INTERNAL EXACT CONTROLLABILITY OF THE LINEAR POPULATION DYNAMICS WITH DIFFUSION

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1 Elecronic Journal of Differenial Equaions, Vol. 24(24), No. 112, pp ISSN: URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu (login: fp) INTERNAL EXACT CONTROLLABILITY OF THE LINEAR POPULATION DYNAMICS WITH DIFFUSION BEDR EDDINE AINSEBA, SEBASTIAN ANIŢA Absrac. We consider he inernal exac conrollabiliy of a linear age and space srucured populaion model wih nonlocal birh process. The conrol acs only in a spaial subdomain and only for small age classes. The mehods we use combine he Carleman esimaes for he backward adjoin sysem, some esimaes in he heory of parabolic boundary value problems in L k and he Banach fixed poin heorem. 1. Inroducion Le be a bounded domain in R n (n 3) wih a smooh boundary. Assume ha a biological populaion is free o move in he environmen. We denoe by y(a,, x) he densiy of individuals of age a a ime and locaion x and assume ha he flux of populaion akes he form k y(a,, x) wih k >, where is he gradien vecor wih respec o he spaial variable. Le A be he life expecancy of an individual and T be a posiive consan. Le β(a) be he naural feriliy rae and µ(a) he naural moraliy rae corresponding o individuals of age a. The dynamics of he populaion is described by he following model Dy + µ(a)y k y = f(a, x) + m(a, x)u(a,, x), y(,, x) = y (a,, x) =, ν (a, A β(a)y(a,, x)da,, x) Σ T (a,, x) Q T (, x) (, T ) y(a,, x) = y (a, x), (a, x) (, A), (1.1) where u is he conrol and m is he characerisic funcion of (, a ) ω, f is he densiy of an infusion of populaion and y is he iniial populaion densiy. Here a (, A] and ω is a nonempy open subse, Q T = (, A) (, T ), Σ T = (, A) (, T ). We denoe by Dy(a,, x) = lim ε y (a + ε, + ε, x) y (a,, x) ε 2 Mahemaics Subjec Classificaion. 93B5, 35K5, 46B7, 92D25. Key words and phrases. Exac conrollabiliy; age-srucured populaion dynamics. c 24 Texas Sae Universiy - San Marcos. Submied January 4, 24. Published Sepember 29, 24. 1

2 2 B. AINSEBA, S. ANIŢA EJDE-24/112 he direcional derivaive of y wih respec o he direcion (1, 1, ). If y is smooh enough hen Dy = y + y a. The conrol acs only in he spaial se ω and for ages beween and a. Le y s be a nonnegaive seady-sae of (1.1), corresponding o u and such ha y s (a, x) ρ > a.e. (a, x) (, a 1), (1.2) where ρ > is consan and a 1 (, A) is a consan which will be defined laer. The main goal of his paper is o prove he exisence of a conrol u such ha he soluion y of (1.1) saisfies y(a, T, x) = y s (a, x) a.e. (a, x) (, A), y(a,, x) a.e. (a,, x) Q T. (1.3) Condiion (1.3) is naural because y represens he densiy of a populaion. We noice ha if y is he soluion o (1.1), hen y y s is he soluion o Dz + µ(a)z k z = m(a, x)u(a,, x), z(,, x) = z (a,, x) =, ν (a, A β(a)z(a,, x)da,, x) Σ T (a,, x) Q T (, x) (, T ) z(a,, x) = z (a, x), (a, x) (, A), (1.4) where z = y y s. The above formulaed problem is equivalen o he exac null conrollabiliy problem wih sae consrains for (1.4). Indeed, if we denoe now by z he soluion o (1.4), hen condiion (1.3) becomes z(a,, x) y s (a, x) a.e. (a,, x) Q T. We recall ha he inernal null conrollabiliy of he linear hea equaion, when he conrol acs on a subse of he domain, was esablished by G. Lebeau and L. Robbiano [13] and was laer exended o some semilinear equaion by A.V. Fursikov and O.Yu. Imanuvilov [6], in he sublinear case and by V. Barbu [4] and E. Fernandez Cara [5], in he superlinear case. The inernal null conrollabiliy of he age-dependen populaion dynamics in he paricular case when he conrol acs in a spaial subdomain ω bu for all ages a (his is he paricular case corresponding o a = A) was invesigaed by B. Ainseba and S. Aniţa [2]. This paper is organized as follows. We firs give he hypoheses and sae he main resul. The exisence of a seady sae of (1.1) wih u is esablished in Secion 3. The proof of he local exac null conrollabiliy is given in Secion 4. The proof is based on Carleman s inequaliy for he backward adjoin sysem associaed wih (1.4). 2. Assumpions and he main resul Assume ha he following hypoheses hold: (H1) β L (, A), β(a) a.e. a (, A) There exiss a, a 1 (, A), a < a 1, such ha β(a) = a.e. a (, a ) (a 1, A) and β(a) > a.e. in (a, a 1 )

3 EJDE-24/112 INTERNAL EXACT CONTROLLABILITY 3 (H2) µ C([, A)), µ(a) a.e. a (, A), A µ(a)da = + (H3) y L ((, A) ), y (a, x) a.e. in (, A) f L ((, A) ), f(a, x) a.e. in (, A). For he biological significance of he hypoheses and he basic exisence resuls for he soluion o (1.1) we refer o [3, 7, 8, 9, 11, 15]. Le y s be a nonnegaive seady-sae of (1.1), corresponding o u and such ha y s (a, x) ρ > a.e. (a, x) (, a 1 ), where ρ > is a consan. Denoe by z = y y s. Then we have he following inernal conrollabiliy resul Theorem 2.1. Le T > A a be arbirary bu fixed. If y y s L ((,A) ) is small enough, hen here exiss u L 2 (Q T ) such ha he soluion y of (1.1) saisfies y(a, T, x) = y s (a, x) a.e. (a, x) (, A) (2.1) y(a,, x) a.e. (a,, x) Q T. If T < A a and if y y s L ((a,a T ) ) >, hen here is no u L 2 (Q T ) such ha he soluion y of (1.1) o saisfy (2.1). This resul can be equivalenly formulaed as follows Theorem 2.2. Le T > A a be arbirary bu fixed. If z L ((,A) ) is small enough, hen here exiss u L 2 (Q T ) such ha he soluion z of (1.4) saisfies z(a, T, x) = a.e. (a, x) (, A) z(a,, x) y s (a, x) a.e. (a,, x) Q T. (2.2) If T < A a and if z L ((a,a T ) ) >, hen here is no u L 2 (Q T ) such ha he soluion z of (1.4) o saisfy (2.2). 3. Exisence of seady saes for (1.1) In his secion we shall remind some resuls (see [2]) concerning he exisence of y s, a nonnegaive seady-sae of (1.1), corresponding o u, which saisfies (1.2). y s should be a soluion o Denoe by y s a + µ(a)y s k y s = f(a, x), (a, x) (, A) y s (a, x) =, (a, x) (, A) ν y s (, x) = R = A A β(a)y s (a, x)da, x. β(a) exp ( a µ(s)dsda ) he reproducive number and consider f a nonnegaive consan. (3.1) Theorem 3.1. If R < 1 and f(a, x) f > a.e. (a, x) (, A), hen here exiss a unique nonnegaive soluion o (3.1), which in addiion saisfies (1.2).

4 4 B. AINSEBA, S. ANIŢA EJDE-24/112 If R = 1 and f, hen here exis infiniely many nonnegaive soluions o (3.1), which saisfy (1.2). If R > 1, hen here is no nonnegaive soluion o (3.1), saisfying (1.2). Proof. If R < 1, hen here exiss a unique and nonnegaive soluion o (3.1) (his follows by Banach s fixed poin heorem). Since f(a, x) f > a.e. (a, x) (, A), hen by he comparison resul in [7](see also [3]) we ge ha y s (a, x) y i (a,, x) a.e. (a,, x) Q = (, A) (, + ), where y i is he soluion o y i (,, x) = Dy i + µy i k y i = f, A y i ν =, β(a)y i (a,, x)da, y i (a,, x) =, (a,, x) Σ (a,, x) Q (, x) (, + ) (a, x) (, A) Noe ha Σ = (, A) (, + ) ); y i does no explicily depend on x. So, we shall wrie y i (a, ) insead of y i (a,, x). I means ha y s (a, x) y i (a, ) [, + ), a.e.(a, x) (, A), and ha y i is he soluion of Dy i + µy i = f, y i (, ) = A (a, ) (, A) (, + ) β(a)y i (a, )da, y i (a, ) =, a (, A). (, + ) For > A we have y i (, ) > and y i (, ) is coninuous wih respec o (see [3]). As a consequence we obain ha here exiss ρ > such ha, for large enough, and for any a (, a 1), y i (a, ) > ρ, and in conclusion we ge ha y s saisfies (1.2). If R = 1 and f, hen all he soluions of (3.1) which are saisfying (1.2) are given by y(a, x) = ce a µ(s)ds, (a, x) (, A), where c R + is an arbirary consan. The conclusion is now obvious. If R > 1 and if i would exis a nonnegaive soluion y s o (3.1) saisfying (1.2), hen y(a,, x) = y s (a, x), (a,, x) Q is he soluion o y(,, x) = Dy + µy k y = f(a, x), A y ν =, β(a)y(a,, x), y(a,, x) = y s (a, x), and for + we have (see [3, 12]) (a,, x) Σ (a,, x) Q (, x) (, + ) (a, x) (, A) lim y() L + 2 ((,A) ) = +.

5 EJDE-24/112 INTERNAL EXACT CONTROLLABILITY 5 On he oher hand y() L2 ((,A) ) = y s L2 ((,A) ), and so y s L2 ((,A) ) = +, which is absurd. 4. Proof of he main resul We shall prove Theorem 2.2 (which is equivalen o Theorem 2.1). We inend o use he general Carleman inequaliy for linear parabolic equaions given in [6]. Namely, le ω ω be a nonempy bounded se, T (, + ) and ψ C 2 () be such ha and se ψ(x) >, x, ψ(x) =, x, ψ(x) >, x \ ω α(, x) = eλψ(x) e 2λ ψ C(), (T ) where λ is an appropriae posiive consan. Denoe by D T = (, T ). Lemma 4.1. There exis posiive consans C 1, s 1 such ha 1 ( (T ) e 2sα w 2 + w 2) dx d s D T DT e 2sα DT e 2sα + s (T ) w 2 dx d + s 3 3 (T ) 3 w 2 dx d [ C 1 e 2sα w + w 2 dx d + s 3 e 2sα ] D T (,T ) ω 3 (T ) 3 w 2 dx d, for all w C 2 (D T ), w ν (, x) =, (, x) (, T ) and s s 1. The proof of his resul can be found in [6]. (4.1) If a = A, he resul has already been proved in [2]. We shall rea now he case a (, A). Consider a 1 := a. Le us choose T (, min{a, a, A a, T A + a, A a 1 }). Define K = L ((, A a + T ) ). In wha follows we shall denoe by he same symbol C, several consans independen of z and all oher variables. For b K arbirary bu fixed and for any ε >, consider he following opimal conrol problem: Minimize { G ϕ(a,, x) u(a,, x) 2 dx d da + 1 ε Γ } z(a,, x) 2 dx dl, (4.2) subjec o (4.3) (u L 2 (G ) and z is he soluion of (4.3) corresponding o u). Here G = (, a ) (, T ) (, T ) (, A a + T ), Γ = {T } (T, A a + T ) (T, a ) {T }, { e 2sα(,x) 3 (T ) 3, if < a, (a, ) G ϕ(a,, x) = e 2sα(a,x) a 3 (T a) 3, if a <, (a, ) G

6 6 B. AINSEBA, S. ANIŢA EJDE-24/112 (See figure 1). Dz + µz k z = m(a, x)u(a,, x), z ν =, z(,, x) = b(, x), (a,, x) G (a,, x) G (, x) (, A a + T ) z(a,, x) = z (a, x), (a, x) (, a ). (4.3) T A a + T A a T O T a a T a a 1 A a Figure 1. Denoe by Ψ ε (u) he value of he cos funcion in u. Since he cos funcion Ψ ε : L 2 (G ) R + is convex, coninuous and lim Ψ ε(u) = +, u L 2 (G ) + hen i follows ha here exiss a leas one minimum poin for Ψ ε and consequenly an opimal pair (u ε, z ε ) for (P ε ). By sandard argumens we have u ε (a,, x) = m(x)q ε (a,, x)ϕ 1 (a,, x) a.e. (a,, x) G, (4.4) where m is he characerisic funcion of ω and q ε is he soluion of Dq µq + k q =, q ν =, q(a,, x) =, (a,, x) G (a,, x) G (a,, x) (Γ \ Γ ) (4.5) q(a,, x) = 1 ε z ε(a,, x), (a,, x) Γ. Here Γ = (, T ) {A a + T } {a } (, T ) Γ.

7 EJDE-24/112 INTERNAL EXACT CONTROLLABILITY 7 Muliplying he firs equaion in (4.5) by z ε and inegraing on G we obain afer some calculaion (and using (4.3) and (4.4)) ha ϕ(a,, x) u ε (a,, x) 2 dx da d + 1 G ω ε A a +T = b(, x)q ε (,, x)dx d Γ a Le S be an arbirary characerisic line of equaion z ε (a,, x) 2 dx dl z (a, x)q ε (a,, x)dx da. S = {(γ +, θ + ); (, T ), (γ, θ) (, a T ) {} {} (, A a )}. Define Noe ha (ũ ε, z ε ) saisfies ũ(, x) = u(γ +, θ +, x), z ε (, x) = z ε (γ +, θ +, x), q ε (, x) = q ε (γ +, θ +, x), (, x) (, T ) µ() = µ(γ + ), (, T ). (, x) (, T ) (, x) (, T ) ( z ε ) + µ z ε k z ε = m(x)ũ ε (, x), (, x) (, T ) z ε ν =, (, x) (, T ) { b(θ, x) γ =, x z ε (, x) = z (γ, x) θ =, x (4.6) By (4.4) we ge ha ũ ε (, x) = m(x) q ε (, x) e 2sα(,x) 3 (T ) 3 (4.7) a.e. (, x) (, T ), ( q ε ) + k q ε = µ q ε, (, x) (, T ) q ε ν =, (, x) (, T ) q ε (T, x) = 1 ε z ε(t, x) x. (4.8) Muliplying he firs equaion in (4.8) by z ε and inegraing on D T, we obain ha T = ω e 2sα(,x) 3 (T ) 3 ũ ε (, x) 2 dx d + 1 z ε (T, x) 2 dx ε z ε (, x) q ε (, x)dx. (4.9)

8 8 B. AINSEBA, S. ANIŢA EJDE-24/112 By Carleman s inequaliy (4.1) we infer ha T e 2sα [ (T ) ( ( q ε ) s 2 + q ε 2) + s (T ) 3 q ε 2 ]dx d [ T C 1 e 2sα µ 2 C([,T q ]) ε 2 dx d + s 3 and consequenly T e 2sα [ (T ) ( ( qε ) s 2 + q ε 2) + s (T ) 3 q ε 2 ]dx d T C ω e 2sα s 3 3 (T ) 3 q ε 2 dx d, s (T ) q ε 2 (,T ) ω e 2sα ] 3 (T ) 3 q ε 2 dx d s (T ) q ε 2 (4.1) for s max(s 1, C µ 2 3 C([,a ]) ). Muliplying he firs equaion in (4.8) by q ε we obain ha 1 d q ε (, x) 2 dx k q ε (, x) 2 dx µ() q ε (, x) 2 dx = 2 d and d q ε (, x) 2 dx a.e. (, T ). d Inegraing he las inequaliy we ge ha T q ε (, x) 2 dx C q ε (, x) 2 e 2sα(x,) 3 (T ) 3 dx. and by Carleman s inequaliy we have ha T q ε (, x) 2 dx C q ε (, x) 2 e 2sα(x,) 3 dx d. (4.11) (T ) 3 By Young s inequaliy, (4.9), (4.11) and (4.7) we obain ha e 2sα 3 (T ) 3 ũ ε (, x) 2 dx d + 1 z ε (T, x) 2 dx ε (,T ) ω C z ε () 2 L 2 (), for s max(s 1, C µ 2 3 C([,a ])). Using now (4.1) we ge T e 2sα [ (T ) ( ( qε ) s 2 + q ε 2) s + (T ) q ε 2 s (T ) 3 q ε 2 ]dx d C z ε () 2 L 2 (), for any ε > and consequenly ω ṽ ε 2 W 1,2 2 ((,T ) ) C z ε() 2 L 2 (),

9 EJDE-24/112 INTERNAL EXACT CONTROLLABILITY 9 where ṽ ε (, x) = e2sα(,x) 3 (T ) 3 q ε, (, x) (, T ). As W 1,2 2 ((, T ) ) L l ((, T ) ) (where l = + for N = 1, 2 and l = 1 for N = 3), we may infer ha ũ ε 2 L 1 ((,T ) ) = mṽ ε 2 L 1 ((,T ) ) C z ε() 2 L 2 (), (4.12) for any ε > and s max(s 1, C µ 2 3 C([,a ]) ). The las esimae and he exisence heory of parabolic boundary value problems in L r (see [1]) imply ha on a subsequence (also denoed by (ũ ε )) we have ha ũ ε ũ weakly in L 1 ((, T ) ) z ε zũ where ( ũ, zũ) saisfies (4.6) and By (4.6) we ge ha zũ 2 ( ) C L (,T ) weakly in W 1,2 1 ((, T ) ), zũ(t, x) = a.e. x. ( ) zũ() 2 L () + mũ 2 L 3 ((,T ) ) (we recall ha W 1,2 3 ((, T ) ) L ((, T ) ) for N {1, 2, 3}; see [1, 1]). So by (4.12) we have zũ 2 L ((,T ) ) C zũ() 2 L (). We exend u given by ũ (on each characerisic line) by. In his manner we ge ha u L 2 (Q T ). Le z u be he soluion o Dz + µz k z = m(a, x)u(a,, x), (a,, x) (, A) (, A a + T ) z ν =, (a,, x) (, A) (, A a + T ) z(,, x) = b(, x), (, x) (, A a + T ) z(a,, x) = z (a, x), (a, x) (, A). Since z u = on Γ and u = ouside G we conclude ha z u (a,, x) = a.e. in {(a,, x); (T, A a +T ), T < a < +a T, x }, z u (a, A a +T, x) = a.e. (a, x) (T, A) and ha z u L (Q A a +T ) C( z L ((,A) ) + b L ((,A a +T ) )). (4.13) We are now ready o prove he exac null conrollabiliy resul. For any b K, we denoe by Φ(b) L 2 ((, A a + T ) ) he se of all A β(a)zu (a,, x)da, such ha u L 2 (Q A a +T ), u = ouside G, where z u saisfies (4.13) and z u (a,, x) = a.e. in {(a,, x); (T, A a + T ), T < a < + a T, x }, z u (a, A a + T, x) =, a.e. (a, x) (T, A). There exiss an elemen in Φ(b) which does no depend on b: If > T, hen A β(a)zu (a,, x)da = A β(a)z u (a,, x)da and does no depend on b.

10 1 B. AINSEBA, S. ANIŢA EJDE-24/112 If (, T ), hen A β(a)zu (a,, x)da = A T T β(a)z u (a,, x)da, and his depends only on z and no on b. We also have ha z u (a, A a + T, x) = a.e. (a, x) (T, A) and A T T β(a)z u (a,, x)da C β L (,A) z L ((,A) ) (4.14) a.e. in (, A a + T ). I also follows ha A β(a)z u (a,, x)da = T β(a)z u (a,, x)da + A A T β(a)z u (a,, x)da = a.e. (, x) (A a, A a + T ) (because β(a) = on (, T ) (A T, A)). So, for any u as above we can ake { a.e. (, x) (A a, A a + T ) b(, x) = A β(a)zu (a,, x)da a.e. (, x) (, A a ) a fixed poin of he mulivalued funcion Φ. In addiion, by (4.13) and (4.14) we have z u L (Q A a +T ) C z L ((,A) ). So, if z L ((,A) ) is small enough, here exiss u L 2 (Q A a +T ), u = on (a, A) (A a, A a + T ), such ha z, he soluion of (1.4) (wih T := A a + T ) saisfies z(a, A a + T, x) = a.e. (a, x) (, A), z L (Q A a +T ) C z L ((,A) ) ρ. In conclusion z(a,, x) ρ a.e. (a,, x) Q A a +T. This implies (via Theorem 3.1) ha z(a,, x) y s (a, x) a.e. (a,, x) (, a ) (, T ). On he oher hand mu = on (a, A) (, T ). The comparison principle for parabolic equaions allows us o conclude ha z(a,, x) y s (a, x) a.e. (a,, x) (a, A) (, T ). For he second asserion of Theorem 2.2 we assume by conradicion ha T < A a (his also implies ha a < A), z L ((a,a T ) ) > and here exiss u L 2 (Q T ) such ha z u he soluion of (1.4) saisfies (2.2) (see figure 2). Since mu = on (a, A) (, T ) we may conclude ha z u does no explicily depend on u on S, where S = {(a, ); a (a, A), (, T ), < a a }. However we have ha z u saisfies Dz u + µ(a)z u k z u =, (a,, x) S z u (a,, x) =, ν (a,, x) S z u (a,, x) = z (a, x), (a, x) (a, A), and since z L ((a,a T ) ) >, we conclude ha z u (, T, ) L ((,A) ) > (his follows via he backward uniqueness heorem); which is in conradicion o (2.2). So, we ge he conclusion.

11 EJDE-24/112 INTERNAL EXACT CONTROLLABILITY 11 A a T O a A a Figure 2. References [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, [2] B. Ainseba, S. Aniţa, Local exac conrollabiliy of he age-dependen populaion dynamics wih diffusion, Absrac Appl. Anal., 6 (21), [3] S. Aniţa, Analysis and Conrol of Age-Dependen Populaion Dynamics, Kluwer Acad. Publ., 2. [4] V. Barbu, Exac conrollabiliy of he superlinear hea equaion, Appl. Mah. Opim., 42 (2), [5] E. Fernandez-Cara, Null conrollabiliy of he semilinear hea equaion, ESAIM:COCV, 2 (1997), [6] A. V. Fursikov, O.Yu. Imanuvilov, Conrollabiliy of Evoluion Equaions, Lecure Noes Series 34, RIM Seoul Naional Universiy, Korea, [7] M. G. Garroni, M. Langlais, Age dependen populaion diffusion wih exernal consrains, J. Mah. Biol., 14 (1982), [8] M. E. Gurin, A sysem of equaions for age dependen populaion diffusion, J. Theor. Biol., 4 (1972), [9] M. Iannelli, Mahemaical Theory of Age-Srucured Populaion Dynamics, Giardini Ediori e Sampaori, Pisa, [1] O. A. Ladyzenskaya, V.A. Solonnikov, N.N. Uralzeva, Linear and Quasilinear Equaions of Paraboic Type, Nauka, Moskow, [11] M. Langlais, A nonlinear problem in age dependen populaion diffusion, SIAM J. Mah. Analysis, 16 (1985), [12] M. Langlais, Large ime behaviour in a nonlinear age dependen populaion dynamics problem wih spaial diffusion, J. Mah. Biol., 26 (1988), [13] G. Lebeau, L. Robbiano, Conrôle exac de l equaion de la chaleur, Comm. P.D.E., 3 (1995), [14] J. L. Lions, Conrôle des sysèmes disribués singuliers, MMI 13, Gauhier Villars, Paris, [15] G. F. Webb, Theory of Nonlinear Age-Dependen Populaion Dynamics, Marcel Dekker, New York, Bedr Eddine Ainseba Mahémaiques Appliquées de Bordeaux, UMR CNRS 5466, Case 26, UFR Sciences e Modélisaion, Universié Vicor Segalen Bordeaux 2, 3376 Bordeaux Cedex, France address: ainseba@sm.u-bordeaux2.fr Sebasian Aniţa Faculy of Mahemaics, Universiy Al.I. Cuza and Insiue of Mahemaics of he Romanian Academy, Iaşi 66, Romania address: sania@uaic.ro