Assia Benabdallah

Cédric Dupaix

Farid Ammar-Khodja

Manuel González-Burgos

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 345 (2007) 543–548http://france.elsevier.com/direct/CRASS1/Partial Differential Equations/Optimal ControlControllability for a class of reaction–diffusion systems:the generalized Kalman’s conditionFarid Ammar-Khodja a, Assia Benabdallah b, Cédric Dupaix a, Manuel González-Burgos ca Laboratoire de mathématiques UMR 6623, Université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, Franceb CMI-LATP, UMR 6632, Université de Provence, technopôle Château-Gombert, 39, rue F. Joliot-Curie, 13453 Marseille cedex 13, Francec Dpto. de Ecuaciones Diferentiales y Análisis Numérico, Universidad Sevilla, Aptdo. 1160, 41080 Sevilla, SpainReceived 15 January 2007; accepted after revision 2 October 2007Available online 9 November 2007Presented by Gilles LebeauAbstractIn this Note, we study the controllability of a class of parabolic systems of the form Yt = (D + A)Y + Bχωu with Dirichletconditions on the boundary of a bounded domain Ω , where ω ⊂ Ω is a subdomain. Here D,A ∈ L(Rn), B ∈ L(Rm;Rn) and weprove that the algebraic Kalman condition extends to such systems. To cite this article: F. Ammar-Khodja et al., C. R. Acad. Sci.Paris, Ser. I 345 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméContrôlabilité d’une classe de systèmes réaction–diffusion : la condition de Kalman généralisée. Dans cette Note, on étudiela contrôlabilité d’une classe de systèmes paraboliques de la forme Yt = (D + A)Y + Bχωu avec des conditions de Dirichlet surle bord d’un domaine borné Ω , où ω ⊂ Ω est un sous-domaine. Ici D,A ∈ L(Rn), B ∈ L(Rm;Rn) et on montre que la conditionalgébrique de Kalman s’étend à de tels systèmes. Pour citer cet article : F. Ammar-Khodja et al., C. R. Acad. Sci. Paris, Ser. I 345(2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeLe critère de Kalman est une condition nécessaire et suffisante de contrôlabilité de systèmes différentiels linéairesà coefficients constants. Ce travail a pour objectif de donner une extension de ce critère aux systèmes d’équationsaux dérivées partielles, plus précisément aux systèmes paraboliques linéaires de la forme (1) dont les coefficients sontindépendants du temps.La contrôlabilité de systèmes paraboliques est étudiée depuis les années 2000, essentiellement pour des systèmesde deux équations (voir [11,1,2,5]). Le résultat principal de ce travail est de démontrer que la condition de Kalman(dans un sens à préciser) est nécessaire et suffisante pour la contrôlabilité aux trajectoires de systèmes (1) satis-E-mail addresses: fammarkh@univ-fcomte.fr (F. Ammar-Khodja), assia@cmi.univ-mrs.fr (A. Benabdallah), cedric.dupaix@univ-fcomte.fr(C. Dupaix), manoloburgos@us.es (M. González-Burgos).1631-073X/$ – see front matter © 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2007.10.023544 F. Ammar-Khodja et al. / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 543–548faisant aux hypothèses suivantes : Ω ⊂ Rd est un ouvert borné régulier, ω ⊂ Ω est sous-ensemble ouvert de Ω ,T > 0, M = PDP −1, où D = diag(dii > 0)n×n ∈ L(Rn), R un opérateur elliptique autoadjoint donné par (2),A = (aij )1i, jn ∈ L(Rn), B ∈ L(Rm,Rn), v ∈ L2(ΩT )m (ΩT = (0, T ) × Ω), y0 = (y0,i )1in ∈ L2(Ω)n.Dans (1), y = (yi)1in est la variable d’état et 1ω la fonction caractéristique de ω. L’opérateur de Kalman, K estdéfini dans (3) et (K∗) désigne son adjoint dans L2(Ω)n. Notre principal résultat est :Théorème. Le système (1) est contrôlable aux trajectoires en tout temps T > 0 si et seulement siKer(K∗) = {0}.La condition nécessaire se vérifie aisément. La démonstration de la partie suffisante se fait en deux étapes. Dansun premier temps, la définition de l’opérateur K permet de transformer le système adjoint de (1) en un système encascade pour lequel on démontre une inégalité d’observabilité de type Carleman. Celle-ci est obtenue via une inégalitéde Carleman pour des opérateurs paraboliques d’ordre n et une inégalité d’observabilité pour des systèmes d’ordre 2.Dans une deuxième partie, on conclut en revenant au système initial grâce à (4).Le contrôle aux trajectoires de tels systèmes par le bord est ouvert.1. IntroductionControllability of linear differential systems is well-known. In particular we have at our disposal the famousKalman rank condition (see for example [8, Chapter 2, p. 35]), that is to say, if A ∈ L(Rn), B ∈ L(Rm,Rn), thenthe system Y ′ = AY + Bu is controllable at time T > 0 if and only if rank[A | B] = rank[B,AB, . . . ,An−1B] = n.Our main goal is to extend this algebraic condition to partial differential systems. For n,m ∈ N∗ and T > 0, weconsider the following n × n parabolic system:{∂ty = (MR + A)y + Bv1ω in ΩT = Ω × (0, T ),y = 0 on ΣT = ∂Ω × (0, T ), y(·,0) = y0(·) in Ω, (1)where Ω ⊂ Rd is a bounded open set with a C2-boundary ∂Ω , ω ⊂ Ω is an open subset, T > 0, M = PDP −1, whereD = diag(dii)n×n ∈ L(Rn), R is the second order elliptic self adjoint operator given byR = Σi,j ∂irij (x)∂j + c(x) (2)with rij ∈ C1(Ω), c ∈ L∞(Ω). A = (aij )1i, jn ∈ L(Rn), B ∈ L(Rm,Rn), v ∈ L2(ΩT )m (ΩT = (0, T ) × Ω), y0 =(y0,i )1in ∈ L2(Ω)n. In (1), y = (yi)1in is the state variable while 1ω denotes the characteristic function of theopen subset ω. We assume that dii > 0, ∀i ∈ {1, . . . , n}. The exact controllability to zero reads as follows: For givenT > 0 and y0 ∈ L2(Ω)n can we find v ∈ L2(ΩT )m such that the corresponding solution y of (1) satisfies y(T ) = 0a.e. in Ω? In the linear case, this property is equivalent to the controllability to trajectories. There are few resultson null controllability of system (1) when n > 1 and most of them are proved for n = 2: see, for instance, [11,1,2]and [5]. Also see [6] where the authors provide a null controllability result for a general cascade parabolic system ofn equations.To be clear, we will present our result in the case where M = D (i.e., P ≡ Id ) and R = . All the proofs canbe adapted to the general case. Let us now introduce the Kalman operator. If we denote by L := D + A withD(L) = D()n = (H 2(Ω) ∩ H 10 (Ω))n, then the Kalman operator associated with (L,B) is the operator matrix{K := [L | B] = [Ln−1B,Ln−2B, . . . ,LB,B] :D(K) ⊂ L2(Ω)nm → L2(Ω)n, withD(K) := {u ∈ L2(Ω)nm: Ku ∈ L2(Ω)n}, (3)Our main result is the following:Theorem 1.1. System (1) is exactly controllable to trajectories at any time T if and only if the Kalman operator KsatisfiesKer(K∗) = {0}. (4)F. Ammar-Khodja et al. / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 543–548 545Of course, this result is still true if  is replaced by R (defined in (2)). If (λp)p∈N∗ is the sequence of eigenvaluesof − with Dirichlet boundary conditions, we consider the matrices Lp = −λpD +A ∈ L(Rn) and Kp = [Lp | B] =[Ln−1p B, . . . ,LpB,B] ∈ L(Rnm,Rn), for every p ∈ N∗. It is not difficult to see that (4) is equivalent torankKp = n, ∀p ∈ N∗. (5)Note that either (5) reduces to a finite number of conditions or it is never satisfied.Remark 1.(i) The closest result of this kind we know is due to H. Leiva [9] which gives a necessary and sufficient conditionfor the approximate controllability of (1) in the case where A ≡ 0, D is not diagonal and B(x) ∈ Rn (m ≡ 1).(ii) A necessary and sufficient condition (if it exists) of controllability to trajectories in the case of controls acting ona subset of the boundary is still open for such systems.(iii) To our knowledge, the null controllability problem of (1) when the operators are time dependent is still open. Wewill address in details this question in a forthcoming paper. On the other hand, the case in which A,B depend onx seems to be much more complicated. A necessary and sufficient condition for the controllability of (1) is openeven if there are some partial results (see [11,2,1,5,6]). The methods used in this paper would hardly providegeneral results for non-autonomous systems since the proofs take advantage in a crucial way of the fact that thematrices considered are constant in time and space variables.2. Sketch of the proof of Theorem 1.1Controllability of System (1) is equivalent (see for instance [10,3], etc.) to the existence of a positive constant Csuch that, for every ϕ0 ∈ L2(Ω)n, the solution ϕ ∈ C0([0, T ];L2(Ω)n) of the following adjoint system{−∂tϕ = (D + A∗)ϕ in ΩT ,ϕ = 0 on ΣT , ϕ(·,T ) = ϕ0(·) in Ω, (6)(where we have used that D∗ = D) satisfies the observability inequality (with the notation ωT = ω × (0, T ))∥∥ϕ(·,0)∥∥2L2(Ω) C∫∫ωT∣∣B∗ϕ(x, t)∣∣2 dx dt. (7)We will need, in an essential way, the following properties of the Kalman operator:Proposition 2.1.(i) If u ∈ D(n−1)nm then Ku ∈ L2(Ω)n and ‖Ku‖L2(Ω)m  C‖u‖D(n−1)nm .(ii) Assume Ker(K∗) = {0}. Let r ∈ R. There exists C > 0 such that for every ϕ ∈ L2(Ω)n satisfying K∗ϕ ∈ D(r)nm,one has:‖ϕ‖2D(r−(n2−1)/2)n C‖rK∗ϕ‖2L2(Ω)nm.2.1. Necessary conditionSuppose that Ker(K∗) = {0}. It follows from (5) that there exists p0 ∈ N∗ such that 0 ∈ σ(Kp0). Therefore, thanksto Kalman’s rank condition applied to the pair (−λp0D + A,B), there exists a non-zero solution zp0(t) ∈ Rn of theadjoint system associated with (−λp0D + A,B){−z′ = (−λp0D + A∗)z in (0, T ),z(0) = zp0 ∈ Rn = 0,satisfying B∗zp0(t) = 0, ∀t ∈ [0, T ]. Then, letting ϕ0 = zp0(T )φp0 , where φp0 is an eigenfunction associatedwith λp0 , it is easy to check that ϕ(t, x) = zp0(t)φp0(x) is the solution of (6). It is non-zero and satisfiesB∗ϕ(t) = 0, ∀t ∈ [0, T ]. Obviously this solution does not satisfy the observability inequality (7), and thus (1) isnot null controllable.546 F. Ammar-Khodja et al. / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 543–5482.2. Sufficient condition2.2.1. A Carleman inequality for a parabolic equation of order nLet us denote by D = ⋂p0 D(p). If we set X = Dn then, one has:Proposition 2.2. If ϕ = (ϕ1, . . . , ϕn)∗ is the solution to (6) corresponding to ϕ0 ∈ X, thendet(Id∂t + D + A∗)ϕi = 0 in ΩT , ∀i: 1  i  n. (8)This proposition can be deduced as in the case of an ordinary differential system (see [7, §6.4, p. 144]).In order to state our second result, let us introduce some additional notations. If s, τ ∈ R, we define the followingfunctionals:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩I (s, ϕ) = τ s−1∫∫ΩTρs−1e−2τα(|ϕt |2 + |ϕ|2 + τ 2ρ2|∇ϕ|2 + τ 4ρ4|ϕ|2)dx dt,I(s,φ) = I (s + 3(n − 1),φ) +n−1∑p=1∑1i1<···<ipnI(s + 3(n − p − 1),Pip . . . Pi1φ),where ρ = ρ(x, t) and α = α(x, t) are suitable weight functions defined in ΩT and where the operator Pi is givenby Pi ≡ ∂t + di (1  i  n). We will obtain the proof of the observability inequality (7) as a consequence of thefollowing result:Theorem 2.3. Let us assume that n, k1, k2 ∈ N and s0 ∈ R are given. Then, there exist two positive functions α0,α1 ∈ C2(Ω) (only depending on Ω and ω), two positive constants C and σ (only depending on Ω , ω, n, s0, k1and k2) and r0 = r0(n) ∈ N such thatk1∑i=0k2∑j=0I(s0 − 4(i + j),i∂jt φ) Cτs0+r0∫∫ωTρs0+r0e−2τα|φ|2, ∀τ  σ(T + T 2), (9)for every φ ∈ C∞((0, T );D) solution to (8). In (9), the functions α and ρ are given by α(x, t) = α0(x)/t (T − t) andρ(x, t) = α1(x)/t (T − t).Sketch of the proof. We will present a sketch of the proof in the particular case in which k1 ≡ k2 ≡ 0. Let us fixs0 ∈ R and φ ∈ C∞((0, T );D) a solution to (8). If we develop the determinant in (8) we deduce that φ also satisfies(Pn · · ·P1 +n−1∑p=2∑1i1<···<ipnαi1,...,ipPi1 · · ·Pip +n∑i=1αiPi + α)φ = 0 in ΩT (10)where the coefficients αi1,...,ip , αi and α are real numbers that only depend on the coefficients of D and A. If weperform the following change of variables ψ1 = φ, ψi = Pi−1ψi−1 = (∂t + di−1)ψi−1, for 2  i  n, it followsfrom (10) that (ψ1, . . . ,ψn)∗ solves the cascade parabolic system⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩(∂t + d1)ψ1 = ψ2 in ΩT ,(∂t + d2)ψ2 = ψ3 in ΩT ,...(∂t + dn)ψn = f in ΩT ,ψi = 0 on ΣT ,(11)withf = f (φ) = −n−1∑p=2∑1i <···<i nαi1,...,ipPi1 · · ·Pipφ +n∑i=1αiPiφ + αφ ∈ L2(ΩT ). (12)1 pF. Ammar-Khodja et al. / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 543–548 547Thanks to the cascade structure of system (11) and following the ideas in [6] (also see [5]), we can prove that thereexist r0 = r0(n) ∈ N and positive constants σ = σ(Ω,ω, s0, n) and C = C(Ω,ω, s0, n) such thatn∑i=1I(s0 + 3(n − i),ψi) C(τ s0+r0∫∫ωTρs0+r0e−2τα|ψ1|2 dx dt + τ s0∫∫ΩTρs0e−2τα|f |2 dx dt), (13)for every τ  σ(T + T 2). The key point in the proof of this last inequality is the following global Carleman estimatefor the heat equationI (s0, ϕ)  C(τ s0∫∫ΩTρs0e−2τα|ϕt ± cϕ|2 dx dt + τ s0+3∫∫ωTρs0+3e−2τα|ϕ|2 dx dt),valid for any τ  σ(T + T 2) and any function ϕ ∈ L2(0, T ;H 10 (Ω)) such that ϕt ± cϕ ∈ L2(ΩT ) (see [4]).Now, if Π denotes any permutation of {1,2, . . . , n}, one can consider the new functions ψ̃1 = φ, ψ̃i =PΠ(i−1)ψ̃i−1, deducing that ψ̃ = (ψ̃1, . . . , ψ̃n)∗ is a solution to system (11) with dΠ(i) instead of di . We can ap-ply the previous argument, deducing that (13) holds for ψ̃ . Finally, using the definition of f (see (12)), we infer (9)in the particular case k1 = k2 = 0. This ends the proof. 2.2.2. Use of the Kalman conditionFrom Proposition 2.1 we deduce that X ⊂ D(K∗). We also have that X is dense in L2(Ω)n and, since the constant Cin (7) is independent of ϕ0, it is sufficient to prove inequality (7) for ϕ0 ∈ X. Under this assumption, the correspondingsolution to (6) satisfies ϕ ∈ C∞((−∞, T );X) andK∗ϕ(·,t) ≡ ((−1)n−1∂n−1t w, (−1)n−2∂n−2t w, . . . ,−∂tw,w)∗(·,t), ∀t ∈ [0, T ),where w = B∗ϕ. From Proposition 2.1 and using inequality (8) for w = B∗ϕ, if Ker(K∗) = {0}, the following in-equality holds:∥∥ϕ(·,0)∥∥2L2(Ω)n C3T/4∫T/4∥∥ϕ(·,t)∥∥2L2(Ω)ndt C3T/4∫T/4∥∥(n−1)(2n−1)K∗ϕ(·,t)∥∥2L2(Ω)nmdt Cn−1∑j=0∫∫ΩTe−2τα|(n−1)(2n−1)∂jt w|2 dx dt.On the other hand, thanks to Proposition 2.2, we can apply Theorem 2.3 with k1 = (n− 1)(2n− 1), k2 = n− 1 ands0 = 8n(n − 1) to each component w = (B∗ϕ), 1    m, obtainingn−1∑j=0∫∫ΩTe−2τα|(n−1)(2n−1)∂jt w|2 dx dt  Cτr0+8n(n−1)∫∫ωTe−2ταρr0+8n(n−1)|B∗ϕ|2 dx dt,which is valid for τ  σ(T + T 2). Choosing τ = σ(T + T 2) and combining these two last inequalities we can inferthat inequality (7) is satisfied. Therefore, system (1) is null controllable at time T . This ends the proof of Theorem 1.1.References[1] F. Ammar Khodja, A. Benabdallah, C. Dupaix, Null-controllability of some reaction–diffusion systems with one control force, J. Math. Anal.Appl. 320 (2) (2006) 928–943.[2] F. Ammar-Khodja, A. Benabdallah, C. Dupaix, I. Kostine, Null controllability of some systems of parabolic type by one control force, ESAIMControl Optim. Calc. Var. 11 (3) (2005) 426–448.548 F. Ammar-Khodja et al. / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 543–548[3] E. Fernández-Cara, E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations 5 (4–6)(2000) 465–514.[4] A. Fursikov, O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser., vol. 34, Seoul National University, Korea, 1996.[5] M. González-Burgos, R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force, AsymptoticAnal. 46 (2006) 123–162.[6] M. González-Burgos, L. de Teresa, Controllability results for cascade systems of m coupled parabolic PDEs by one control force, SystemsControl Lett., submitted for publication.[7] E.L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944.[8] R.E. Kalman, P.L. Falb, M.A. Arbib, Topics in Mathematical Control Theory, McGraw-Hill Book Co., New York, 1969.[9] H. Leiva, Controllability of a system of parabolic equations with non-diagonal diffusion matrix, IMA J. Math. Control Inform. 22 (2) (2005)187–199.[10] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAMRev. 20 (4) (1978) 639–739.[11] L. de Teresa, Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations 25 (1–2) (2000) 39–72.

Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition

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Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition

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