Null controllability of a cascade model in population dynamics
aa r X i v : . [ m a t h . O C ] J a n NULL CONTROLLABILITY OF A CASCADE MODEL INPOPULATION DYNAMICS
BEDR’EDDINE AINSEBA, YOUNES ECHARROUDI AND LAHCEN MANIAR
Abstract.
In this paper, we are concerned with the null controllability of a linearpopulation dynamics cascade systems (or the so-called prey-predator models) with twodifferent dispersion coefficients which degenerate in the boundary and with one controlforce. We develop first a Carleman type inequality for its adjoint system, and then anobservability inequality which allows us to deduce the existence of a control acting on asubset of the space domain which steers both populations of a certain age to extinctionin a finite time. Introduction
We consider the coupled population cascade system ∂y∂t + ∂y∂a − ( k ( x ) y x ) x + µ ( t, a, x ) y = ϑχ ω in Q, (1.1) ∂p∂t + ∂p∂a − ( k ( x ) p x ) x + µ ( t, a, x ) p + µ ( t, a, x ) y = 0 in Q,y ( t, a,
1) = y ( t, a,
0) = p ( t, a,
1) = p ( t, a,
0) = 0 on (0 , A ) × (0 , T ) ,y (0 , a, x ) = y ( a, x ); p (0 , a, x ) = p ( a, x ) in Q A ,y ( t, , x ) = Z A β ( t, a, x ) y ( t, a, x ) da in Q T ,p ( t, , x ) = Z A β ( t, a, x ) p ( t, a, x ) da in Q T , where Q = (0 , T ) × (0 , A ) × (0 , Q A = (0 , A ) × (0 , Q T = (0 , T ) × (0 ,
1) and we willdenote q = (0 , T ) × (0 , A ) × ω . The system (1.1) models the dispersion of a gene in twogiven populations which are in interaction. In this case, x represents the gene type and y ( t, a, x ) and p ( t, a, x ) as the distributions of individuals of age a at time t and of genetype x of both populations. The parameters β ( t, a, x ) (respectively β ( t, a, x )), µ ( t, a, x ) Mathematics Subject Classification.
Key words and phrases.
Degenerate population dynamics model, cascade systems, Carleman estimate,observability inequality, null controllability.Institut de Math´ematiques de Bordeaux, UMR-CNRS 5251, Universit´e Bordeaux Segalen, 3 Place dela Victoire, 33076 Bordeaux Cedex, France, e-mail: [email protected] university of Marrakesh, Km 13 Route d’Amizmiz, Marrakesh, Morocco,e-mail: [email protected],D´epartement de Math´ematiques, Facult´e des Sciences Semlalia, Laboratoire LMDP, UMMISCO (IRD-UPMC), B. P. 2390 Marrakech 40000, Maroc, e-mail: [email protected]. (respectively µ ( t, a, x )) are respectively the natural fertility and mortality rates of indi-viduals of age a at time t and of gene type x of the population whose distribution is y (respectively p ), µ can be interpreted as the interaction coefficient between two popula-tions (cancer cells and healthy cells for instance) which depends on x , t and a , the subset ω is the region where a control ϑ is acting. Such a control corresponds to an externalsupply or to removal of individuals on the subdomain ω . Finally, R A β ( t, a, x ) y ( t, a, x ) da and R A β ( t, a, x ) p ( t, a, x ) da are the distributions of the newborns of the two populationsthat are of gene type x at time t .The control problems of (1.1) or in general of coupled systems take an intense inter-est and are widely investigated in many papers, among them we find [3], [7], [17] and thereferences therein. In fact, in [3] the authors studied a coupled reaction-diffusion equationsdescribing interaction between a prey population and predator population. The goal ofthis work was to look for a suitable control supported on a small spatial subdomain whichguarantees the stabilization of the predator population to zero. In [17], the objective wasdifferent. More precisely, the authors considered an age-dependent prey-predator systemand they proved the existence and uniqueness for an optimal control (called also ”optimaleffort”) which gives the maximal harvest via the study of the optimal harvesting problemassociated to their coupled model.However, the previous results were found in the case when the diffusion coefficients areconstants. This leads Ait Ben Hassi et al. in [7] to generalize the model of [3] and in-vestigate a semilinear parabolic cascade systems with two different diffusion coefficientsallowed to depend on the space variable and degenerate at the left boundary of the spacedomain. Moreover, the purpose of this paper was to show the null controllability via aCarleman type inequality of the adjoint problem of the associated linearized system usingthe results of [8] (or [12]) and with the help of the Schauder fixed point theorem. Onthe other hand, a massive interest was given to the question of null controllability of thepopulation dynamics models in the case of one equation both in the case without diffu-sion (see for example [9]) and with diffusion (see for instance [1, 2, 4, 5, 15] in the caseof a constant diffusion coefficient). Recently, a more general case was investigated by B.Ainseba and al. in [6] and [13]. Indeed, in [6] the authors allowed the dispersion coeffi-cient to depend on the variable x and verifies k (0) = 0 (i.e, the coefficient of dispersion k degenerates at 0) and they tried to obtain the null controllability in such a situation with β ∈ L ∞ basing on the work done in [8] for the degenerate heat equation to establish anew Carleman estimate for a suitable full adjoint system and afterwards his observabilityinequality. However, the main controllability result of [6] was shown under the condition T ≥ A (as in [9]) and this constitutes a restrictiveness on the ”optimality” of the controltime T since it means, for example, that for a pest population whose the maximal age A may equal to a many days (may be many months or years) we need much time to bringthe population to the zero equilibrium. In the same trend and to overcome the condition T ≥ A , L. Maniar et al in [13] suggested the fixed point technique implemented in [15]and which requires that the fertility rate must belong to C ( Q ) and consists briefly todemonstrate in a first time the null controllability for an intermediate system with a fer-tility function b ∈ L ( Q T ) instead of R A β ( t, a, x ) y ( t, a, x ) da and to achieve the task via aLeray-Schauder theorem.But up now, little is known about the null controllability question of population dynamics ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 3 cascade systems both in degenerate and nondegenerate cases to our knowledge and thework done in this paper will address to such a control problem and it will be a general-ization of the results established in [6] and [13]. More precisely, following the strategy of[7] we expect in this contribution to prove the null controllability of system (1.1) when T ∈ (0 , δ ) where δ ∈ (0 , A ) small enough in the case of one control force. That is, we showthat for all y , p ∈ L ( Q A ) and δ ∈ (0 , A ) small enough, there exists a control ϑ ∈ L ( q )such that the associated solution of (1.1) verifies ( y ( T, a, x ) = 0 , a.e. in ( δ, A ) × (0 , ,p ( T, a, x ) = 0 , a.e in ( δ, A ) × (0 , . (1.2)Such a result is gotten under the conditions that all the natural rates possess an L ∞ − regularity(see (2.4) beneath) and the dispersion coefficients are different and depend on the genetype with a degeneracy in the left hand side of its domain, i.e k i (0) = 0; i = 1 , k i = x α , α > Well-posedness and Carleman estimates
Well-posedness result.
For this section and for the sequel, we assume that thedispersion coefficients k i , i = 1 , (cid:26) k i ∈ C ([0 , ∩ C ((0 , , k i > ,
1] and k i (0) = 0 , ∃ γ ∈ [0 ,
1) : xk ′ i ( x ) ≤ γk i ( x ) , x ∈ [0 , . (2.3)The last hypothesis on k i means in the case of k ( x ) = x α i that 0 ≤ α i <
1. Similarly,all results of this paper can be obtained also in the case of 1 ≤ α i < x = 0, the Newmann condition ( k i ( x ) u x )(0) = 0. On the otherhand, we assume that the rates µ , µ , µ , β and β verify ( µ , µ , µ , β , β ∈ L ∞ ( Q ) , µ , µ , µ , β , β ≥ Q,β i ( ., , . ) ≡ , T ) × (0 , , for i = 1 , . (2.4)The third assumption in (2.4) on the fertility rates β and β is natural since the newbornsare not fertile.As in [13], we discuss the well-posedness of (1.1) by introducing the weighted spaces H k i (0 ,
1) and H k i (0 ,
1) defined by ( H k i (0 ,
1) := { u ∈ L (0 ,
1) : u is abs. cont. in [0 ,
1] : √ k i u x ∈ L (0 , , u (1) = u (0) = 0 } ,H k i (0 ,
1) := n u ∈ H k (0 ,
1) : k i ( x ) u x ∈ H (0 , o , BEDR’EDDINE AINSEBA, YOUNES ECHARROUDI AND LAHCEN MANIAR endowed respectively with the norms ( k u k H ki (0 , := k u k L (0 , + k√ k i u x k L (0 , , u ∈ H k i (0 , , k u k H ki := k u k H ki (0 , + k ( k i ( x ) u x ) x k L (0 , , u ∈ H k i (0 , , with i = 1 , C i u := ( k i ( x ) u x ) x , u ∈ D ( C i ) = H k i (0 , , i =1 , L (0 , H = ( L ((0 , A ) × (0 , , the system (1.1) canbe rewritten abstractly as an inhomogeneous Cauchy problem in the following way X ′ ( t ) = A X ( t ) + B ( t ) X ( t ) + f ( t ) ,X (0) = y p ! , (2.5)where X ( t ) = (cid:18) y ( t ) p ( t ) (cid:19) , A = (cid:18) A A (cid:19) ; D ( A ) = D ( A ) × D ( A ), f ( t ) = (cid:18) ϑ ( t, ., · ) χ ω ( . )0 (cid:19) , B ( t ) = (cid:18) M µ ( t ) M µ ( t ) M µ ( t ) (cid:19) , where M µ j ( t ) w = − µ j ( t ) w , theoperators A : L ((0 , A ) × (0 , → L ((0 , A ) × (0 , A : L ((0 , A ) × (0 , → L ((0 , A ) × (0 , ( A θ ( a, x ) = − ∂θ∂a + ( k ( x ) θ x ) x , ∀ θ ∈ D ( A ) ,D ( A ) = { θ ( a, x ) : θ, A θ ∈ L ((0 , A ) × (0 , , θ ( a,
0) = θ ( a,
1) = 0 , θ (0 , x ) = R A β ( a, x ) θ ( a, x ) da } , (2.6)and ( A θ ( a, x ) = − ∂θ∂a + ( k ( x ) θ x ) x , ∀ θ ∈ D ( A ) ,D ( A ) = { θ ( a, x ) : θ, A θ ∈ L ((0 , A ) × (0 , , θ ( a,
0) = θ ( a,
1) = 0 , θ (0 , x ) = R A β ( a, x ) θ ( a, x ) da } . (2.7)It is well-known, from [16] and the references therein that the operators A and A definedabove generate a C − semigroups. On the other hand, one can see that the operator A is diagonal and B ( t ) is a bounded perturbation. Therefore, the following well-posednessresult holds (see for instance [7] for a similar result of cascade parabolic equations). Theorem 2.1. i ) The operator A generates a C − semigroup. ii ) Under the assumptions (2.3) and (2.4) and for all ϑ ∈ L ( Q ) and ( y , p ) ∈ ( L ( Q A )) ,the system (1.1) admits a unique solution ( y, p ) . This solution belongs to E := C ([0 , T ] , ( L ((0 , A ) × (0 , ) ∩ C ([0 , A ] , ( L ((0 , T ) × (0 , ) ∩ L ((0 , T ) × (0 , A ) , H k (0 , × H k (0 , . More-over, the solution of (1.1) satisfies the followinginequality sup t ∈ [0 ,T ] k ( y ( t ) , p ( t )) k L ( Q A ) × L ( Q A ) + sup a ∈ [0 ,A ] k ( y ( a ) , p ( a )) k L ( Q T ) × L ( Q T ) + Z Z A Z T (( p k y x ) + ( p k p x ) ) dtdadx ≤ C (cid:18)Z q ϑ dtdadx + k ( y , p ) k L ( Q A ) × L ( Q A ) (cid:19) . (2.8) ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 5
Carleman inequality results.
In this paragraph, we show a Carleman type in-equality for the following adjoint system of (1.1) ∂u∂t + ∂u∂a + ( k ( x ) u x ) x − µ ( t, a, x ) u − µ ( t, a, x ) v = − β ( t, a, x ) u ( t, , x ) in Q, (2.9) ∂v∂t + ∂v∂a + ( k ( x ) v x ) x − µ ( t, a, x ) v = − β ( t, a, x ) v ( t, , x ) in Q,u ( t, a,
1) = u ( t, a,
0) = v ( t, a,
1) = v ( t, a,
0) = 0 on (0 , T ) × (0 , A ) ,u ( T, a, x ) = u T ( a, x ) in Q A ,v ( T, a, x ) = v T ( a, x ) in Q A ,u ( t, A, x ) = v ( t, A, x ) = 0 in Q T . To do this, we prove firstly the Carleman estimate for the following intermediate system ∂u∂t + ∂u∂a + ( k ( x ) u x ) x − µ ( t, a, x ) u − µ ( t, a, x ) v = h in Q, (2.10) ∂v∂t + ∂v∂a + ( k ( x ) v x ) x − µ ( t, a, x ) v = h in Q,u ( t, a,
1) = u ( t, a,
0) = v ( t, a,
1) = v ( t, a,
0) = 0 on (0 , T ) × (0 , A ) ,u ( T, a, x ) = u T ( a, x ) in Q A ,v ( T, a, x ) = v T ( a, x ) in Q A ,u ( t, A, x ) = v ( t, A, x ) = 0 in Q T , with ( u T , v T ) ∈ ( L ( Q A )) and h , h ∈ L ( Q ). Such a system can be rewritten in thefollowing way ∂u∂t + ∂u∂a + ( k ( x ) u x ) x − µ ( t, a, x ) u = h + µ ( t, a, x ) v in Q, (2.11) u ( t, a,
1) = u ( t, a,
0) = 0 on (0 , T ) × (0 , A ) ,u ( T, a, x ) = u T ( a, x ) in Q A ,u ( t, A, x ) = 0 in Q T , where v is the solution of ∂v∂t + ∂v∂a + ( k ( x ) u x ) x − µ ( t, a, x ) v = h in Q, (2.12) v ( t, a,
1) = v ( t, a,
0) = 0 on (0 , T ) × (0 , A ) ,v ( T, a, x ) = v T ( a, x ) in Q A ,v ( t, A, x ) = 0 in Q T . Classically, the proof of such a kind of estimates is based tightly on the choice of theso-called weight functions. In our case, these functions are set in the following way ϕ i ( t, a, x ) := Θ( t, a ) ψ i ( x ) , i = 1 , , Θ( t, a ) := 1( t ( T − t )) a ,ψ i ( x ) := λ i (cid:16)R x rk i ( r ) dr − d i (cid:17) ,φ ( t, a, x ) = Θ( a, t ) e κσ ( x ) , Φ( t, a, x ) = Θ( a, t )Ψ( x ) , Ψ( x ) = e κσ ( x ) − e κ k σ k ∞ , (2.13) BEDR’EDDINE AINSEBA, YOUNES ECHARROUDI AND LAHCEN MANIAR where σ is the function given by (cid:26) σ ∈ C ([0 , , σ ( x ) > , , σ (0) = σ (1) = 0 ,σ x ( x ) = 0 in [0 , \ ω , (2.14) ω ⋐ ω is an open subset. The existence of this function is proved in [14, Lemma 1.1]. λ i , d i for i = 1 , κ are supposed to verify following assumptions ( d > k (1)(2 − γ ) , λ λ ≥ d d − R rk r ) dr ,κ ≥ k σ k ∞ , d ≥ k (1)(2 − γ ) , (2.15)with λ ∈ I = [ k (1)(2 − γ )( e κ k σ k∞ − d k (1)(2 − γ ) − , e κ k σ k∞ − e κ k σ k∞ )3 d ) which can be shown not empty (seeLemma 4.3 in the appendix). On other hand, in the light of the first and the fourthconditions in (2.15) on d and d , one can observe that ψ i ( x ) < x ∈ [0 , t, a ) → + ∞ as t → + , T − and a → + .Now, we state the first result of this section which is the intermediate Carleman estimatesatisfied by solution of system (2.10). Theorem 2.2.
Assume that k i satisfy the hypotheses (2.3) and let A > and T > begiven. Then, there exist two positive constants C and s , such that every solution ( u, v ) of (2.10) satisfies, for all s ≥ s , the following inequality Z Q (cid:18) s Θ x k ( x ) u + s Θ k ( x ) u x (cid:19) e sϕ dtdadx + Z Q (cid:18) s Θ x k ( x ) v + s Θ k ( x ) v x (cid:19) e sϕ dtdadx ≤ C (cid:18)Z Q ( h + h ) e s Φ dtdadx + Z q s Θ ( u + v ) e s Φ dtdadx (cid:19) . (2.16)The proof of Theorem 2.2 needs two basic results. These results are concerned withCarleman type inequalities in both cases degenerate and nondegenerate. The first one isstated in the following proposition Proposition 2.3.
Consider the following system with h ∈ L ( Q ) , µ ∈ L ∞ ( Q ) and k verifies the hypotheses (2.3) ∂u∂t + ∂u∂a + ( k ( x ) u x ) x − µ ( t, a, x ) u = h, (2.17) u ( t, a,
1) = u ( t, a,
0) = 0 ,u ( T, a, x ) = u T ( a, x ) ,u ( t, A, x ) = 0 . Then, there exist two positive constants C and s , such that every solution of (2.17) satisfies, for all s ≥ s , the following inequality s Z Q Θ x k ( x ) u e sϕ dtdadx + s Z Q Θ k ( x ) u x e sϕ dtdadx (2.18) ≤ C (cid:18)Z Q | h | e sϕ dtdadx + sk (1) Z A Z T Θ u x ( a, t, e sϕ ( a,t, dtda (cid:19) , ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 7 where ϕ and Θ are the weight functions defined by ϕ ( t, a, x ) := Θ( t, a ) ψ ( x ) with : Θ( t, a ) := t ( T − t )) a ,ψ ( x ) := c ( R x rk ( r ) dr − c ) . (2.19) with c > k (1)(2 − γ ) , c > and γ is the parameter defined by (2.3) . For the proof of this proposition, we refer the reader to [13, Proposition 3.1]. Thesecond result is the following
Proposition 2.4.
Let us consider the following system ∂z∂t + ∂z∂a + ( k ( x ) z x ) x − c ( t, a, x ) z = h in Q b , (2.20) z ( t, a, b ) = z ( t, a, b ) = 0 on (0 , T ) × (0 , A ) , where Q b := (0 , T ) × (0 , A ) × ( b , b ) , ( b , b ) ⊂ [0 , , h ∈ L ( Q b ) , k ∈ C ([0 , is astrictly positive function and c ∈ L ∞ ( Q b ) . Then, there exist two positive constants C and s , such that for any s ≥ s , z verifies the following estimate Z Q b ( s φ z + sφz x ) e s Φ dtdadx ≤ C (cid:18)Z Q b h e s Φ dtdadx + Z ω Z A Z T s φ z e s Φ dtdadx (cid:19) , (2.21) where φ , Θ and Φ are defined by (2.13) and σ by (2.14) . For the proof of Proposition 2.4, a careful computations allow us to adapt the sameprocedure of [2, Lemma 2.1] to show (2.21) in case where k is a positive general nonde-generate coefficient, with our weight function Θ( t, a ) = t ( T − t ) a and the source term h .Besides the two Propositions 2.3 and 2.4, we must bring out another important result Lemma 2.5.
Under assumptions (2.15) , the functions ϕ , ϕ and Φ defined by (2.13) satisfy the following inequalities (cid:26) ϕ ≤ ϕ , Φ < ϕ ≤ Φ . (2.22) Proof.
By the definitions of ϕ , ϕ and Φ and taking into account that Θ is positive,showing the results of (2.22) is equivalent to show (cid:26) ψ ≤ ψ , Ψ < ψ ≤ Ψ . (2.23)The first inequality in (2.23) is assured by the second assumption in (2.15) while the secondone is deduced from λ ∈ I = [ k (1)(2 − γ )( e κ k σ k∞ − d k (1)(2 − γ ) − , e κ k σ k∞ − e κ k σ k∞ )3 d ) and this achieves theproof. (cid:3) Now, we can address the proof of Theorem 2.2.
BEDR’EDDINE AINSEBA, YOUNES ECHARROUDI AND LAHCEN MANIAR
Proof.
Let us introduce the smooth cut-off function ξ : R → R defined as follows ≤ ξ ( x ) ≤ , x ∈ R ,ξ ( x ) = 1 , x ∈ [0 , x + x ] ,ξ ( x ) = 0 , x ∈ [ x +2 x , . (2.24)Let u and v be respectively the solutions of (3.75) and (3.76). Set w := ξu , z := ξv andput ω ′ = ( x + x , x +2 x ). Then, ( w, z ) satisfies the following system ∂w∂t + ∂w∂a + ( k ( x ) w x ) x − µ ( t, a, x ) w = µ ( t, a, x ) z + ξh + ( k ξ x u ) x + ξ x k u x in Q, (2.25) ∂z∂t + ∂z∂a + ( k ( x ) z x ) x − µ ( t, a, x ) z = ξh + ( k ξ x v ) x + ξ x k v x in Q,w ( t, a,
1) = w ( t, a,
0) = z ( t, a,
1) = z ( t, a,
0) = 0 on (0 , T ) × (0 , A ) ,w ( T, a, x ) = w T ( a, x ) in Q A ,z ( T, a, x ) = z T ( a, x ) in Q A ,w ( t, A, x ) = z ( t, A, x ) = 0 in Q T . Using Proposition2.3 for the inhomogeneous term ξ ( h + µ v ) + ( k ξ x u ) x + ξ x k u x , thedefinition of ξ and Young inequality, we get the following inequality Z Q ( s Θ k w x + s Θ x k w ) e sϕ dtdadx ≤ C ( Z Q [ ξ ( h + µ v ) + (( k ξ x u ) x + ξ x k u x ) ] e sϕ dtdadx + sk (1) Z A Z T Θ w x ( t, a, e sϕ ( t,a, dtda ) ≤ C Z Q [ µ z + ξ h + (( k ξ x u ) x + ξ x k u x ) ] e sϕ dtdadx. (2.26)Thanks again to the definition of ξ , we have Z (( k ξ x u ) x + ξ x k u x ) e sϕ dx ≤ Z ω ′ (8( k ξ x ) u x + 2(( k ξ x ) x ) u ) e sϕ dx ≤ C Z ω ′ ( u + u x ) e sϕ dx. (2.27)On the other hand, since x k ( x ) is non-decreasing, with the help of Hardy-Poincar´e inequal-ity stated in [8] and since ϕ ≤ ϕ we get Z µ z e sϕ dx ≤ k µ k ∞ k (1) Z k ( x ) x ( ze sϕ ) dx ≤ C k µ k ∞ k (1) Z k ( x )(( ze sϕ ) x ) dx. Thus, from the definition of ψ , we obtain Z µ z e sϕ dx ≤ C Z k ( x ) z x e sϕ dx + C Z s Θ x k ( x ) z e sϕ dx. ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 9
Hence, for s quite large we get Z µ z e sϕ dx ≤ Z s Θ k ( x ) z x e sϕ dx + 12 Z s Θ x k ( x ) z e sϕ dx. (2.28)Combining (2.26), (2.27) and (2.28), for s quite large the following inequality holds Z Q ( s Θ k w x + s Θ x k w ) e sϕ dtdadx (2.29) ≤ C Z Q h e sϕ dtdxda + 12 Z Q ( s Θ k ( x ) z x + s Θ x k ( x ) z ) e sϕ dtdadx + C Z ω ′ Z A Z T ( u + u x ) e sϕ dtdadx. Applying the same way with ξh + ( k ξ x v ) x + ξ x k v x we obtain Z Q ( s Θ k z x + s Θ x k z ) e sϕ dtdadx ≤ C Z Q h e sϕ dtdxda + C Z ω ′ Z A Z T ( v + v x ) e sϕ dtdadx. (2.30)Therefore, for s quite large we conclude by inequalities (2.29) and (2.30) and again ϕ ≤ ϕ that Z Q ( s Θ k w x + s Θ x k w ) e sϕ dtdadx + Z Q ( s Θ k z x + s Θ x k z ) e sϕ dtdadx ≤ C Z Q ( h + h ) e sϕ dtdadx + C Z ω ′ Z A Z T ( u + v + u x + v x ) e sϕ dtdadx. Using Caccioppoli’s inequality (4.87), the last inquality becomes Z Q ( s Θ k w x + s Θ x k w ) e sϕ dtdadx + Z Q ( s Θ k z x + s Θ x k z ) e sϕ dtdadx ≤ C Z Q ( h + h ) e sϕ dtdadx + C Z q s Θ ( u + v ) e sϕ dtdadx. (2.31)Now, let W := ηu and Z := ηv with η = 1 − ξ . Then W and Z are supported in ( x , ∂W∂t + ∂W∂a + ( k ( x ) W x ) x − µ ( t, a, x ) W = µ ( t, a, x ) Z + ηh + ( k η x u ) x + η x k u x in Q x , (2.32) ∂Z∂t + ∂Z∂a + ( k ( x ) Z x ) x − µ ( t, a, x ) Z = ηh + ( k η x v ) x + η x k v x in Q x ,W ( t, a,
1) = W ( t, a, x ) = Z ( t, a,
1) = Z ( t, a, x ) = 0 on (0 , T ) × (0 , A ) ,W ( t, a, x ) = W T ( a, x ) in Q A ,Z ( t, a, x ) = Z T ( a, x ) in Q A ,W ( t, A, x ) = Z ( t, A, x ) = 0 in Q T , where, Q x := (0 , T ) × (0 , A ) × ( x , W and Z is non-degenerate. Hence, applying Proposition 2.4 on the first equation of (2.32) for b = x , b = 1 and h := η ( h + µ v ) + ( k η x u ) x + η x k u x , with the aid of Caccioppoli’s inequalitystated in [13, Lemma 5.1], thanks to the definition of η and Young inequality and taking s quite large we obtain the following estimate Z Q ( s φ W + sφW x ) e s Φ dtdadx ≤ C (cid:18)Z Q ( η ( h + µ v ) + ( kη x u ) x + kη x u x ) e s Φ dtdadx + Z ω Z A Z T s Θ u e s Φ dtdadx (cid:19) ≤ e C (cid:18)Z Q η ( h + µ v ) e s Φ + (( kη x u ) x + kη x u x ) e s Φ dtdadx + Z ω Z A Z T s Θ u e s Φ dtdadx (cid:19) ≤ e C ( Z Q η ( h + µ v ) e s Φ dtdadx + Z ω ′ Z A Z T (8( kη x ) u x + 2(( kη x ) x ) u ) e s Φ dtdadx + Z ω Z A Z T s Θ u e s Φ dtdadx ) ≤ e C (cid:18)Z Q η ( h + µ v ) e s Φ dtdadx + Z ω ′ Z A Z T ( u x + u ) e s Φ dtdadx + Z ω Z A Z T s Θ u e s Φ dtdadx (cid:19) ≤ e C (cid:18)Z Q η ( h + µ v ) e s Φ dtdadx + Z ω Z A Z T s Θ u e s Φ dtdadx (cid:19) ≤ e C (cid:18)Z Q ( h + µ Z ) e s Φ dtdadx + Z q s Θ u e s Φ dtdadx (cid:19) , (2.33)with Φ and φ are defined in (2.13) and ω ′ is defined in the beginning of the proof. On theother hand, using the fact that x x k ( x ) is non-decreasing, Hardy-Poincar´e inequalityfor the function Ze s Φ and the definition of ψ we have for s quite large the followinginequality Z Q µ Z e s Φ dx ≤ c (cid:18)Z Q k ( x ) Z x e s Φ dtdadx + Z Q s Θ x k ( x ) Z e s Φ dtdadx (cid:19) ≤ Z Q ( s φ Z + sφZ x ) e s Φ dtdadx. (2.34)Therefore, injecting (2.34) in (2.33) we get Z Q ( s φ W + sφW x ) e s Φ dtdadx (2.35) ≤ C (cid:18)Z Q h e s Φ dtdadx + Z q s Θ u e s Φ dtdadx (cid:19) + 12 Z Q ( s φ Z + sφZ x ) e s Φ dtdadx. Replying the same argument for the source term h := ηh + ( k η x v ) x + η x k v x we inferthat Z Q ( s φ Z + sφZ x ) e s Φ dtdadx ≤ C (cid:18)Z Q h e s Φ dtdadx + Z q s Θ v e s Φ dtdadx (cid:19) . (2.36) ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 11
Subsequently, combining (2.35) and (2.36) we arrive to Z Q [ s φ ( W + Z ) + sφ ( W x + Z x )] e s Φ dtdadx ≤ C (cid:18)Z Q ( h + h ) e s Φ dtdadx + Z q s Θ ( u + v ) e s Φ dtdadx (cid:19) . (2.37)Using the fact that u = w + W and v = z + Z , ϕ ≤ ϕ ≤ Φ, the estimates (2.31) and(2.37)lead to estimate (2.16). (cid:3)
Using the Theorem 2.2 for a special functions h and h , we are ready to deduce thefollowing result Theorem 2.6.
Assume that the assumptions (2.3) and (2.4) hold. Let
A > and T > be given such that T ∈ (0 , δ ) with δ ∈ (0 , A ) small enough. Then, there exist positiveconstants C (independent of δ ) and s such that for all s ≥ s , every solution ( u, v ) of (2.9) satisfies Z Q (cid:18) s Θ x k ( x ) u + s Θ k ( x ) u x (cid:19) e sϕ dtdadx + Z Q (cid:18) s Θ x k ( x ) v + s Θ k ( x ) v x (cid:19) e sϕ dtdadx ≤ C (cid:18)Z q s Θ ( u + v ) e s Φ dtdadx + Z Z δ ( u T ( a, x ) + v T ( a, x )) dadx (cid:19) . (2.38) Proof.
Let h := − β ( t, a, x ) u ( t, , x ) and h := − β ( t, a, x ) v ( t, , x ).Therefore, thanks to (2.16) and (2.4) we have the existence of two positive constants C and s such that, for all s ≥ s , the following inequality holds s Z Q Θ (cid:18) x k ( x ) u e sϕ + x k ( x ) v e sϕ (cid:19) dtdadx + s Z Q Θ (cid:0) k ( x ) u x e sϕ + k ( x ) v x e sϕ (cid:1) dtdadx ≤ C (cid:18)Z Q (( β ) u ( t, , x ) + ( β ) v ( t, , x )) e s Φ dtdadx + Z q s Θ ( u + v ) e s Φ dtdadx (cid:19) ≤ e C (cid:18)Z Z T ( u ( t, , x ) + v ( t, , x )) dtdadx + Z q s Θ ( u + v ) e s Φ dtdadx (cid:19) (2.39)Set U ( t, a, x ) = u ( T − t, A − a, x ) and V ( t, a, x ) = v ( T − t, A − a, x ). Then, one has ∂U∂t + ∂U∂a − ( k ( x ) U x ) x + µ ( T − t, A − a, x ) U + µ ( T − t, A − a, x ) V = β ( T − t, A − a, x ) U ( t, A, x ) ,U ( t, a,
1) = U ( t, a,
0) = 0 , (2.40) U (0 , a, x ) = U ( a, x ) = u T ( A − a, x ) ,U ( t, , x ) = 0 , where V is the solution of ∂V∂t + ∂V∂a − ( k ( x ) V x ) x + µ ( T − t, A − a, x ) V = β ( T − t, A − a, x ) V ( t, A, x ) ,V ( t, a,
1) = V ( t, a,
0) = 0 , (2.41) V (0 , a, x ) = V ( a, x ) = v T ( A − a, x ) ,V ( t, , x ) = 0 . Integrating along the characteristic lines, we get respectively the implicit formulas for thesolutions U of (2.40) and V of (2.41) given by U ( t, a, · ) = R a S ( a − l )( β ( T − t, A − l, · ) U ( t, A, · ) − µ ( T − t, A − l, · ) V ( t, l, · )) dl, if t > aU ( t, a, · ) = S ( t ) U ( a − t, · ) + R t S ( t − l )( β ( T − l, A − a, · ) U ( l, A, · ) − µ ( T − l, A − a, · ) V ( l, a, · )) dl, if t ≤ a, (2.42)and ( V ( t, a, · ) = R a L( a − l ) β ( T − t, A − l, · ) V ( t, A, · ) dl, if t > aV ( t, a, · ) = L( t ) V ( a − t, · ) + R t L( t − l ) β ( T − l, A − a, · ) V ( l, A, · ) dl, if t ≤ a, (2.43)where ( S ( t )) t ≥ and (L( t )) t ≥ are the bounded semigroups generated respectively by theoperators A U = − ( k U x ) x + µ ( T − t, A − a, x ) U and A V = − ( k V x ) x + µ ( T − t, A − a, x ) V .Hence, after a careful computations, (2.42) and (2.43) become respectively u ( t, a, · ) = R A − a S ( A − a − l )( β ( t, A − l, · ) u ( t, , · ) − µ ( t, A − l, · ) v ( t, A − l, · )) dl, if a > t + ( A − T ) u ( t, a, · ) = S ( T − t ) u T ( T + ( a − t ) , · ) + R Tt S ( l − t )( β ( l, a, · ) u ( l, , · ) − µ ( l, a, · ) v ( l, a, · )) dl, if a ≤ t + ( A − T ) , (2.44) ( v ( t, a, · ) = R A − a L( A − a − l ) β ( t, A − l, · ) v ( t, , · ) dl, if a > t + ( A − T ) v ( t, a, · ) = L( T − t ) v T ( T + ( a − t ) , · ) + R Tt L( l − t ) β ( l, a, · ) v ( l, , · ) dl, if a ≤ t + ( A − T ) , (2.45)Thus, by the third hypothesis in (2.4) on β and β one has ( u ( t, , · ) = S ( T − t ) u T ( T − t, · ) − R Tt S ( l − t ) µ ( l, , · ) v ( l, , · ) dl,v ( t, , · ) = L( T − t ) v T ( T + ( a − t ) , · ) . (2.46)Subsequently, by (2.39) we deduce that s Z Q Θ (cid:18) x k ( x ) u e sϕ + x k ( x ) v e sϕ (cid:19) dtdadx + s Z Q Θ (cid:0) k ( x ) u x e sϕ + k ( x ) v x e sϕ (cid:1) dtdadx ≤ b C (cid:18)Z q s Θ ( u + v ) e s Φ dtdadx + Z Z δ ( u T ( a, x ) + v T ( a, x )) dadx (cid:19) , (2.47)since ( S ( t )) t ≥ and (L( t )) t ≥ are a bounded semigroups, µ ∈ L ∞ ( Q ) and T ∈ (0 , δ ).Then the thesis follows. (cid:3) We come now to the more challenging point and the novelty of this contribution whichis the following ω -Carleman type inequality. Such an estimate plays a crucial role toobtain the null controllability of population dynamics cascade system with one controlforce. ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 13
Theorem 2.7.
Let (2.3) and (2.4) be verified. Let
A > and T > be given such that T ∈ (0 , δ ) with δ ∈ (0 , A ) small enough. Assume that there exists a positive constant ν such that µ ≥ ν on [0 , T ] × [0 , A ] × ω for some ω ⋐ ω, (2.48) Then every solution ( u, v ) of (2.9) satisfies Z Q (cid:18) s Θ x k ( x ) u + s Θ k ( x ) u x (cid:19) e sϕ dtdadx + Z Q (cid:18) s Θ x k ( x ) v + s Θ k ( x ) v x (cid:19) e sϕ dtdadx ≤ C δ (cid:18)Z q u dtdadx + Z Z δ ( u T ( a, x ) + v T ( a, x )) dadx (cid:19) . (2.49)This inequality is an immediate outcome of Theorem 2.6 applied to ω and the followinglemma (see for instance [7] and the references therein). Lemma 2.8.
Assume that (2.3) and (2.4) hold and let
A > and T > be given suchthat T ∈ (0 , δ ) with δ ∈ (0 , A ) small enough. we suppose also that (2.48) holds. Then, forall ǫ > there exist two positive constants C and M ǫ such that for every solution ( u, v ) of (2.9) the following inequality is satisfied Z ω Z A Z T s Θ v e s Φ dtdadx ≤ ǫC (cid:18)Z Q s Θ x k v e sϕ dtdadx + Z Q s Θ k ( x ) v x e sϕ dtdadx (cid:19) + M ǫ (cid:18)Z ω Z A Z T u dtdadx + Z Z δ ( u T ( a, x ) + v T ( a, x )) dadx (cid:19) . (2.50) Proof.
Let χ : R → R be the non-negative cut-off function defined as follows χ ∈ C ∞ (0 , ,supp ( χ ) ⊂ ω,χ ≡ ω . (2.51)Recall that ω = ( x , x ). Multiplying the first equation of (2.9) by χs Θ ve s Φ and afteran integration by parts, we get Z Q χs Θ ve s Φ u t dtdadx = − Z Q (3 + 2 s Φ) χs Θ t Θ uve s Φ dtdadx − Z Q χs Θ uv t e s Φ dtdadx. Z Q χs Θ ve s Φ u a dtdadx = − Z Q (3 + 2 s Φ) χs Θ a Θ uve s Φ dtdadx − Z Q χs Θ uv a e s Φ dtdadx. Z Q χs Θ ve s Φ ( k u x ) x dtdadx = − Z Q χs Θ k e s Φ u x v x dtdadx + Z Q s Θ k ( χe s Φ ) x uv x dtdadx + Z Q s Θ ( k ( χe s Φ ) x ) x uvdtdadx. − Z Q χs Θ ve s Φ µ udtdadx = − Z Q χs Θ µ uve s Φ dtdadx. − Z Q χs Θ ve s Φ µ vdtdadx = − Z Q χs Θ µ v e s Φ dtdadx. Then, summing all these identities side by side, using the second equation of (2.9) andintegrating again by parts Z Q χs Θ µ v e s Φ dtdadx = I + I + I + I + I , (2.52)where, I := R Q χs Θ β vu ( t, , x ) e s Φ dtdadx , I := − R Q ((3 + 2 s Φ) s Θ t Θ + (3 + 2 s Φ) s Θ a Θ + µ s Θ + µ s Θ ) χe s Φ uvdtdadx + R Q s Θ ( k ( χe s Φ ) x ) x uvdtdadx , I := R Q χs Θ β uv ( t, , x ) e s Φ dtdadx , I := R Q s Θ ( k − k )( x ) uv x ( χe s Φ ) x dtdadx , I := − R Q χs Θ ( k + k )( x ) u x v x e s Φ dtdadx .On one hand, we have by Young inequality and definition of χI ≤ ǫ Z Q s Θ k ( x ) v x e sϕ dtdadx + 14 ǫ Z Q χ s Θ ( k + k ) u x e s (2Φ − ϕ ) k dtdadx ≤ ǫ Z Q s Θ k ( x ) v x e sϕ dtdadx + max [0 , ( k + k ) ǫ min ω k Z Q χs Θ u x e s (2Φ − ϕ ) dtdadx. (2.53)Put L := R Q χs Θ u x e s (2Φ − ϕ ) dtdadx . To increase I , we will find an upper bound of L .To do this, we multiply the first equation of (2.9) by χs Θ e s (2Φ − ϕ k u and after integrationby parts Z Q χs Θ e s (2Φ − ϕ ) k uu t dtdadx = − Z Q s χk Θ Θ t (5 + 2 s (2Φ − ϕ )) e s (2Φ − ϕ ) u dtdadx. Z Q χs Θ e s (2Φ − ϕ ) k uu a dtdadx = − Z Q s χk Θ Θ a (5 + 2 s (2Φ − ϕ )) e s (2Φ − ϕ ) u dtdadx. Z Q χs Θ e s (2Φ − ϕ ) k u ( k u x ) x dtdadx = − Z Q χs Θ u x e s (2Φ − ϕ ) dtdadx + 12 Z Q s Θ (cid:18) k (cid:18) χe s (2Φ − ϕ ) k (cid:19) x (cid:19) x u dtdadx. − Z Q χs Θ e s (2Φ − ϕ ) k uµ udtdadx = − Z Q χs Θ e s (2Φ − ϕ ) k µ u dtdadx. − Z Q χs Θ e s (2Φ − ϕ ) k uµ vdtdadx = − Z Q χs Θ e s (2Φ − ϕ ) k µ uvdtdadx. Hence, adding these equalities side by side we get L = L + L + L , (2.54)where, L := R Q χs Θ e s (2Φ − ϕ k β uu ( t, , x ) dtdadx.L := − R Q χs Θ e s (2Φ − ϕ k µ uvdtdadx.L := − R Q (cid:16) χs Θ k µ + s χk Θ Θ t ( + 2 s (2Ψ − ψ )) + s χk Θ Θ a ( + 2 s (2Ψ − ψ )) (cid:17) e s (2Φ − ϕ ) u dtdadx + R Q s Θ (cid:16) k (cid:16) χe s (2Φ − ϕ k (cid:17) x (cid:17) x u dtdadx. ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 15
The assumptions in (2.4) on β together with Young inequality, Lemma ?? , the defini-tions of χ and Θ, the fact that the function x k x is non-increasing, | Θ t | ≤ C Θ and | Θ a | ≤ e C Θ and sup ( t,a,x ) ∈ Q s p Θ p e s (2Φ − ϕ ) < + ∞ for p ∈ R , (2.55)lead to L ≤ ǫ Z Q χs Θ e s (2Φ − ϕ ) ( k ) u dtdadx + ǫ Z Q χs Θ e s (2Φ − ϕ ) ( β ) u ( t, , x ) dtdadx ≤ e K ǫ Z Q χs Θ e s (2Φ − ϕ ) u dtdadx + ǫK Z Z A Z TT − δ χu ( t, , x ) dtdadx ≤ e K ǫ Z Q χs Θ e s (2Φ − ϕ ) u dtdadx + ǫK Z Z δ χu T ( a, x ) dadx (2.56)and L ≤ ǫ Z Q x k s Θ e sϕ v dtdadx + 14 ǫ Z Q χ s Θ ( k ) e s (4Φ − ϕ ) k x ( µ ) u dtdadx ≤ ǫ Z Q x k s Θ e sϕ v dtdadx + K ǫ Z ω Z A Z T s Θ e s (4Φ − ϕ ) u dtdadx, (2.57)and | L | ≤ K Z ω Z A Z T s Θ e s (2Φ − ϕ ) u dtdadx, (2.58)where K = k µ k ∞ k ( x )( x ) min ω k . On the other hand, by Lemma 2.5 we have e s (2Φ − ϕ ) ≤ e s (4Φ − ϕ ) . (2.59)Then, combining relations (2.54), (2.56), (2.57) and (2.58) we conclude L ≤ ǫ Z Q x k s Θ e sϕ v dtdadx + K ǫ Z ω Z A Z T s Θ e s (4Φ − ϕ ) u dtdadx + ǫK Z Z δ u T ( a, x ) dadx. (2.60)Hence, by (2.53) and (2.60) we deduce I ≤ ǫC (cid:18)Z Q x k s Θ e sϕ v dtdadx + Z Q s Θ k ( x ) v x e sϕ dtdadx (cid:19) + K ǫ Z ω Z A Z T s Θ e s (4Φ − ϕ ) u dtdadx + K Z Z δ u T ( a, x ) dadx. (2.61)where K ǫ is a positive constants that depend on ǫ . Similarly, we will find an upper boundsof I , I , I and I . Firstly, we will start by I . One has the following relations (cid:12)(cid:12)(cid:12)(cid:12)Z Q χ (3 + 2 s Φ) s Θ t Θ e s Φ uvdtdadx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Q χ | s Φ | s | Θ t | Θ e s Φ | uv | dtdadx ≤ C Z Q χ | s Φ | s Θ e s Φ | uv | dtdadx ≤ ǫ Z Q s Θ x k e sϕ v dtdadx + C ǫ Z ω Z A Z T s Θ e s (2Φ − ϕ ) u dtdadx, (2.62) (cid:12)(cid:12)(cid:12)(cid:12)Z Q χ (3 + 2 s Φ) s Θ a Θ e s Φ uvdtdadx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ Z Q s Θ x k e sϕ v dtdadx + C ǫ Z ω Z A Z T s Θ e s (2Φ − ϕ ) u dtdadx, (2.63) (cid:12)(cid:12)(cid:12)(cid:12)Z Q χ ( µ + µ ) s Θ e s Φ uvdtdadx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ Z Q s Θ x k e sϕ v dtdadx + C ǫ Z ω Z A Z T s Θ e s (2Φ − ϕ ) u dtdadx, (2.64) (cid:12)(cid:12)(cid:12)(cid:12)Z Q s Θ ( k ( χe s Φ ) x ) x uvdtdadx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ Z Q s Θ x k e sϕ v dtdadx + 14 ǫ Z Q s Θ k x ( k ( χe s Φ ) x ) x e − sϕ u dtdadx ≤ ǫ Z Q s Θ x k e sϕ v dtdadx + C ǫ Z Q s Θ k x ( χ + χ x + χ xx ) e s (2Φ − ϕ ) u dtdadx ≤ ǫ Z Q s Θ x k e sϕ v dtdadx + C ǫ Z ω Z A Z T s Θ e s (2Φ − ϕ ) u dtdadx, (2.65)Hence, summing inequalities (2.62), (2.63), (2.64) and (2.65) we obtain I ≤ ǫ Z Q s Θ x k e sϕ v dtdadx + C ǫ Z ω Z A Z T s Θ e s (2Φ − ϕ ) u dtdadx. (2.66)For the rest of integrals, I = Z Q χs Θ β vu ( t, , x ) e s Φ dtdadx ≤ ǫ Z Q s Θ x k e sϕ v dtdadx + C ǫ Z Z δ u T ( a, x ) dadx. (2.67) I = Z Q χs Θ β uv ( t, , x ) e s Φ dtdadx ≤ ǫ Z Z δ v T ( a, x ) dadx + 14 ǫ Z ω Z A Z T s Θ e s (2Φ − ϕ ) u dtdadx. (2.68) I = Z Q s Θ ( k − k )( x ) uv x ( χe s Φ ) x dtdadx ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 17 = Z Q s Θ ( k − k )( x ) uv x ( χ x + 2 s Φ x χ ) e s Φ dtdadx ≤ ǫ Z Q s Θ k v x e sϕ dadx + 14 ǫ Z Q s Θ ( k − k ) k ( χ x + 2 s Φ x χ ) e s (2Φ − ϕ ) u dtdadx ≤ ǫ Z Q s Θ k v x e sϕ dadx + C ǫ Z ω Z A Z T s Θ e s (2Φ − ϕ ) u dtdadx. (2.69)Subsequently, combining (2.61), (2.66), (2.67), (2.68), (2.69) and using again (2.59) Z Q χs Θ µ v e s Φ dtdadx ≤ ǫC (cid:18)Z Q s Θ x k v e sϕ dtdadx + Z Q s Θ k ( x ) v x e sϕ dtdadx (cid:19) + C ǫ Z ω Z A Z T s Θ e s (4Φ − ϕ ) u dtdadx + C ǫ Z Z δ ( u T ( a, x ) + v T ( a, x )) dadx. Finally, the hypothesis (2.48), the definition of χ and the relationsup ( t,a,x ) ∈ Q s p Θ p e s (4Φ − ϕ ) < + ∞ for p ∈ R , (2.70)yield Z ω Z A Z T s Θ v e s Φ dtdadx ≤ ǫC (cid:18)Z Q s Θ x k v e sϕ dtdadx + Z Q s Θ k ( x ) v x e sϕ dtdadx (cid:19) + C ǫ (cid:18)Z ω Z A Z T u dtdadx + Z Z δ ( u T ( a, x ) + v T ( a, x )) dadx (cid:19) , (2.71)which finishes the proof. (cid:3) The above Carleman estimate can be used in a standard way to obtain the null con-trollability of the cascade system with one control force. This will be reached showing anobservability inequality of the adjoint system. Observability inequality and null controllability results
This paragraph is devoted to the observability inequality of system (2.9) and then the nullcontrollability result of system (1.1). We start to show our observability inequality whoseproof is based essentially on Carleman estimate (2.49) and Hardy-Poincar´e inequality.
Proposition 3.1.
Assume that (2.3) and (2.4) hold. Suppose also that (2.48) is fulfilledand let
A > and T > be given such that T ∈ (0 , δ ) with δ ∈ (0 , A ) small enough.Then, there exists a positive constant C δ such that for every solution ( u, v ) of (2.9) , thefollowing observability inequality is satisfied Z Z A ( u (0 , a, x ) + v (0 , a, x )) dadx ≤ C δ (cid:18)Z q u dtdadx + Z Z δ ( u T ( a, x ) + v T ( a, x )) dadx (cid:19) . (3.72) Proof.
Then for κ > e u = e κt u and e v = e κt v are respectively asolutions of ∂ e u∂t + ∂ e u∂a + ( k ( x ) e u x ) x − µ ( t, a, x ) e u = µ ( t, a, x ) e v − β e u ( t, , x ) in Q, (3.73) e u ( t, a,
1) = e u ( t, a,
0) = 0 on (0 , T ) × (0 , A ) , e u ( T, a, x ) = e κT u T ( a, x ) in Q A , e u ( t, A, x ) = 0 in Q T , and ∂ e v∂t + ∂ e v∂a + ( k ( x ) e v x ) x − µ ( t, a, x ) e v = − β e v ( t, , x ) in Q, (3.74) e v ( t, a,
1) = e v ( t, a,
0) = 0 on (0 , T ) × (0 , A ) , e v ( T, a, x ) = e κT v T ( a, x ) in Q A , e v ( t, A, x ) = 0 in Q T , where, u and v are respectively the solutions of ∂u∂t + ∂u∂a + ( k ( x ) u x ) x − µ ( t, a, x ) u = µ ( t, a, x ) v − β u ( t, , x ) in Q, (3.75) u ( t, a,
1) = u ( t, a,
0) = 0 on (0 , T ) × (0 , A ) ,u ( T, a, x ) = u T ( a, x ) in Q A ,u ( t, A, x ) = 0 in Q T , and ∂v∂t + ∂v∂a + ( k ( x ) v x ) x − µ ( t, a, x ) v = − β v ( t, , x ) in Q, (3.76) v ( t, a,
1) = v ( t, a,
0) = 0 on (0 , T ) × (0 , A ) ,v ( T, a, x ) = v T ( a, x ) in Q A ,v ( t, A, x ) = 0 in Q T . Multiplying the first equations of (3.73) and (3.74) respectively by e u and e v and integratingby parts on Q t = (0 , t ) × (0 , A ) × (0 ,
1) one obtains12 Z Q A u (0 , a, x ) dadx + 12 Z Z t e u ( τ, , x ) dτ dx + κ Z Z A Z t e u ( τ, a, x ) dτ dadx ≤ k β k ∞ + 14 ǫ ′ Z Z A Z t e u ( τ, a, x ) dτ dadx + ǫ ′ A Z Z t e u ( τ, , x ) dτ dx + ǫ ′ Z Q t µ e v dτ dadx + 12 Z Q A e u ( t, a, x ) dadx. (3.77)and 12 Z Q A v (0 , a, x ) dadx + 12 Z Z t e v ( τ, , x ) dτ dx + κ Z Z A Z t e v ( τ, a, x ) dτ dadx ≤ k β k ∞ + 14 ǫ ′ Z Z A Z t e v ( τ, a, x ) dτ dadx ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 19 + ǫ ′ A Z Z t e v ( τ, , x ) dτ dx + 12 Z Q A e v ( t, a, x ) dadx. (3.78)Summing (3.77) and (3.78) side by side and taking κ = max( k β k ∞ +14 ǫ ′ , k β k ∞ +14 ǫ ′ + ǫ ′ k µ k ∞ )and ǫ ′ < A , on gets Z Q A u (0 , a, x ) dadx + Z Q A v (0 , a, x ) dadx ≤ Z Q A e u ( t, a, x ) dadx + Z Q A e v ( t, a, x ) dadx. (3.79)Arguing as in [2]and integrating over ( T , T ) we conclude Z Q A u (0 , a, x ) dadx + Z Q A v (0 , a, x ) dadx ≤ C e κT (cid:18)Z Z δ u T ( a, x ) dadx + Z Z δ v T ( a, x ) dadx (cid:19) + 2 e κT T Z Z Aδ Z T T u ( t, a, x ) dtdadx + Z Z Aδ Z T T v ( t, a, x ) ! dtdadx. (3.80)Hence, Hardy-Poincar´e inequality and the definitions of ϕ i , i = 1 , Z Q A u (0 , a, x ) dadx + Z Q A v (0 , a, x ) dadx ≤ C e κT (cid:18)Z Z δ u T ( a, x ) dadx + Z Z δ v T ( a, x ) dadx (cid:19) + C δ Z Z Aδ Z T T s Θ k ( x ) u ( t, a, x ) e sϕ dtdadx + Z Z Aδ Z T T s Θ k ( x ) v ( t, a, x ) e sϕ dtdadx ! . Finally, using the Carleman estimate (2.49) we deduce the observability inequality (3.72).and then the proof is finished. (cid:3)
Now, obtaining our observability inequality, following a standard argument, we are nowready to prove our main result.
Theorem 3.2.
Assume that (2.3) and (2.4) are verified. Let
A > and T > be givensuch that T ∈ (0 , δ ) with δ ∈ (0 , A ) small enough. Then, for all ( y , p ) ∈ L ( Q A ) × L ( Q A ) , there exists a control ϑ ∈ L ( q ) such that the associated solution of (1.1) verifies ( y ( T, a, x ) = 0 , a.e. in ( δ, A ) × (0 , ,p ( T, a, x ) = 0 , a.e in ( δ, A ) × (0 , . (3.81) Proof.
Let ε > J ε ( ϑ , ϑ ) = 12 ε Z Z Aδ ( y ( T, a, x ) + p ( T, a, x )) dadx + 12 Z q ϑ ( t, a, x ) dtdadx. We can prove that J ε is continuous, convex and coercive. Then, it admits at least oneminimizer ϑ ε and we have ϑ ε = − u ε ( t, a, x ) χ ω ( x ) in Q, (3.82)with u ε is the solution of the following system ∂u ε ∂t + ∂u ε ∂a + ( k ( x )( u ε ) x ) x − µ ( t, a, x ) u ε − µ v ε = − β u ε ( t, , x ) in Q, (3.83) u ε ( t, a,
1) = u ε ( t, a,
0) = 0 on (0 , T ) × (0 , A ) ,u ε ( T, a, x ) = 1 ε y ε ( T, a, x ) χ ( δ,A ) ( a ) in Q A ,u ε ( t, A, x ) = 0 in Q T , where v ε is the solution of ∂v ε ∂t + ∂v ε ∂a + ( k ( x )( v ε ) x ) x − µ ( t, a, x ) v ε = − β v ε ( t, , x ) in Q, (3.84) v ε ( t, a,
1) = v ε ( t, a,
0) = 0 on (0 , T ) × (0 , A ) ,v ε ( T, a, x ) = 1 ε p ε ( T, a, x ) χ ( δ,A ) ( a ) in Q A ,v ε ( t, A, x ) = 0 in Q T , and ( y ε , p ε ) is the solution of the system (1.1) associated to the control ϑ ε .Multiplying the first equation of (3.83) by y ε and the second equation of (1.1) by v ε ,integrating over Q , using (3.82) and the Young inequality we obtain1 ε Z Z Aδ ( y ε ( T, a, x ) + p ε ( T, a, x )) dadx + Z q ϑ ε ( t, a, x ) dtdadx = Z Q A ( y ( a, x ) u ε (0 , a, x ) + p ( a, x ) v ε (0 , a, x )) dadx ≤ C δ Z Q A ( u ε (0 , a, x ) + v ε (0 , a, x )) dadx + C δ Z Q A ( y ( a, x ) + p ( a, x )) dadx, with C δ is the constant of the observability inequality (3.72). Hence, using relation (3.82),the observability inequality leads to1 ε Z Z Aδ ( y ε ( T, a, x ) + p ε ( T, a, x )) dadx + 34 Z q ϑ ε ( t, a, x ) dtdadx ≤ C δ Z Q A ( y ( a, x ) + p ( a, x )) dadx. (3.85)Hence, it follows that R R Aδ y ε ( T, a, x ) dadx ≤ C δ ε R Q A ( y ( a, x ) + p ( a, x )) dadx, R R Aδ p ε ( T, a, x ) dadx ≤ C δ ε R Q A ( y ( a, x ) + p ( a, x )) dadx, R q ϑ ε ( t, a, x ) dtdadx ≤ C δ R Q A ( y ( a, x ) + p ( a, x )) dadx. (3.86)Then, we can extract two subsequences of ( y ε , p ε ) and ϑ ε denoted also by ϑ ε and( y ε , p ε )that converge weakly towards ϑ and ( y, p ) in L ( q ) and L ((0 , T ) × (0 , A ); H k (0 , × H k (0 , y, p ) is a solutionof (1.1) corresponding to the controls ϑ and, by the first and second estimates of (3.86),( y, p ) satisfies (1.2). (cid:3) ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 21 Appendix
As is mentioned in the introduction, this section is devoted to the proofs of some inter-mediate results useful to show the Carleman type inequality (2.49). Firstly, we begin bythe Caccioppoli’s inequality stated in the following lemma
Lemma 4.1.
Let ω ′ be a subset of ω such that ω ′ ⊂⊂ ω . Then, there exists a positiveconstant C such that Z ω ′ Z A Z T ( u x + v x ) e sϕ i dtdadx ≤ C (cid:18)Z q s Θ ( u + v ) e sϕ i dtdadx + Z q ( h + h ) e sϕ i dtdadx (cid:19) , (4.87) where ( u, v ) is the solution of (2.10) and the weight functions ϕ i , i = 1 , are defined by (2.13) .Proof. The proof of this result is similar to the one of [13, Lemma 5.1]. Indeed, considerthe cut-off function ζ defined by ≤ ζ ( x ) ≤ , x ∈ R ,ζ ( x ) = 0 , x < x and x > x ,ζ ( x ) = 1 , x ∈ ω ′ , (4.88)For ( u, v ) solution of (2.10) one has0 = Z T ddt (cid:20)Z Z A ζ e sϕ i ( u + v ) dadx (cid:21) dt = 2 s Z Z A Z T ζ ( ϕ i ) t ( u + v ) e sϕ i dtdadx + 2 Z Z A Z T ζ ww t e sϕ i dtdadx = 2 s Z Z A Z T ζ ( ϕ i ) t ( u + v ) e sϕ i dtdadx +2 Z Z A Z T ζ u ( − ( k u x ) x − u a + h + µ u + µ v ) e sϕ i dtdadx +2 Z Z A Z T ζ v ( − ( k v x ) x − v a + h + µ v ) e sϕ i dtdadx. Then, integrating by parts we obtain2 Z Q ζ ( k u x + k v x ) e sϕ i dtdadx = − s Z Q ζ ( u + v ) ψ i (Θ a + Θ t ) e sϕ i dtdadx − Z Q ζ ( uh + vh ) e sϕ i dtdadx − Z Q ζ ( µ u + µ v ) e sϕ i dtdadx + Z Q ( k ( ζ e sϕ i ) x ) x u dtdadx + Z Q ( k ( ζ e sϕ i ) x ) x v dtdadx − Z Q ζ µ uve sϕ i dtdadx. On the other hand, by the definitions of ζ , ψ and Θ, using Young inequality and taking s quite large there is a constant c such that2 Z Q ζ ( k u x + k v x ) e sϕ i dtdadx ≥ x ∈ ω ′ k ( x ) , min x ∈ ω ′ k ( x )) Z ω ′ Z A Z T ( u x + v x ) e sϕ i dtdadx, Z Q ( k ( ζ e sϕ i ) x ) x u dtdadx ≤ c Z ω Z A Z T s Θ u e sϕ i dtdadx, Z Q ( k ( ζ e sϕ i ) x ) x v dtdadx ≤ c Z ω Z A Z T s Θ v e sϕ i dtdadx, − s Z Q ζ ( u + v ) ψ i (Θ a + Θ t ) e sϕ i dtdadx ≤ c Z ω Z A Z T s Θ ( u + v ) e sϕ i dtdadx, − Z Q ζ uh e sϕ i dtdadx ≤ c (cid:18)Z ω Z A Z T s Θ u e sϕ i dtdadx + Z ω Z A Z T h e sϕ i dtdadx (cid:19) , − Z Q ζ vh e sϕ i dtdadx ≤ c (cid:18)Z ω Z A Z T s Θ v e sϕ i dtdadx + Z ω Z A Z T h e sϕ i dtdadx (cid:19) , − Z Q ζ ( µ u + µ v ) e sϕ i dtdadx ≤ c Z ω Z A Z T s Θ ( u + v ) e sϕ i dtdadx, − Z Q ζ µ uve sϕ i dtdadx ≤ c Z ω Z A Z T s Θ ( u + v ) e sϕ i dtdadx. Combining all these inequalities, we can see that there is
C > Z ω ′ Z A Z T ( u x + v x ) e sϕ i dtdadx ≤ C (cid:18)Z q s Θ ( u + v ) e sϕ i dtdadx + Z q ( h + h ) e sϕ i dtdadx (cid:19) . Thus, the proof is achieved. (cid:3)
Remark 4.2.
In Lemma 4.1, the set ω ′ is chosen so that which is exactly the point ofdegeneracy of the dispersion coefficients k and k does not belong to ω ′ . More generally,if the degeneracy occurs at a point x ∈ (0 , , one must take x out of ω ′ in the case ofinterior degeneracy to establish a Caccioppoli’s type inequality (see [10] for more detailsin this context). We close this section by the following result
Lemma 4.3.
Assume that the conditions (2.15) hold. Then, I = [ k (1)(2 − γ )( e κ k σ k∞ − d k (1)(2 − γ ) − , e κ k σ k∞ − e κ k σ k∞ )3 d ) is not empty.Proof. Indeed, one has 4( e κ k σ k ∞ − e κ k σ k ∞ )3 d − k (1)(2 − γ )( e κ k σ k ∞ − d k (1)(2 − γ ) −
1= 4( e κ k σ k ∞ − e κ k σ k ∞ )( d k (1)(2 − γ ) − − d k (1)(2 − γ )( e κ k σ k ∞ − d ( d k (1)(2 − γ ) − e κ k σ k ∞ ( d k (1)(2 − γ ) − − e κ k σ k ∞ ( d k (1)(2 − γ ) − d ( d k (1)(2 − γ ) −
1) + k (1)(2 − γ ) d k (1)(2 − γ ) − ULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 23 = e κ k σ k ∞ [ e κ k σ k ∞ ( d k (1)(2 − γ ) − − d k (1)(2 − γ ) − d ( d k (1)(2 − γ ) −
1) + k (1)(2 − γ ) d k (1)(2 − γ ) − . Using the fact that d ≥ k (1)(2 − γ ) , we can conclude that d k (1)(2 − γ ) − d k (1)(2 − γ ) − ≤ κ ≥ k σ k ∞ , then we have e κ k σ k ∞ ≥
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