Groupe de travail «Contrôle» Laboratoire J.L. Lions 6 Mai 2011

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1 Groupe de travail «Contrôle» Laboratoire J.L. Lions 6 Mai 2011

2 Un modèle de dynamique des populations : Contrôlabilité approchée par contrôle des naissances

3 Otared Kavian Département de Mathématiques Université de Versailles 45, avenue des Etats Unis Versailles cedex (France) 6 mai 2011 Travail en commun avec Oumar Traoré (Université de Ouagadougou, Burkina Faso) Publié dans ESAIM COCV (2010) sous le titre Approximate Controllability by Birth Control for a Nonlinear Population Dynamics Model.

4 Today s talk The problem Main result The adjoint system Unique continuation: idea of proof Fixed point result

5 The problem

6 Reaction-diffusion equations Ω R N a bounded domain where N 1 0 t T the time variable, u(t, x) density of population of one species Ω u u = f (u) t u(t, σ) = 0 u(0, x) = u 0 (x) in (0, T) Ω on (0, T) Ω in Ω Here, the function f : R R describes the total balance of mortality and birth. For instance one may consider f (u) := au(b u)

7 Reaction-diffusion equations Ω Ω R N a bounded domain where N 1 0 t T the time variable, u(t, x) := (u 1 (t, x), u 2 (t, x),..., u m (t, x)) R m, densities of populations of m species, a given diffusion matrix, D := diag(d 1, d 2,..., d m ) with d j > 0. u t D u = f (u) u(t, σ) = 0 u(0, x) = u 0 (x) in (0, T) Ω on (0, T) Ω in Ω Here, the function f : R m R m describes the total balance of mortality, birth and reactions between the different species. In these models, the populations are ageless... Hence the need to modify the model so that the behaviour of individuals might be age dependent.

8 Population dynamics Ω y t + y a Ω R N a bounded domain where N 1 0 t T the time variable, 0 a A the age variable, A < given y(t, a, x) density of population of one species µ(a) the mortality rate β(a) the birth rate y + µy = 0 in (0, T) (0, A) Ω y(t, a, σ) = 0 on (0, T) (0, A) Ω y(0, a, x) = y 0 (a, x) in (0, A) Ω y(t, 0, x) = F (y) in (0, T) Ω. Here, with a certain given birth rate β, the function F is defined by ( ) F (y)(t, x) := F t, x, A 0 β(a)y(t, a, x)da

9 Population dynamics A simple example of the function F is the case F(t, x, s) := s, so that F (y)(t, x) := A 0 β(a)y(t, a, x)da. The net reproduction rate is defined as R := A 0 ( β(a) exp a 0 ) µ(s)ds da. With the above F, when R < 1 one may show that y goes to zero, lim y(t) L t + 2 ((0,A) Ω) = 0. While if R > 1, lim y(t) L t + 2 ((0,A) Ω) = +.

10 Population dynamics + Birth control We consider now ω Ω a small subdomain, where a certain control v(t, x)1 ω (x) applies ω Ω y t + y a y + µy = 0 As before, the function F is defined by ( in (0, T) (0, A) Ω y(t, a, σ) = 0 on (0, T) (0, A) Ω y(0, a, x) = y 0 (a, x) in (0, A) Ω y(t, 0, x) = F (y) + v(t, x)1 ω in (0, T) Ω. F (y)(t, x) := F t, x, A 0 β(a)y(t, a, x)da )

11 Population dynamics + Birth control Approximate control: given ε > 0 and a target function h L 2 ((0, A) Ω) find v L 2 ((0, T) Ω) such that the solution of y t + y y + µy = 0 in (0, T) (0, A) Ω a (1.1) y(t, a, σ) = 0 on (0, T) (0, A) Ω y(0, a, x) = y 0 (a, x) in (0, A) Ω y(t, 0, x) = F (y) + v(t, x)1 ω in (0, T) Ω. satisfies at time T y(t,, ) h L 2 ((0,A) Ω) ε. Previous works on other models (for instance without diffusion, control over Ω, etc): B. Ainseba, S. Anita, V. Barbu, M. Langlais, A. Ouédraogo, O. Traoré.

12 Main result

13 Main result We make the following assumptions: µ 0 and A 0 µ(a)da = +. β 0 and there exists 0 < A 0 < A 1 < A such that supp(β) [A 0, A 1 ]. F : (0, T) Ω R R is a Caratheodory function such that a.e. in (t, x) (0, T) Ω the function s F(t, x, s) is in C 1 (R), verifies F(t, x, 0) = 0 and moreover is globally Lipschitz. If π(a) := exp ( a 0 µ(s)ds) is the survival likelihood, we assume that We assume also that T > A. π 1 y 0 L 2 ((0, A) Ω).

14 Main result Theorem. For any h L 2 ((0, A) Ω) and ε > 0, there exists v L 2 ((0, T) ω) such that the corresponding solution of the system (1.1) satisfies y(t,, ) h L 2 ((0,T) Ω) ε. The proof can be adapted for a control on the boundary, or other types of boundary conditions The theorem is an existence result: it is not entirely satisfactory (for instance one cannot ensure that y(t, a, x) remains nonnegative during the control process)

15 Main result: idea of the method First set H(t, x, s) := s 1 F(t, x, s), and consider the linear version of (1.1): for Y 0 L 2 ((0, T) Ω) y t + y a y = 0 in (0, T) (0, A) Ω (2.1) y(t, a, σ) = 0 on (0, T) (0, A) Ω y(0, a, x) = 0 in (0, A) Ω y(t, 0, x) = v(t, x)1 ω + H(t, x, Y 0 )Y in (0, T) Ω. Here Y(t, x) := A 0 β(a)y(t, a, x)da. Once it is shown that a control v exists, one considers the mapping Y 0 Y and seeks a fixed point of this mapping... which is multivalued...

16 The adjoint system

17 The adjoint system First we consider the linear problem. Recall that if L : E F is a linear bounded operator, then L : F E and one has N(L ) = R(L). Apply this with E := L 2 ((0, T) ω), and F := L 2 ((0, A) Ω), while L is the mapping v y(t,, ) = Lv. One has to determine L : L 2 ((0, A) Ω) L 2 ((0, T) ω)... For g L 2 ((0, A) Ω) and v L 2 ((0, T) ω) we have (L g v) = (g Lv), so that one tries to express Lv in terms of u: one multiplies equation (2.1) by a function p(t, a, x) and integrates by parts...

18 The adjoint system After some manipulations one ends up with the following adjoint problem to the linear system p t p a p = β 0(t, a, x)p(t, 0, x) in (0, T) (0, A) Ω (3.1) p(t, a, σ) = 0 on (0, T) (0, A) Ω p(t, a, x) = g(a, x) in (0, A) Ω p(t, A, x) = 0 in (0, T) Ω. Then L : L 2 ((0, A) Ω) L 2 ((0, T) ω) is given by g p(t, 0, x)1 ω =: L g for g L 2 ((0, A) Ω). The question is whether N(L ) = {0}, in which case R(L) is dense in L 2 ((0, A) Ω) (thus the approximate controllability of the linear equation).

19 The adjoint system We end up to show the following unique continuation result: Theorem. Let g L 2 ((0, A) Ω). If the solution p L 2 ((0, T) (0, A); H 1 (Ω)) of 0 the equation p t p a p = β 0(t, a, x)p(t, 0, x) in (0, T) (0, A) Ω p(t, a, σ) = 0 on (0, T) (0, A) Ω p(t, a, x) = g(a, x) in (0, A) Ω p(t, A, x) = 0 in (0, T) Ω. verifies p(t, 0, x) = 0 in (0, T) ω then p 0 in (0, T) (0, A) Ω.

20 Unique continuation: idea of proof

21 Unique continuation: idea of proof It is more convenient to prove the result for the forward system, that is setting z(t, a, x) := p(t t, A a, x), to prove that z 0 whenever z(t, A, x) = 0 in (0, T) ω. Clearly z satisfies the system: z t + z a z = β 1(t, a, x)z(t, A, x) in (0, T) (0, A) Ω z(t, a, σ) = 0 on (0, T) (0, A) Ω z(0, a, x) = k(a, x) in (0, A) Ω z(t, 0, x) = 0 in (0, T) Ω where β 1 (t, a, x) := β 0 (T t, A a, x), k(a, x) := g(a a, x).

22 Unique continuation: idea of proof (ϕ j, λ j ) j being the eigenfunctions and eigenvalues of on H 1 (Ω) we write 0 z(t, a, x) = z j (t, a)ϕ j (x). j 1 With k j (a) := k(a, x)ϕ j (x)dx, and γ j (t, a) := ϕ j (x)β 1 (t, a, x)z(t, A, x)dx, one should have z j t + z j a + λ jz j = γ j (t, a) in (0, T) (0, A) z j (0, a) = k j (a) in (0, A) z j (t, 0) = 0 in (0, T). One finds z j (t, a) = e λjt k j (a t) + t 0 a 0 e λ js γ j (t s, a s)ds, if a t e λ js γ j (t s, a s)ds, if a < t.

23 Unique continuation: idea of proof Now, knowing the coefficients z j, one notices that for any fixed t such that 0 < t < A 0 (for instance), then we have z(t, A, x) = c j ϕ j 0 on ω, c j := e λjt k j (A t). j 1 We point out the following elementary result: Lemma. If the sequence (c j ) j is such that for some τ > 0 one has e 2λjτ c j 2 <. j 1 Then the function Z := j 1 c j ϕ j L 2 (Ω) is well defined and if Z 0 on ω, then Z 0 on Ω and c j = 0 for all j 1. And one shows that actually the sequence (c j ) j (which depends on z j ) does indeed satisfy the above growth condition. For other values of t, a one proceeds analogously.

24 Fixed point result

25 Fixed point result Recall that H(t, x, s) := s 1 F(t, x, s), and we consider (1.1) in the form: for Y 0 L 2 ((0, T) Ω) y t + y a y = 0 in (0, T) (0, A) Ω (5.1) y(t, a, σ) = 0 on (0, T) (0, A) Ω y(0, a, x) = 0 in (0, A) Ω y(t, 0, x) = v(t, x)1 ω + H(t, x, Y 0 )Y in (0, T) Ω. Here (5.2) y(t, ) h L 2 ((0,A) Ω) ε, and Y(t, x) := A 0 β(a)y(t, a, x)da. We define a multivalued map K defined on L 2 ((0, T) Ω) K(Y 0 ) := {Y as in (5.1) (5.2)}

26 Fixed point result and we seek a fixed point of K, that is Y 0 L 2 ((0, T) Ω), Y 0 K(Y 0 ). For this, we use the Kakutani-Fan-Glicksberg fixed point theorem. Kakutani s theorem (1941) is in finite dimension, and the version we use is due to Ky Fan (1952), and to I. Glicksberg (1952). It is equivalent to Brouwer s theorem (1912). See Eberhard Zeidler s book Nonlinear Functional Analysis and its Applications (volume IV: Applications to Mathematical Physics), section In particular in chapter 77 of this book there is a very nice account of various forms of fixed point theorems.

27 Fixed point result Theorem. Let X be a reflexive Banach space and K : X 2 X a multivalued mapping such that: For all Y 0 X the set K(Y 0 ) is a nonempty convex closed subset of X. There exists a compact convex set X c such that K(X c ) X c. For all Z X the mapping Y 0 sup P K(Y 0 ) T 0 Ω Z(t, x)p(t, x)dxdt is upper semicontinuous. Then the mapping K has at least one fixed point, that is there exists Y X such that Y K(Y).