 (x x 2(2(x 1 (x x (xν 2 dt da dx (x sc 1 A T [Θe 2sϕ (x x w 2 x] x=1 x= dt da. (2.13 Thans to the third assumption in (2.1, we have S 1 s (Θ aa + Θ tt ψν 2 dt da dx + s 2 + sc 1 Θ(xνx 2 dt da dx 2s 2 + s 3 Θc 2 1 (x x 2 (Θ a + Θ t ν 2 dt da dx (x Θ 3 c 3 (x x 2 1 ν 2 dt da dx. (x Θ ta ψν 2 dt da dx (2.1 Observe that Θ(Θ a + Θ t cθ 3 ; for s large we infer that 2s 2 Θc 2 1 2s 2 c 2 1c c3 1 s3 (x x 2 (Θ a + Θ t ν 2 dt da dx (x (x x 2 Θ 3 ν 2 dt da dx (x (x x 2 Θ 3 ν 2 dt da dx. (x (2.15 On the other hand, the mapping r r x γ (r is nondecreasing at the right of x. Then x r x ψ(x = c 1 l(x c 1 c 2 c 1 x (r dr + c 1c 2 c 1 c 2 + c 1 (1 x 2 (1(2 γ c 1 (2 γ(1 + c 1c 2. (2.16

8 8 I. BOUTAAYAMOU, Y. ECHARROUDI EJDE-217/131 A simple computations shows that Θ aa + Θ tt + Θ ta C 1 Θ 3/2. This yields s (Θ aa + Θ tt ψν 2 dt da dx + s Θ ta ψν 2 dt da dx 2 ( c ( 1 s (2 γ(1 + c Θaa + Θ tt 1c 2 2 ( c 1 Ms (2 γ(1 + c 1c 2 Θ 3/2 ν 2 dt da dx. + Θ ta ν 2 dt da dx (2.17 It remains now to bound the term Θ3/2 ν 2 dt da dx. Using the generalized Young ineuality we obtain 1 Θ 3/2 ν 2 dx 1 = (Θ 1/3 ν 2 3/( Θ 3 x x 2 ν 2 1/ dx x x 2 3 3ɛ 1 Θ 1/3 ν 2 dx Θ 3 x x ( ν 2 dx x x 2 3 ɛ Put p(x = ((x x x 1/3. (2.19 By hypothesis (2.5, one can chec that x p(x x x, with := +θ 3 (1, 2 is nonincreasing on the left of x = x and nondecreasing on the right of x = x. Furthermore, we have 1/3 = p(x x x 2 3 (x x and there exists C 2 2 > such that p(x < C 2 (x. Hence, by Hardy-Poincaré ineuality (see [22, Proposition 2.3], 1 Θ 1/3 ν 2 dx = x x C Θ p(x (x x 2 ν2 dx 1 Θpν 2 xdx 1 CC 2 Θνxdx, 2 (2.2 where C > is the constant of Hardy-Poincaré. Combining (2.18 and (2.2, we obtain 1 Θ 3/2 ν 2 dx 3ɛ C 3 Θνx 2 dt da dx + 1 Θ 3 x x 2 ν 2 dt da dx. (2.21 ɛ Hence, (2.17 and (2.21 lead to s (Θ aa + Θ tt ψν 2 dt da dx + s Θ ta ψν 2 dt da dx 2 sc 1 C ɛ Θνx 2 dt da dx + sc 1C 5 Θ 3 x x 2 ν 2 dt da dx. ɛ Taing ɛ small enough and s large, we conclude that s (Θ aa + Θ tt ψν 2 dt da dx + s Θ ta ψν 2 dt da dx 2 sc 1 Θνx 2 dt da dx + s3 c 3 1 Θ 3 x x 2 ν 2 dt da dx. (2.22 (2.23

9 EJDE-217/131 NULL CONTROLLABILITY 9 Taing into account relations (2.1 and (2.15 we arrive at S 1 K 1 s 3 Θ 3 x x 2 ν 2 dt da dx + K 2 s Θνx 2 dt da dx (2.2 Hence, 2 L + s ν, L s ν m (s 3 Θ 3 x x 2 A T 2sc 1 ν 2 dt da dx + s Θνx 2 dt da dx [Θe 2sϕ (x x ν 2 x] x=1 x= dt da (2.25 This yields to the following Carleman estimate satisfied by the solution ν of (2.9, s 3 Θ 3 x x 2 ν 2 dt da dx + s Θνx 2 dt da dx ( A T (2.26 C 6 h 2 e 2sϕ dt da dx + s [Θe 2sϕ (x x νx] 2 x=1 x= dt da By the definition of ν we infer that ν x = sϕ x e sϕ w + e sϕ w x, e 2sϕ w 2 x 2(ν 2 x + s 2 ϕ 2 xν 2. (2.27 Finally, the Carleman estimate (2.8 of (2.6 is obtained. Now, If we apply the same ineuality of Hardy-Poincaré in a similar way as before to the function ν := e sϕ w, taing into account the hypothesis on µ assumed in (2.2, using the Carleman type ineuality (2.8 for the function h+µw and taing s uite we achieve the Proposition 2.2. With the aid of the estimate (2.8 and Caccioppoli s ineuality (.1, we can now show a -local Carleman estimate for the system (2.7. This result will be useful to show our main Carleman estimate replacing the second term h by β(t, a, xw(t,, x. To this end, we introduce the weight functions Φ(t, a, x := Θ(t, aψ(x, Ψ(x = e κσ(x e 2κ σ, where Θ is given by (2., κ >, and σ is the function given by (2.28 σ C 2 ([, 1], σ(x > in (, 1, σ( = σ(1 =, σ x (x in [, 1]\, (2.29 where is an open subset. The existence of the function σ is proved in [26]. On the other hand by the definition of ϕ (2. and taing ( (1(2 γ(e 2κ σ 1 c 1 max c 2 (1(2 γ (1 x 2, ((2 γ(e2κ σ 1 c 2 ((2 γ x 2, (2.3 one can prove that Our main theorem is stated as follows. ϕ Φ. (2.31

10 1 I. BOUTAAYAMOU, Y. ECHARROUDI EJDE-217/131 Theorem 2.. Assume that assumptions (2.1, (2.2 and (2.5 hold. Let A > and T > be given. Then, there exist positive constants C and s such that for all s s, every solution w of (2.7 satisfies (sθwx 2 + s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx ( C h 2 e 2sΦ dt da dx + s 3 Θ 3 w 2 e 2sΦ dt da dx, (2.32 To prove this theorem, we need the following result which represents the Carleman estimate of nondegenerate population dynamics systems. The ineuality is stated as follows. Proposition 2.5. Let us consider the system z t + z a + ((xz x x c(t, a, xz = h in b, z(t, a, b 1 = z(t, a, b 2 = on (, T (, A, (2.33 where b := (, T (, A (b 1, b 2, (b 1, b 2 [, x, or (b 1, b 2 (x, 1], h L 2 ( b, C 1 ([, 1] is a strictly positive function and c L ( b. Then, there exist two positive constants C and s, such that for any s s, z satisfies (s 3 φ 3 z 2 + sφzxe 2 2sΦ dt da dx b ( A T (2.3 C h 2 e 2sΦ dt da dx + s 3 φ 3 z 2 e 2sΦ dt da dx, b where φ(t, a, x = Θ(t, ae κσ(x, (2.35 Θ and Φ are defined by (2.28, and σ by (2.29. Before giving the proof of Theorem 2., we note that a similar result was demonstrated in [2, Lemma 2.1] in the case when is a positive constant, for any dimension 1 n without the source term h and with the weight function Θ(t, a = t(t ta. By careful computations, the same proof can be adapted to (2.3 where is a positive 1 general nondegenerate coefficient, with our weight function Θ(t, a = t (T t a and the source term h. Proof of Theorem 2.. Let us introduce the smooth cut-off function ξ : R R defined by ξ(x 1, x [, 1], ξ(x = 1, x [λ 1, λ 2 ], ξ(x =, x [, 1]\, (2.36 where λ 1 = x1+2x 3 and λ 2 = x+2x2 3. Let w be the solution of (2.7 and define v := ξw. Then, v satisfies the system v t + v a + ((xv x x µ(t, a, xv = h, v(t, a, 1 = v(t, a, =, (2.37 v(t, a, x = ξw T (a, x, v(t, A, x =,

11 EJDE-217/131 NULL CONTROLLABILITY 11 where h := ξh + ((xξ x w x + (xw x ξ x. Using Carleman estimate (2.8 and the definition of ξ, one has (sθv 2 x + s 3 Θ 3 (x x 2 v 2 e 2sϕ dt da dx C h 2 e 2sϕ dt da dx. (2.38 On the other hand, using again the definition of ξ we can chec readily that λ2 A T λ 1 λ2 A T = λ 1 (sθvx 2 + s 3 Θ 3 (x x 2 v 2 e 2sϕ dt da dx (sθwx 2 + s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx. (2.39 Therefore, combining (2.38 and (2.39 we have λ2 A T λ 1 C (sθwx 2 + s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx h 2 e 2sϕ dt da dx. (2. Hence by Caccioppoli s ineuality (.1 and (2., we conclude that λ2 A T λ 1 ( C (sθwx 2 + s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx h 2 e 2sϕ dt da dx + s 2 Θ 2 w 2 e 2sϕ dt da dx, (2.1 where of Lemma.1 here is exactly (x 1, λ 1 (λ 2, x 2. Now, let z := ηw, with η is the smooth cut-off function defined by η(x 1, x [, 1], η(x =, x [, λ 3 + 2λ 2 ], 3 η(x = 1, x [λ 2, 1], (2.2 where λ 3 = x2+2x 3. We can observe easily that λ 3 < λ3+2λ2 3 < λ 2. Then, z satisfies the population dynamics euation z t + z a + (xz xx + (xz x µ(t, a, xz = h, in (λ 3, 1 z(t, a, 1 = z(t, a, λ 3 =, in, (, T (, A, (2.3 where h := ηh + ((xη x w x + (xw x η x. By assumption on, we have (x >, x (λ 3, 1. Hence, (2.3 is a non-degenerate model. In this case, applying Proposition 2.5 to the function h with b 1 = λ 3 and b 2 = 1 and using again

12 12 I. BOUTAAYAMOU, Y. ECHARROUDI EJDE-217/131 Caccioppoli s ineuality (.1, we infer that 1 A T λ 3 ( 1 C + C ( + C( + (s 3 φ 3 z 2 + sφz 2 xe 2sΦ dt da dx A T λ 3 A T A T (ηh + (η x w x + η x w x 2 e 2sΦ dt da dx s 3 Θ 3 w 2 e 2sΦ dt da dx h 2 e 2sΦ + ((η x w x + η x w x 2 e 2sΦ dt da dx A T s 3 Θ 3 w 2 e 2sΦ dt da dx h 2 e 2sΦ dt da dx + A T s 3 Θ 3 w 2 e 2sΦ dt da dx C ( 1 h 2 e 2sΦ dt da dx + + A T A T s 3 Θ 3 w 2 e 2sΦ dt da dx C ( 2 h 2 e 2sΦ dt da dx + A T (8(η x 2 w 2 x + 2((η x x 2 w 2 e 2sΦ dt da dx (w 2 x + w 2 e 2sΦ dt da dx s 3 Θ 3 w 2 e 2sΦ dt da dx, (2. with := ( λ3+2λ2 3, λ 2. By the restriction (2.31 there exists c 3 > such that, for (t, a, x [, T ] [, A] [λ 3, 1], we have Θ(xe 2sϕ c 3 φe 2sΦ, Θ 3 (x x 2 e 2sϕ c 3 φ 3 e 2sΦ. (x Then 1 A T λ 3 c 3 1 (sθzx 2 + s 3 Θ 3 (x x 2 z 2 e 2sϕ dt da dx A T λ 3 (s 3 φ 3 z 2 + sφz 2 xe 2sΦ dt da dx. (2.5 This ineuality and (2. lead to 1 A T (sθzx 2 + s 3 Θ 3 (x x 2 z 2 e 2sϕ dt da dx λ 3 ( A T c 3 h 2 e 2sΦ dt da dx + s 3 Θ 3 w 2 e 2sΦ dt da dx. (2.6

13 EJDE-217/131 NULL CONTROLLABILITY 13 Taing into account the definition of η (2.2, we can say that Hence, 1 A T λ 2 1 A T = λ 2 1 A T (sθzx 2 + s 3 Θ 3 (x x 2 z 2 e 2sϕ dt da dx (sθwx 2 + s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx. (sθwx 2 + s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx λ 2 ( A T c 3 h 2 e 2sΦ dt da dx + s 3 Θ 3 w 2 e 2sΦ dt da dx, (2.7 (2.8 as a conseuence of (2.6 and (2.7. Arguing in the same way for (, λ 1, one can show that λ1 A T (sθwx 2 + s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx ( A T (2.9 c 3 h 2 e 2sΦ dt da dx + s 3 Θ 3 w 2 e 2sΦ dt da dx, Finally, summing the ineualities (2.1, (2.8 and (2.9 side by side, taing s large and using again the restriction on c 1, (2.31, we arrive at 1 A T (sθwx 2 + s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx ( A T c 3 h 2 e 2sΦ dt da dx + and this is exactly the desired estimate (2.32. s 3 Θ 3 w 2 e 2sΦ dt da dx, (2.5 Before providing the main Carleman estimate, we mae the following remars. Remar The proof of our distributed-carleman estimate (2.32 is based on the cut-off functions and given by two different weighted functions ϕ and Φ, in addition by (2.31 there is no positive constant C such that e 2sΦ Ce 2sϕ. 2. Our proof is not based on the reflection method used for the proof of [22, Lemma.1] which is needed to eliminate the boundary term arising in the classical Carleman estimate for nondegenerate heat euation. By the Carleman estimate (2.32, we are able to show the following -Carleman estimate for the full adjoint system (2.3. Theorem 2.7. Assume that (2.1, (2.2 and (2.5 hold. Let A > and T > be given such that < T < δ, where δ (, A small enough. Then, there exist positive constants C and s such that for all s s, every solution w of (2.7 satisfies (sθwx 2 + s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx ( 1 δ (2.51 C s 3 Θ 3 w 2 e 2sΦ dt da dx + wt 2 (a, x dx dx,

14 1 I. BOUTAAYAMOU, Y. ECHARROUDI EJDE-217/131 for all s s and δ verifying (2.2. Proof. Applying ineuality (2.32 to the function h(t, a, x = β(t, a, xw(t,, x, we have the existence of two positive constants C and s such that, for all s s, the following ineuality holds s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx + s Θ(xw (x xe 2 2sϕ dt da dx ( C β 2 w 2 (t,, xe 2sΦ dt da dx + s 3 Θ 3 w 2 e 2sΦ dt da dx (2.52 ( C A β 2 1 T w 2 (t,, x dt dx + s 3 Θ 3 w 2 e 2sΦ dt da dx, using (2.2. On the other hand, integrating over the characteristics lines and after a careful calculus we obtain the following implicit formula of w, the solution of (2.3, A a S(A a lβ(t, A l, w(t,, dl if a > t + (A T w(t, a, = S(T tw T (T + (a t, + (2.53 T S(l tβ(l, a, w(l,, dl t if a t + (A T, where (S(t t is the semi-group generated by the operator A 2 w = (w x x µw. Thus, w(t,, = S(T tw T (T t,, (2.5 using the last hypothesis in (2.2 on β. Inserting this formula in (2.52 and using the fact (S(t t is a bounded semi-group, we obtain s 3 Θ 3 (x x 2 w 2 e 2sϕ dt da dx + s Θ(xw (x xe 2 2sϕ dt da dx ( 1 T Ĉ wt 2 (T t, x dt dx + s 3 Θ 3 w 2 e 2sΦ dt da dx ( 1 Ĉ ( 1 Ĉ T δ wt 2 (m, x dm dx + wt 2 (m, x dm dx + s 3 Θ 3 w 2 e 2sΦ dt da dx s 3 Θ 3 w 2 e 2sΦ dt da dx, (2.55 since T (, δ with δ (, A small enough and this achieves the proof of ( Observability ineuality and null controllability 3.1. Observability ineuality. The objective of this paragraph is to reach the observability ineuality of the adjoint system (2.3. To attain this purpose, we combine the Carleman estimate (2.51 with the Hardy-Poincaré ineuality stated in [22, Proposition 2.3] and arguing in a similar way as in [2]. Our observability ineuality is given by the following proposition. Proposition 3.1. Assume that (2.1, (2.2 and (2.5 hold. Let A > and T > be given such that < T < δ, where δ (, A is small enough. Then, there

15 EJDE-217/131 NULL CONTROLLABILITY 15 exists a positive constant C δ such that for every solution w of (2.3, the following observability ineuality holds 1 A ( 1 δ w 2 (, a, x dx dx C δ w 2 dt da dx + wt 2 (a, x dx dx. (3.1 Proof. Let w be a solution of (2.3. Then for κ > to be defined later, let w = e κt w be a solution of w t + w a + ((x w x x (µ(t, a, x + κ w = β w(t,, x, w(t, a, 1 = w(t, a, =, (3.2 w(t, a, x = e κt w T (a, x, w(t, A, x =. We point out that the parameter κ considered here is not the same as in (2.28. Multiplying the first euation of (3.2 by w and integrating by parts on t = (, t (, A (, 1. Then, one obtains 1 w 2 (t, a, x dx dx + 1 w 2 (, a, x dx dx t w 2 (τ,, xdτdx 2 A 2 A 2 + κ 1 A t β 2 ɛ 1 A t w 2 (τ, a, xdτ dx dx w 2 (τ, a, xdτ dx dx + ɛ A 1 t w 2 (τ,, xdτdx. (3.3 Thus, for κ = β 2 ɛ and ɛ < 1 2A, integrating over ( T, 3T we obtain 3T/ w 2 (, a, x dx dx C 12 e 2κT w 2 (t, a, x dx dx. (3. A On the other hand, let us prove that there exists a positive constant C δ such that 1 δ 3T 3T/ T/ A T/ w 2 (t, a, x dt da dx C δ 1 δ w 2 T (a, x dx dx. (3.5 For this purpose, we use the implicit formula of w defined by (2.53 and we discuss the two cases: when a > t + (A T and when a t + (A T. In fact, if a > t + (A T one has w(t, a, = = A a A a S(A a lβ(t, A l, w(t,, dl S(A a lβ(t, A l, S(T tw T (T t, dl, (3.6 using (2.5. Since (S(t t is a bounded semi-group and β L (, one can see that for T (, δ 1 δ 3T 3T/ T/ w 2 (t, a, x dt da dx C 1 1 C 1 1 3T/ T/ δ w 2 T (T t, x dt dx w 2 T (m, x dm dx, (3.7

16 16 I. BOUTAAYAMOU, Y. ECHARROUDI EJDE-217/131 Now, if a t + (A T one has w(t, a, = S(T tw T (T + (a t, + = S(T tw T (T + (a t, + T t T S(l tβ(l, a, S(T lw T (T l, dl. t S(l tβ(l, a, w(l,, dl Thans to the same argument employed to get (3.7, we conclude that 1 δ 3T 2 C 11 ( + 3T/ T/ 1 δ 3T 3T/ 1 T On the one hand, we can chec that 1 t 1 δ δ 3T 1 1 w 2 (t, a, x dt da dx T/ w 2 T (T l, x dl dx. δ 3T δ 3T δ 3T 1 δ 3T/ T/ 3T/ T/ a+ 3T a+ T δ w 2 T (T + (a t, x dt da dx w 2 T (T + (a t, x dt da dx w 2 T (a + m, xdm dx dx w 2 T (z, xdz dx dx w 2 T (z, xdz dx dx w 2 T (z, x dz dx. On the other hand, we have the ineuality 1 T t w 2 T (T l, x dl dx = 1 T t 1 δ Combining ineualities (3.9, (3.1 and (3.11 we obtain 1 δ 3T 3T/ T/ w 2 (t, a, x dt da dx C 12 1 w 2 T (z, x dz dx w 2 T (z, x dz dx. δ (3.8 (3.9 (3.1 (3.11 w 2 T (z, x dz dx. (3.12 Subseuently, (3.5 occurs in both studied cases. Therefore, in light of ineuality (3. we conclude that w 2 (, a, x dx dx A C 13 1 δ w 2 T (a, x dx dx + 2e2κT T 1 A δ 3T 3T/ T/ (3.13 w 2 (t, a, x dt da dx.

17 EJDE-217/131 NULL CONTROLLABILITY 17 Now, let p defined by (2.19. Then, using the hypotheses (2.1 on the function x (x x2 p(x is non-increasing at the left of x and nondecreasing at the right of x. Hence, applying Hardy-Poincaré ineuality (see [22, Proposition 2.3] and taing into account the definition of ϕ stated in (2. we have 1 δ w 2 (, a, x dx dx C 13 wt 2 (a, x dx dx A + C 13 δ 1 A δ 3T 3T/ T/ Therefore, using Carleman estimate (2.51 we infer that ( w 2 15 (, a, x dx dx C δ s 3 Θ 3 w 2 e 2sΦ dt da dx + A sθ(xw 2 x(t, a, xe 2sϕ dt da dx. 1 δ wt 2 (a, x dx dx, and then the proof is complete using the fact that sup (t,a,x s d Θ d e 2sΦ < + for all d inr Null controllability. In the previous paragraph, we obtained the observability ineuality of system (2.3. Such a tool will be very useful to prove the null controllability of the model (1.1 in the case where T (, δ as we emphasized in the introduction. Our main result is provided in the following theorem. Theorem 3.2. Assume that the dispersion coefficient satisfies (2.1 and the natural rates β and µ verify (2.2. Let A, T > be given such that < T < δ, where δ (, A small enough. For all y L 2 ( A, there exists a control ϑ L 2 ( such that the associated solution of (1.1 satisfies y(t, a, x =, a.e. in (δ, A (, 1. (3.1 Furthermore, there exists a positive constant C 1 which depends on δ such that ϑ satisfies the ineuality ϑ 2 (t, a, x dt da dx C 1 y(a, 2 x dx dx. (3.15 A The constant C 1 is called the control cost. Before proving, we mae the following remar. Remar 3.3. Ineuality (3.15 shows us clearly that the control that we want depends on δ and the initial distribution y. Proof Theorem 3.2. Let ε > and consider the cost function J ε (ϑ = 1 1 A y 2 (T, a, x dx dx + 1 ϑ 2 (t, a, x dt da dx. 2ε 2 δ We can prove that J ε is continuous, convex and coercive. Then, it admits at least one minimizer ϑ ε and we have ϑ ε = w ε (t, a, xχ (x in, (3.16

18 18 I. BOUTAAYAMOU, Y. ECHARROUDI EJDE-217/131 with w ε is the solution of the system w ε t + w ε a + ((x(w ε x x µ(t, a, xw ε = βw ε (t,, x in, w ε (t, a, 1 = w ε (t, a, = on (, T (, A, w ε (T, a, x = 1 ε y ε(t, a, xχ (δ,a (a in A, w ε (t, A, x = in T, (3.17 and y ε is the solution of the system (1.1 associated to the control ϑ ε. Multiplying (3.17 by y ε, integrating over, using (3.16 and the Young ineuality we obtain 1 1 A y ε ε(t, 2 a, x dx dx + ϑ 2 ε(t, a, x dt da dx δ = y (a, xw ε (, a, x dx dx (3.18 A 1 w C ε(, 2 a, x dx dx + C δ y(a, 2 x dx dx, δ A A with C δ is the constant of the observability ineuality (3.1. This again leads to 1 1 A y ε ε(t, 2 a, x dx dx + ϑ 2 ε(t, a, x dt da dx δ 1 (3.19 w 2 dt da dx + C δ y (a, 2 x dx dx. A Keeping in the mind (3.16, we conclude that 1 1 A y 2 ε ε(t, a, x dx dx + 3 ϑ 2 ε(t, a, x dt da dx C δ y(a, 2 x dx dx. A δ Hence, it follows that 1 A yε(t, 2 a, x dx dx εc δ y(a, δ 2 x dx dx A ϑ 2 ε(t, a, x dt da dx C δ y 3 (a, 2 x dx dx. A (3.2 Then, we can extract two subseuences of y ε and ϑ ε denoted also by ϑ ε and y ε that converge wealy towards ϑ and y in L 2 ( and L 2 ((, T (, A; H 1 (, 1 respectively. Now, by a variational technic, we prove that y is a solution of (1.1 corresponding to the control ϑ and, by the first estimate of (3.2, y satisfies (3.1 for T (, δ and this shows our claim.. Appendix As we said in the introduction, this Appendix concerns a result which plays an important role to show the -Carleman estimate associated to the full adjoint system (2.3 namely the Caccioppoli s ineuality which is stated in the following lemma.

19 EJDE-217/131 NULL CONTROLLABILITY 19 Lemma.1. Let and w be the solution of (2.7. Suppose that x /. Then, there exists a positive constant C such that w satisfies A T wxe 2 2sϕ dt da dx ( (.1 C s 2 Θ 2 w 2 e 2sϕ dt da dx + h 2 e 2sϕ dt da dx. Proof. Define the smooth cut-off function ζ : R R by For the solution w of (2.7, we have = T = 2s = 2s d [ 1 dt A 1 A T A T 1 A T ζ(x 1, x R, ζ(x =, x < x 1 and x > x 2, ] ζ 2 e 2sϕ w 2 dx dx dt ζ(x = 1, x. ζ 2 ϕ t w 2 e 2sϕ dt da dx + 2 ζ 2 ϕ t w 2 e 2sϕ dt da dx 1 A T ζ 2 w( (w x x w a + h + µwe 2sϕ dt da dx. ζ 2 ww t e 2sϕ dt da dx Then, integrating by parts we obtain 2 ζ 2 e 2sϕ wx 2 dt da dx = 2s ζ 2 w 2 ψ(θ a + Θ t e 2sϕ dt da dx 2 ζ 2 whe 2sϕ dt da dx 2 ζ 2 µw 2 e 2sϕ dt da dx + ((ζ 2 e 2sϕ x x w 2 dt da dx. (.2 (.3 On the other hand, by the definitions of ζ, ψ and Θ, thans to Young ineuality, taing s uite large and using the fact that x /, one can prove the existence of a positive constant c such that A T 2 ζ 2 e 2sϕ wx 2 dt da dx 2 min (x wxe 2 2sϕ dt da dx, x A T ((ζ 2 e 2sϕ x x w 2 dt da dx c s 2 Θ 2 w 2 e 2sϕ dt da dx, 2s ζ 2 w 2 ψ(θ a + Θ t e 2sϕ dt da dx c 2 ζ 2 whe 2sϕ dt da dx ( c A T s 2 Θ 2 w 2 e 2sϕ dt da dx + A T A T s 2 Θ 2 w 2 e 2sϕ dt da dx, h 2 e 2sϕ dt da dx,

20 2 I. BOUTAAYAMOU, Y. ECHARROUDI EJDE-217/131 2 ζ 2 µw 2 e 2sϕ dt da dx c A T s 2 Θ 2 w 2 e 2sϕ dt da dx. This implies that there is C > such that A T ( wxe 2 2sϕ dt da dx C s 2 Θ 2 w 2 e 2sϕ dt da dx + Thus, the proof is complete. h 2 e 2sϕ dt da dx. Remar.2. Lemma.1 remains valid for any function π C([, 1], (, C 1 ([, 1]\{x }, (, satisfying π x c, for x [, 1]\{x }, where c >. see [22, Proposition.2] for more details. Acnowledgements. The authors would lie to than the anonymous referee and the Professors B. Ainseba and L. Maniar for their fruitful remars which allow us to realize this wor. References [1] B. Ainseba; Corrigendum to Exact and approximate controllability of the age and space population dynamics structured model [J. Math. Anal. Appl. 275 (22, ], J. Math. Anal. Appl., 393 (212, 328. [2] B. Ainseba; Exact and approximate controllability of the age and space population dynamics structured model, J. Math. Anal. Appl., 275 (22, [3] B. Ainseba, S. Anita; Internal exact controllability of the linear population dynamics with diffusion, Electronic Journal of Differential Euations, 2 (2, [] B. Ainseba, S. Anita; Local exact controllability of the age-dependent population dynamics with diffusion, Abstr. Appl. Anal., 6 (21, [5] B. Ainseba, Y. Echarroudi, L. Maniar; Null controllability of a population dynamics with degenerate diffusion, Journal of Differential and Integral Euations, 26 (213, [6] F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli; Carleman estimates for degenerate parabolic operators with applications to null controllability, J. evol. eu., 6 (26, [7] V. Barbu, M. Iannelli, M. Martcheva; On the controllability of the Lota-McKendric model of population dynamics, J. Math. Anal. Appl., 253 (21, [8] A. Bátai, P. Csomós, B. Faras, G. Nicel; Operator splitting for non-autonomous evolution euations, Journal of Functional Analysis, 26 (211, [9] A. Bátai, S. Piazzera; Semigroups for delay euations, Research Notes in Mathematics; 1, Wellesley, MA : Peters, 25, 259 pp. [1] I. Boutaayamou, G. Fragnelli, L. Maniar; Carleman estimates for parabolic euations with interior degeneracy and Neumann boundary conditions, Accepted for publication in J. Anal. Math. [11] P. Cannarsa, P. Martinez and J. Vancostenoble; Null controllability of degenerate heat euations, Adv. Differential Euations. 1 (25, pp [12] P. Cannarsa, G. Fragnelli; Null controllability of semilinear degenerate parabolic euations in bounded domains, Electron. J. Differential Euations, 26 (26, No. 136, 1 2. [13] P. Cannarsa, G. Fragnelli, D. Rocchetti; Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Eu., 8 (28, [1] P. Cannarsa, G. Fragnelli, D. Rocchetti; Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (27, [15] P. Cannarsa, G. Fragnelli, J. Vancostenoble; Regional controllability of semilinear degenerate parabolic euations in bounded domains, J. Math. Anal. Appl. 32 (26, [16] P. Cannarsa, G. Fragnelli, J. Vancostenoble; Linear degenerate parabolic euations in bounded domains: controllability and observability, IFIP Int. Fed. Inf. Process. 22 (26, , Springer, New Yor.

21 EJDE-217/131 NULL CONTROLLABILITY 21 [17] Y. Echarroudi, L. Maniar; Null controllability of a model in population dynamics, Electronic Journal of Differential Euations, 21 (21, No. 2, 1 2. [18] G. Fragnelli; An age dependent population euation with diffusion and delayed birth process, International Journal of Mathematics and Mathematical Sciences, 2 (25, [19] G. Fragnelli; Null controllability of degenerate parabolic euations in non divergence form via Carleman estimates, Discrete Contin. Dyn. Syst. Ser. S, 6 (213, [2] G. Fragnelli, A. Idrissi, L. Maniar; The asymptotic behaviour of a population euation with diffusion and delayed birth process, Discrete and Continuous Dynamical Systems-Series B, 7 (27, No., [21] G. Fragnelli, P. Martinez, J. Vancostenoble; ualitative properties of a population dynamics describing pregnancy, Math. Models Methods Appl. Sci., 15, 57 (25. DOI: 1.112/S [22] G. Fragnelli, D. Mugnai; Carleman estimates and observability ineualities for parabolic euations with interior degeneracy, Advances in Nonlinear Analysis 8/213; 2(: DOI: /anona [23] G. Fragnelli, D. Mugnai; Carleman estimates, observability ineualities and null controllability for interior degenerate non smooth parabolic euations, to appear in Mem. Amer. Math. Soc. ArXiv: [2] G. Fragnelli, G. Ruiz Goldstein, J. A. Goldstein, S. Romanelli; Generators with interior degeneracy on spaces of L2 type, Electron. J. Differential Euations 212 (212, 1 3. [25] G. Fragnelli, L. Tonetto; A population euation with diffusion, J. Math. Anal. Appl., 289 (2, [26] A. V. Fursiov, O. Yu. Imanuvilov; Controllability of Evolution Euations, Lecture Notes Series, vol. 3, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, [27] M. Langlais; A nonlinear problem in age-dependent population diffusion, Siam J. Math. Anal., 16 (1985, [28] S. Piazzera; An age-dependent population euation with delayed birth process, Mathematical Methods in the Applied Sciences, Vol. 27, Issue (2, [29] M. A. Pozio, A. Tesei; Degenerate parabolic Problems in population dynamics, Japan Journal of Applied Mathematics, December 1985, 2:351 [3] G. Nicel; Evolution semigroups for nonautonomous Cauchy problems, Abstr. Appl. Anal. 2 ( [31] A. Rhandi, R. Schnaubelt; Asymptotic behaviour of a non-autonomous population euation with diffusion in L 1, Discrete Contin. Dynam. Systems, 5 (1999, [32] G. F. Webb; Population models structured by age, size, and spatial position. Structured population models in biology and epidemiology, 1 9, Lecture Notes in Math. 1936, Springer, Berlin, 28. Idriss Boutaayamou Faculté polydisciplinaire de Ouarzazate, Université Ibn Zohr, B.P. 638, Ouarzazate 5, Morocco address: dsboutaayamou@gmail.com Younes Echarroudi Private University of Marraesh, Km 13 Route d Amizmiz, Marraesh, Morocco address: yecharroudi@gmail.com, y.echarroudi@upm.ac.ma