Internal Stabilizability of Some Diffusive Models

Transcription

1 Journal of Mathematical Analysis and Applications 265, (22) doi:1.16/jmaa , available online at on Internal Stabilizability of Some Diffusive Models Bedr Eddine Ainseba and Michel Langlais Mathématiques Appliquées de Bordeaux, UMR CNRS 5466, case 26, Université Victor Segalen Bordeaux 2, 3376 Bordeaux Cedex, France and Sebastian Aniţa Faculty of Mathematics, University Al.I. Cuza, Iaşi 66, Romania Submitted by Hélène Frankowska Received February 6, 21 We consider a single species population dynamics model with age dependence, spatial structure, and a nonlocal birth process arising as a boundary condition. We prove that under a suitable internal feedback control, one can improve the stabilizability results given in Kubo and Langlais [J. Math. Biol. 29 (1991), ]. This result is optimal. Our proof relies on an identical stabilizability result of independent interest for the heat equation, that we state and prove in Section Elsevier Science 1. INTRODUCTION AND MAIN RESULTS We consider a linear model describing the dynamics of a single species population with age dependence and spatial structure. Let p = p a t x be the distribution of individuals having age a at time t and location x in, a bounded open set in d d 1 2 3, having a suitably smooth boundary Ɣ. Thus P t x = A 91 p a t x da (1) X/2 $ Elsevier Science All rights reserved.

2 92 ainseba, langlais, and aniţa is the total population density at time t and location x A being the maximum life expectancy of an individual. Let β a be the natural fertility rate and let µ a be the natural death rate of individuals having age a. Under these conditions the evolution of the distribution p is governed by the partial differential equation t p + a p x p + µ a p = a A t + x (2) see [6, 7, 11]. The birth process is given by the renewal equation p t x = A β a p a t x da t + x (3) We assume hostile ends corresponding to homogeneous Dirichlet boundary conditions p a t x = a A t + x Ɣ (4) The dynamics of the solution to (2) (4) starting at time t = from p a x =p a x a A x (5) depends on the root r of the characteristic equation A ( 1 = β a π a exp ra da π a =exp a ) µ s ds (6) as well as on λ 1, the first eigenvalue of the Dirichlet problem for in. When β satisfies H 1 below and p a x on a 1, the large time behavior of the solutions to (2) (5) is the following: for each finite A <A<A, one has (1) If r <λ 1, then p t 2 A ast +. (2) If r >λ 1, then p t 2 A + as t +. (3) If r = λ 1, then p t C p π a as t +, for some constant C p. Herein 2 A is the norm in L 2 A. When p x a on a 1, then p a t x for t>a. A quick proof can be found in [9]. A natural question to address is the following: is it possible to stabilize in a suitable fashion the dynamics of p, a solution to (2) (4)? An answer in this direction is found in [8] for nonnegative data. Assuming that an input u a t x is supplied, the new system becomes t p+ a p x p+µ a p=u a t x in A T p a x =p a x p t x = A p a t x = β a p a t x da on A on T on A T Ɣ (7)

3 internal stabilizability of diffusive models 93 First, when r >λ 1, assuming u a t x and p a x (8) and using a comparison principle [2, 5], then the solution to 7 is bounded from below by the nonnegative solution to (2) (5); therefore its L 2 A norm is still diverging. Next, when r <λ 1, in order to prevent the decay to of the solution to (2) (5), a basic idea in [8] is to supply a nonnegative and periodic, in time, input of individuals. Let T > be fixed; when r <λ 1 p a x u a t + T x =u a t x (9) then the solution to 7 stabilizes as t + toward a unique (stable) nonnegative T periodic solution. Last, in the limiting case r = λ 1 p a x u a t + T x =u a t x (1) the L 2 A norm of solutions to 7 is still diverging. In this paper we are interested in the internal stabilizability of 7 under feedback control lying on ω, where ω is a subdomain of ; that is, we have to find a feedback control u such that lim t p a t x = a.e. a x A. Another approach, i.e., approximate controllability, is devised in [1]; see also [2]. Let ω be a nonempty subdomain of satisfying ω ω. We denote by λ ω 1 the first eigenvalue of the Dirichlet problem for in \ ω, w = λw on \ ω It follows that λ 1 <λ ω 1. Our main result will be proved in Section 4. w x = on Ɣ ω (11) Theorem 1.1. If r <λ ω 1, then there exists a feedback control u a t x with support in A ω that stabilizes (7). Remark 1.2. Note that when r <λ 1, this can be done with u a t x =, while for λ 1 r <λ ω 1 a nontrivial control is required. On the other hand, if r λ ω 1 >λ 1 and p a x p, then there is no control with support in ω such that lim t + p t = in an appropriate space.

4 94 ainseba, langlais, and aniţa 2. ASSUMPTIONS Let be a bounded open set in d with a smooth boundary Ɣ so that locally lies on one side of Ɣ. Next, ω is a nonempty open subset of having a positive d-dimensional Lebesgue measure, such that ω ω, and χ ω is the characteristic function of ω. We define u if u sgn L 1 ω u = u L 1 ω, L ω m L 1 ω m L 1 ω 1 if u L 1 ω =, { 1 if u>, sgn u = if u =, 1 if u<. (12) Let A be a finite positive number. One assumes the fertility rate β, the mortality rate µ, and the initial data satisfy H 1 β L A β a a.e. a A, and a 1 = max supp β <A. H 2 there exists a a 1 such that β a = a.e. a a a 1 A, H 3 µ L loc A µ a a.e. a A, H 4 H 5 A µ a da =+, p L A p a x a.e. in A. The divergence condition in H 4 means that each individual in the population dies before age A. For each A a A,wesetQT A = A T. 3. STABILIZABILITY FOR THE HEAT EQUATION In this section y is an initial condition and q is a potential satisfying H 6 y q L y x a.e. in. Along the characteristic lines of t + a, problem (7) resembles a parabolic problem (but with the integral condition at a = ). Hence we begin by proving a related stabilization result for the heat equation, having its own interest, using controls supported in ω.

5 internal stabilizability of diffusive models Partial Zero Exact Feedback Controllabilityfor the Heat Equation Consider the problem y t y + q x y ρ sgn L 1 ω yχ ω t T x y t x = t T x Ɣ (13) y x =y x x We introduce the following definition for a solution of this parabolic problem. Definition 3.1. An element y in C T L 1 is a weak solution of (13) if there exists an element u in L T L 1 such that u t sgn L 1 ω y t a e t T, and such that y T x ζ T x dx y x ζ x dx T y t x ζ t + x ζ qζ t x dx dt = ρ T ω u t x ζ t x dx dt ζ W = h C 1 2 T h t x = on T Ɣ Theorem 3.2. Given y in L 1 y x a.e. in, and q in L, there exists a ρ y q such that, for ρ>ρ y q, there exists a T = T ρ > such that problem 13 has a unique weak solution y in T, and in addition and Proof. y T x = y t L 1 ω > The operator defined via a.e. x ω t T D = y W 1 1 y L 1 y = y q y is the generator of a compact C -semigroup in L 1 ; see [4]. For each ε>, consider the regularized problem y t x y t y + q x y = ρ y t L 1 ω + ε χ ω x t> x y t x = t> x Ɣ (14) y x =y x x

6 96ainseba, langlais, and aniţa The mapping from L 1 to L 1 y y y L 1 ω + ε χ ω is locally Lipschitz continuous. As a consequence, problem 14 has a unique mild solution y ε C + L 1, in the sense of Pazy [1]. Since ρ y ε t / y ε t L 1 ω + ε χ ω L 1 ρ a.e. t> and using the Baras compactness theorem, see [3], one gets that, for some subsequence ε, for any T>. Hence, as ε, y ε y in C T L 1 y ε t L 1 ω y t L 1 ω (and in L T T>). Now there exists a T + such that y t L 1 ω > y T L 1 ω = t T t T (15) see a proof below. Passing to the limit ε in 14, one gets that y is a solution of 13 on T. Let us now prove (15). Indeed, assume that y t L 1 ω > for any t>. Then, multiplying 13 by sgn y and integrating over t, one has y t L 1 y L 1 q L t y s L 1 ds tρ Set z t = y L 1 t and α> q L, a constant; it follows that Hence and z t = y t L 1 y L 1 + αz t tρ e αt z t y L 1 e αt ρte αt e αt 1 t z t y L 1 α 1 e αt ρ se αs ds t T = y L 1 1 e αt ρ ( t α α + 2 ρe αt α + 1 ) α 2 As a consequence, z t ρ α 2 eαt + ρt α + ρ α + y L1 e αt 1 2 α

7 internal stabilizability of diffusive models 97 and y t L 1 ρ α eαt + ρ α + y L 1 e αt We now observe that for ρ large enough, with respect to q and y, there exists a t such that y t L 1 < ; a contradiction. So is nonempty, and T = inf B. B = t + y t L 1 ω = Remark 3.3. From the proof, it should be clear that this solution is nonnegative and bounded from above on T by the classical solution of the control free problem y t y + q x y = y t x = y x =y x t + x t + x Ɣ 3.2. Characterization x In order to extend after time T the solution y = lim ε y ε constructed in the proof of Theorem 3.2, we introduce the following parabolic problem in \ ω, y t y + q x y = t T + x \ ω y t x = t T + x Ɣ ω (16) y T + x =y 1 x x \ω y 1 x y 1 L \ ω, supplemented by the condition in ω y t x = t T + x ω (17) Note that y = lim ε y ε satisfies (16) with y 1 x =y T x, but the boundary conditions along T + ω. The initial condition at T being bounded, the solution of (16) is smooth on T + \ ω. Definition 3.4. Given any nonnegative y 1 in L \ ω, let µ y be the measure defined on T + by + y µ y ϕ = ϕ dσ dt T ω ν ϕ C T + where ν is the unit outward normal to ω and y ν x y x εν y x =lim for x ε ε ω

8 98 ainseba, langlais, and aniţa Then we have the following result: Lemma 3.5. For t T and for any τ> y satisfying (16) and (17) belongs to L T T + τ L 2 T T + τ H 1 H2 \ ω, and is a solution to y t y + q x y = µ y t T + x y t x = t T + x Ɣ { (18) y1 x x \ ω, y T + x = x ω. Proof. First, (16) has a unique solution lying in L T T + τ \ ω and L 2 T T + τ H 1 \ ω. Then this solution extended by on T + ω satisfies the condition listed in the lemma. Multiplying (16) by a smooth test function vanishing on T + Ɣ and integrating on T T \ ω, one gets y T x ϕ T x dx y 1 x ϕ T x dx \ ω \ ω y ϕ t + ϕ qϕ t x dx dt T T \ ω y ϕ t x dσ dt = T T ω ν One observes that, from (17), one may replace the integrals on \ ω by integrals on in the latter relation. This shows that any y satisfying (16) (17) is a weak solution to (18). As a first consequence, one has Proposition 3.6. Let y be the solution of (13) found in Theorem 3.2, and let T = T ρ y q be such that y x T = a.e. x ω. Setting y 1 x = y T x a.e. x \ω, the measure µ y defined in Definition 3.4 stabilizes the system if and only if λ q ω 1, the first eigenvalue of y + q x y = λy y x = x \ ω x Ɣ ω is positive; µ y is a feedback control with support in T + ω. (19) Proof. Going back to (16), one may deduce that for any y 1 L \ω, the solution y t ast + in L \ω provided λ q ω 1 >. Now, solving (13) along the lines of Theorem 3.2 on T ρ y q, one gets y T ρ y q L. Choosing y 1 x =y x T ρ y q, the solution of (16) (17) goes to as t +.

9 internal stabilizability of diffusive models 99 This can be expressed as: the corresponding measure µ y is a feedback control with support in T + ω. Now we are able to state our main results concerning the stabilizability of the heat equation with a bounded potential. Assuming the first eigenvalue of (19) is nonpositive, then a solution of (16) does not converge to as t +. It follows Theorem 3.7. System (13) is stabilizable if and only if the first eigenvalue of (19) is positive. If it is stabilizable, then the feedback control µ y in Proposition 3.6 stabilizes the system. 4. PROOF OF THE STABILIZABILITY RESULT FOR THE AGE AND SPACE DEPENDENT POPULATION DYNAMICS MODEL Let us consider the problem t p + a p x p + µ a p ρsgn L ω pχ 1 ω a t x QT A p a x =p a x p t x = A β a p a t x da a x A t x T p a t x = a t x A T Ɣ (2) where ρ is a nonnegative number to be chosen later, T is a positive number, and A a 1 A. Definition 4.1. A function p L A T L 1 is a solution of 2 if there exists a u in L 1 loc T L1 A u ρsgn L ω pχ 1 ω, such that t p + a p x p + µ a p = u a t x a t x Q A T lim p a + ε ε = p a in L 1 a.e. a A ε + A (21) lim p ε t + ε = β a p a t da in L 1 a.e. t T ε + p a t x = a t x A T Ɣ and p C S L 1 AC S L 1, for almost any characteristic line S of equation S = γ + t θ + t t T γ θ A T T T

10 1 ainseba, langlais, and aniţa 4.1. Existence Theorem 4.2. Under the hypotheses H 1 H 5, there exists a nonnegative number ρ such that 2 has a solution. Proof. Let p 1 be an arbitrary fixed nonnegative function in L T, and let us study the problem p t +p a +µp x p ρsgn L 1 ω p a t x χ ω p a t x = p t x =p 1 t x p a x =p a x a t x A T Ɣ a t x A T t x T a x A (22) We can view 22 as a collection of parabolic systems on the characteristic lines S. Define p t x =p γ + t θ + t x µ t =µ γ + t t x T t x T Then, p satisfies p t + µ p x p ρsgn L 1 ω p t x χ ω p t x = t x T Ɣ { p1 θ x γ =, p x = x p γ x θ =, t x T (23) Using the results from the previous section, we may conclude that for ρ large enough, system 23 has a unique weak solution on every characteristic line of equation a t =constant, because p L A L 1 p 1 L T L 1, and µ L A. Using now H 2, we infer by an iterative procedure the existence of a unique weak and nonnegative solution of (2) with u = ρsgn L ω pχ 1 ω. For T = a, the solution of (2) on A a is actually given by (22) when substituting p 1 t x by A a β a p a t x da. Note this integral depends only on p for <t<a. More precisely, for a t satis- fying <t<a<a and t<a, one finds that p a t depends only on p and one derives from a comparison principle that p a t x p L A. Next, in the small triangle wherein <a<t<a, one can check that p a t depends only on A a β a p a t x da; again p a t x C 1 p L A a. Assume now that the solution of (2) is known on A na, where n is a positive integer with p a t x C n p L A a

11 internal stabilizability of diffusive models 11 on A na. Then, proceeding as in the first step, the solution of (2) on A na n + 1 a is the solution of (22) when substituting p a x =p na t x and p 1 t x = A a β a p a t x da. One also gets p a t x C n+1 p L A a on A n + 1 a Proof of Theorem 1.1 Let p be a solution of (2). Assume that p t L 1 ω A > for any t>. Multiplying (2) by sgn p and integrating over t A, we obtain A t ( t + ) a + µ a p a s L 1 ds da ρat So p t L 1 A C + β L A t A p a s x da dx ds ρat and once again, for ρ large enough, as in the end of the proof for Theorem 3.2, there exists T> such that the solution to (2) vanishes on ω A at t = T. Now, on \ ω p satisfies, for t T, t p + a p x p + µ a p = p a T + x =p a T x p t x = A p a t x = β a p a t x da a t x A T + \ ω a x A \ ω (24) t x T + \ ω a t x A T + Ɣ ω Assume for a while that p a T x is separable; i.e., there exists a function v in L 2, and a function φ defined below such that p a T x =v x φ a. Then one can consider separable solutions of (24); that is, solutions of the form where v satisfies v t v = p a t x =e r t v x t φ a v x =v x v t x = t T + x \ ω x \ ω t T + x Ɣ ω

12 12 ainseba, langlais, and aniţa and ( φ a =exp a ) µ s ds r a Let us denote by λ ω i y i the eigenvalues and eigenfunctions of (11), and by the scalar product in L 2 \ ω. Then ( a ) p a t x =exp r t exp λ ω i t v y i y i x exp µ s ds r a i=1 As a conclusion, when r λ ω 1 <, then p a t x, in \ ω, ast. Last, from its construction, for each given initial data p a x satisfying H 5, the solution is bounded at time T ; hence one can find a function v x and a nonnegative number α sufficiently large such that p a T x αv x φ a a x A \ ω Using comparison results [2, 5], one has p a t x p a T x p a t x αv x φ a for a t x A T + + \ ω. The stabilizability result follows. REFERENCES 1. B. E. Ainseba and M. Langlais, On a population dynamics control problem with age dependence and spatial structure, J. Math. Anal. Appl. 248 (2), S. Anita, Analysis and Control of Age-Dependent Population Dynamics, Kluwer Academic Publishers, Dordrecht, P. Baras, Compacité de l opérateur f u solution d une équation non linéaire du/dt + Au f, C.R. Acad. Sci. Paris, Série A 286 (1978), H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Dunod, Paris, M.-G. Garroni and M. Langlais, Age dependent population diffusion with external constraints, J. Math. Biol. 14 (1982), M.-E. Gurtin, A system of equations for age dependent population diffusion, J. Theor. Biol. 4 (1972), M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Applied Mathematics Monographs, No. 7, C.N.R., Pisa, M. Kubo and M. Langlais, Periodic solutions for a population dynamics problem with age dependence and spatial structure, J. Math. Biol. 29 (1991), M. Langlais, Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion, J. Math. Biol. 26 (1988), A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.