A new relation between the condensation index of complex sequences and the null controllability of parabolic systems

A new relation between the condensation index of complex sequences and the null controllability of parabolic systems Farid Ammar Khodja aAssia Benabdallah bManuel Gonz´alez-Burgos cLuz de Teresa d aLaboratoire de Math´ematiques, Universit´e de Franche-Comt´e, 16 route de Gray, 25030 Besan¸con cedex, France bUniversit´e de Provence, C.M.I. 39, rue F. Joliot Curie 13 453 Marseille Cedex 13, France. cDpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. dInstituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Exterior, C.U. 04510 D.F., M´exico. Received *****; accepted after revision +++++ Presented by Abstract In this note we present a new result that relates the condensation index of a sequence of complex numbers with the null controllability of parabolic systems. We show that a minimal time is required for controllability. The results are used to prove the boundary controllability of some coupled parabolic equations. To cite this article: F. Ammar-Khodja, A. Benabdallah, M. Gonz´alez-Burgos, L. de Teresa, C. R. Acad. Sci. Paris, Ser. I 340 (2013). R´esum´e Une nouvelle r´elation entre l’indice de condensation de s´equences complexes et la nulle contrˆolabilit´ee des syst`emes paraboliques On annonce un r´esultat qui connecte l’indice de condensation des suites complexes et la nulle controlabilit´ee des syst`emes paraboliques. On montre qu’un temps minimal est n´ecessaire pour controller, puis on voit le controle a z´ero sur le bord de quelques ´equations paraboliques coupl´ees. Pour citer cet article : F. Ammar-Khodja, A. Benabdallah, M. Gonz´alez-Burgos, L. de Teresa, C. R. Acad. Sci. Paris, Ser. I 340 (2013). 1. Notation and main results Let Xbe a Hilbert space on Cwith norm and inner product respectively denoted by k·k and (·,·). Let us consider {φk}k≥1a Riesz basis of Xand denote {ψk}k≥1the corresponding biorthogonal sequence to {φk}k≥1. Also consider a sequence Λ = {λk}k≥1⊂C, with λi6=λkfor all i6=k, satisfying for a δ > 0, Email addresses: [email protected] (Farid Ammar Khodja), [email protected] (Assia Benabdallah), [email protected] (Manuel Gonz´alez-Burgos), [email protected] (Luz de Teresa). Preprint submitted to the Acad´emie des sciences <(λk)≥δ|λk|>0,∀k≥1,and X k≥1 1 |λk|<∞.(1) Denote by X−1the completion of Xwith respect to the norm: kyk−1:= Pk≥1 |(y,ψk)|2 |λk|21/2.Also the Hilbert space (X1,k·k1) is defined by X1:= {y∈X:kyk1<∞} with kyk2 1=Pk≥1|λk|2|(y, ψk)|2. Furthermore, let A:D(A) = X1⊂X→Xbe the operator given by: A=−X k≥1 λk(·, ψk)φk.(2) Let us fix T > 0 a real number and B ∈ L(C,X−1) (so B∗∈ L((X−1)0,C)≡X−1). We consider: y0=Ay+Buon (0, T) ; y(0) = y0∈X.(3) In System (3), u∈L2(0, T;C) is the control which acts on the system by means of the operator B. We assume that Bis an admissible control operator for the semigroup generated by A, i.e., for a positive time T∗one has R (LT∗)⊂X,where LTu=RT 0e(T−s)ABu(s)ds. System (3) is approximately controllable in Xat time T > 0 if for every y0∈X,R(T) = {y(T) = eTAy0+LTuwith u∈L2(0, T;C)}is dense in Xand System (3) is null controllable in Xat time T > 0 if for all y0∈X, 0 ∈ R(T). It is well-known that the controllability properties of System (3) amount to appropriate properties of the so-called adjoint system to System (3). This adjoint system has the form: −ϕ0=A∗ϕon (0, T) ; ϕ(T) = ϕ0∈X.(4) Observe that, for any ϕ0∈X, System (4) admits a unique weak solution ϕ∈C0([0, T]; X). Classical results (see e.g. [6, Theorem 11.2.1]) imply: Theorem 1.1 Assume that B ∈ L(C,X−1)is an admissible control operator for the semigroup etAt>0 generated by A, with Agiven by (2), and Λ = {λk}k≥1is a complex sequence satisfying (1). Then, system (3) is approximately controllable in Xat time Tif and only if bk:= B∗ψk6= 0,∀k≥1.(5) Moreover, (3) is null controllable in Xat time Tif and only if there exists a constant CT>0such that X k≥1 e−2T<(λk)|ak|2≤CT T Z 0 X k≥1 bke−λk(T−t)ak 2 ,∀{ak}k≥1∈`2(C).(6) Our main result reads as follows: Theorem 1.2 Assume that B ∈ L(C,X−1)is an admissible control operator for the semigroup etAt>0 and Λ = {λk}k≥1is a complex sequence satisfying respectively (5) and (1). For z∈C, let us introduce E(z) = Q∞ k=1 1−z2 λ2 kand T0= lim sup log 1 |bk| <(λk)+log 1 |E0(λk)| <(λk).Then System (3) is null controllable for T > T0and is not null controllable for T < T0. The index of condensation of a sequence Λ = {λk}k≥1⊂Csatisfying (1) is the real number c(Λ) = lim sup log 1 |E0(λk)| <(λk), where the function Eis given in Theorem 1.2. The condensation index is related to the overconvergence of Dirichlet series (see [5]). Observe that when lim log|bk| <(λk)= 0, then, T0=c(Λ). 2 2. Idea of the proof of Theorem 1.2 The proof is technical and long and the details are given in [2]. For the proof of the positive result we transform the control problem into a problem of moments. So we need to study the existence of biorthogonal families to complex exponentials and study some properties of these families. We have the following result: Theorem 2.1 Let Λ = {λk}k≥1⊂Cbe a sequence satisfying (1) and fix T∈(0,∞]. Let A(Λ, T) = span {e−λkt:k≥1}L2(0,T ;C). Then, there exists a biorthogonal family {qk}k≥1⊂A(Λ, T )to e−λktk≥1 such that for any ε > 0one has C1,ε e−ε<(λk) |E0(λk)|≤ kqkkL2(0,T ;C)≤C2,ε eε<(λk) |E0(λk)|,∀k≥1,(7) where Eis the function given in Theorem 1.2 and C1,ε, C2,ε >0are constants only depending on ε,Λ and T. The null controllability problem for System (3) reduces to the following moment problem: Find u∈ L2(0, T;C) such that, for bkgiven by (5), we have bkRT 0e−λktu(T−t)dt =−e−λkT(y0, ψk),∀k≥1. We can solve this equality using the characterization of the biorthogonal family given above. So: u(t) = v(T−t) = −Pk≥1e−λkT bk(y0, ψk)qk(T−t).It follows that if T > T0, with T0given in Theorem 1.2, the previous series is absolutely convergent in L2(0, T;C) and thus u∈L2(0, T;C). Indeed, if we choose ε∈ (0, T −T0), then (7) leads to:    e−λkT bk(y0, ψk)qk   2 L2(0,T ;C)≤Cεe−2<(λk)(T−T0−ε)|(y0, ψk)|2,∀k≥kε≥1. We prove that System (3) is not null controllable at time T, when T < T0, showing that inequality (6) does not hold. Without loss of generality, we can assume that the sequence Λ = {λk}k≥1⊂Cis normally ordered, i.e., |λk|≤|λk+1|for any k≥1 and arg (λk)<arg (λk+1) when |λk|=|λk+1|. The negative part of Theorem 1.2 is a consequence of the following result: Theorem 2.2 Let Λ = {λk}k≥1⊂Cbe a normally ordered sequence satisfying condition (1). Then, there exists a sequence of sets ∆ = {Gk}k≥1such that ∪k≥1Gk∩Λ=Λand for any subsequence {λnk}k≥1⊆Λ, one has: lim   log 1 |E0(λnk)| <(λnk)−1 <(λnk)log  qk! P0 Dk(λnk)  = 0,(8) where {Dk}k≥1⊆∆is a subsequence of sets satisfying λnk∈Dkand qk+ 1 is the cardinal of the set Dk∩Λ. In the previous equality PAis the polynomial function PA(z) = Qλ∈A(z−λ). Suppose that the observability inequality (6) holds. Using the previous result, we introduce a(k) n= pk! bnP0 Gk(λn)if λn∈Gkand 0 otherwise (pk+ 1 is the cardinal of Gk∩Λ). Clearly, the (finite) sequence {a(k) n}n≥1lies in `2(C). From (6), we can write: σ(1) k:= X λn∈Gk  pk! bnP0 Gk(λn)e−λnT 2 ≤CT T Z 0 X λn∈Gk pk! P0 Gk(λn)e−λnt 2 dt := σ(2) k,∀k≥1.(9) Using the Lebesgue Theorem, it can be shown that lim σ(2) k= 0. On the other hand, from the definition of T0(see Theorem 1.2) and (8), there exists {nk}k≥1such that T0= lim 1 <(λnk)log  1 bnk+ log  qk! P0 Dk(λnk), where {Dk}k≥1⊆∆ is a subsequence of sets satisfying λnk∈Dk, for any k, and qk+ 1 is the cardinal of 3 the set Dk∩Λ. Observe that σ(1) nk≥ qk! bnkP0 Dk(λnk)e−λnkT 2 =e2<(λnk)h1 <(λnk)log 1 bnk+log qk! P0 Dk(λnk)−Ti. This last inequality shows lim σ(1) nk=∞. This contradicts (9). For the details, see [2]. 3. An application: A boundary controllability problem For T > 0 and Q= (0, π)×(0, T), consider the one-dimensional controlled (non-scalar) system    ∂y ∂t −   1 0 0d  ∂2 ∂x2+  0 1 0 0   y= 0,in Q y(0,·) =   b1 b2  v, y(π, ·) = 0 on (0, T ),(10) and initial datum y(·,0) = y0in (0, π), y0∈H−1(0, π;R2) and d > 0. Observe that v∈L2(0, T) is a scalar boundary control which acts on the Dirichlet boundary condition of the state at point x= 0 by means of the vector (b1, b2)>. The aim is to control the whole system (two states) with a control force v. The control problem (10) has been completely solved in [3] when d= 1. For a general system of n≥2 coupled equations with M=In, see [1]. The controllability problem for System (10) when d6= 1 is more intricate and only few results are known. For b1= 0 and b2= 1: Firstly, System (10) is approximately controllable in H−1(0, π;R2) at time Tif and only if √d6∈ Q(see [3]). Secondly, there exists d∈(0,∞) with √d6∈ Qsuch that System (10) is not null controllable at any time T > 0 (see [4]). To our knowledge and apart from the previous results, the controllability properties of System (10) are completely open in the case d6= 1. As a consequence of Theorem 1.2, we have: Theorem 3.1 Assume d6= 1 and let c(Λd)be the index of condensation of the sequence Λd:= k2, dk2k≥1. Then, (i)System (10) is approximately controllable in X=H−1(0, π;R2)at any time T > 0if and only if √d /∈Qand b2(d−1) k2b1+db26= 0.(ii)System (10) is null controllable in Xat any time T > c(Λd) and is not null controllable in Xfor T < c(Λd).(iii)For any τ0∈[0,∞], there exists d∈(0,∞)with √d /∈Qsuch that c(Λd) = τ0. Acknowledgements The authors where want to thank the partial support of grants: ANR-07-JCJC-0139-01 (Agence Nationale de la Recherche (France)), MTM2010-15592 (D.G.E.S. (Spain)), IN108211 (PAPIIT(Mexico)). References [1] F. Ammar-Khodja, A. Benabdallah, M. Gonz´alez-Burgos, L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl. (9) 96 (2011), no. 6, 555–590. [2] F. Ammar-Khodja, A. Benabdallah, M. 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