NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE

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1 ESAIM: COCV July 5, Vol. 11, DOI: 1.151/cocv:513 ESAIM: Control, Optimisation and Calculus of Variations NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE Farid Ammar Khodja 1, Assia Benabdallah,Cédric Dupaix 1 and Ilya Kostin 3 Abstract. We study the null controllability by one control force of some linear systems of parabolic type. We give sufficient conditions for the null controllability property to be true and, in an abstract setting, we prove that it is not always possible to control. Mathematics Subject Classification. 93B5, 93C, 93C5, 35K9. Received March 3, 3. Revised May 17 and October 14, 4. Introduction Null-controllability of linear and semilinear parabolic equations has been extensively studied these last ten years. In their work, Lebeau and Robbiano [6] considered the heat equation in bounded domains Ω R n and for positive times T u t = u + fχ ω in Q T =,T Ω u = on Σ T =,T Ω u,. = u L Ω 1 where ω is an open set which satisfies ω Ω. They proved its null-controllability by a localized in space control: for any u L Ω, there exists f L Q T such that the associated solution of 1 satisfies ut,. in Ω. The main tool to establish such a result is a Carleman estimate. Fursikov and Imanuvilov [7] proved the same result for the problem: Lu = fχ ω in Q T u = on Σ T u,. = u L Ω Keywords and phrases. Control, parabolic systems. 1 Université de Franche-Comté Département de Mathématiques CNRS-UMR Route de Gray, 53 Besançon Cedex, France; ammar@math.univ-fcomte.fr; dupaix@math.univ-fcomte.fr Université de Provence, CMI - LATP Technopôle Château-Gombert, 39 rue F. Joliot Curie, Marseille Cedex 13, France; assia@cmi.univ-mrs.fr 3 Université de Saint-Etienne, Équipe d Analyse Numérique, 3 rue Paul MICHELON, 43 Saint-Etienne Cedex, France; kostin@free.fr c EDP Sciences, SMAI 5 Article published by EDP Sciences and available at or

2 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE 47 where L is a general linear second order parabolic operator. Again, the main tool is a global Carleman estimate satisfied by the solutions of. This estimate allowed these authors to prove the null-controllability controllability to the trajectories of semilinear equations: Lu = hu+fχ ω in Q T u = on Σ T u,. = u L Ω 3 when h is a globally Lipschitz-continuous function. Fernandez-Cara and Zuazua [8] and Barbu [4] generalized this result to some superlinearities using a fixed point method and a precise expression of the constants appearing in the global Carleman estimate. In contrast, very few results are published for linear or semilinear parabolic systems. Anita and Barbu [3], for a reaction-diffusion system, and Barbu [5] for the phase field system, proved local exact controllability results by two localized in space control forces. In a recent paper [1], the authors give local exact controllability and exact controllability results for phase field systems by only one localized control force. All these papers pass first through the study of the controllability of a linearized version of the system and use some Carleman estimate in order to apply a fixed point theorem. As it is by now well-known, the controllability result of this linearized system amounts to prove an observability estimate for its adjoint system. This part is in a way the corner-stone for the proof of the controllability of the initial semilinear problem when using a fixed point method. In particular if a control problem by only one force is considered, it is to obtain this estimate that the coupling operator between the equations of the system plays role. So in view of these facts, a natural question arises: is it always possible to control coupled linear parabolic systems by a reduced number of control forces. It is the aim of this paper to give some partial answers to this question. To address this question, this article is organized as follows. We first consider in Section a linear system which naturally appears when dealing with the controllability of some nonlinear reaction-diffusion systems. In this case, the observability estimate is related to the existence of a definite, non-negative quadratic form Φ, which is introduced in the proof of Lemma 1. This Φ takes into account the way in which the equations of the system are coupled and it leads to the desired observability estimate. The goal of this section is to illustrate, in a simple case, the main ideas contained in the proof of such aresult. In the light of Section, one may ask if it is possible to adapt this approach to general abstract linear systems of parabolic type at least to obtain some sufficient condition for their controllability. This is the goal of Section 3 where it turns out that the observability estimate is actually related to the existence of such a Φ which is itself related to the existence of a multiplier M as shown in Theorem.1. Note that this relation between Φ and M washiddeninsectionsinceinthiscasem was equal to the identity. Finally, to conclude this third section we exhibit some examples of linear parabolic-like systems showing that it is not always possible to control such coupled systems by a reduced number of control forces. In this case the non-controllability is related to the compactness of the coupling operator this operator being in some sense too compact. Note that the operators in these examples are not of partial differential type and thus do not seem to be physically relevant. In order to illustrate the relevance of the assumption concerning the existence of a multiplier M in Theorem.1, we present in Section 4 an example of linear parabolic system for which it is not possible to take the identity for M. We show how the abstract theorem of Section 3 can be used for this example to deduce the controllability of the system by a single but non-localized control force. Finally, using this first result together with suitable Carleman estimates we obtain the controllability of the system by a single localized control force.

3 48 F. AMMAR KHODJA ET AL. 1. An example: Reaction-diffusion systems We consider a general reaction-diffusion system which arises in mathematical biology: ψ t = ψ + hψ, w+w on Ω R + w t = w + gψ, w +f on Ω R + ψ δω = w δω = inr + ψx, = ψ, wx, = w 4 where Ω is a bounded domain of R n with smooth boundary, h, g smooth functions and f asourceterm. Let T >, f in L Q T withq T =Ω ],T[, and ψ,w L Ω. Suppose that there exists a ψ,w satisfying 4 in C ],T] L Ω with ψ,w = ψ,w and consider the controllability question of finding a function f L Q T such that the corresponding solution pair ψ, w of 4 satisfies ψt =ψ T and wt =w T. Note that this is still an open question with a single control force even for a non-localized control which will be address in a future work []. To our knowledge, the only result in this direction is the one of Anita and Barbu [3] for a localized control acting on both equations. The sketch of the proof of such a result, is to set first where ψ, w satisfies 4, to get: ψ = ψ ψ w = w w ψ t = ψ + hψ, w hψ,w + w on Ω R + w t = w + gψ, w gψ,w + f f on Ω R + ψ δω = w δω = inr + ψx, = ψ ψ, wx, = w w. The main task is now to prove the controllability of a linearization of this system with suitable estimates on the control with the help of Carleman estimates if the control is localized to be able to use a fixed point argument. As mentioned in the introduction, the question under interest here is the study of the controllability of this linearized system. This system takes here the following form: ψ t = ψ + aψ + bw on Ω R + w t = w + dψ + cw + f on Ω R + ψ δω = w δω = inr + ψx, = ψ, wx, = w. We assume a, b, c, d L Q T andf L Q T. The rest of this section is now devoted to the study of the controllability of 5. To begin with we recall for completeness why this controllability property is actually equivalent to the so called observability estimate. Then we prove in Lemma 1. this estimate. Existence and uniqueness for 5 follow from classical results on parabolic systems see [9]. System 5 can bewritteninthefollowingabstractform: Y t = AtY + Bf with +ai bi At =, ψ Y = w di, B = +ci I. 5

4 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE 49 One gets that for all initial data ψ,w H = L Ω L Ω and all f in L Q T, there exists a unique solution ψ, w C[,T]; H. ψ Our aim is to prove that for all initial data Y = H, one can find f L w Q T such that the corresponding solution of 5 satisfies Y T =. 6 As in [13], for any t>, we define the operators S t which associates with an initial data Y H the solution of the homogeneous system 5 f =attimet and L t which associates with f L Q T the solution at time t of 5 corresponding to null initial data. Then, the controllability property is equivalent to the existence, for all initial data in H, off L Ω such that S T Y + L T f =, 7 which is equivalent to R S T R L T. 8 This last inclusion holds if and only if see for instance [13], Th.., p. 8 there exists C T > such that S T Y H C T L T Y L,T ;H Y H, where, for any bounded operator M, M denotes the adjoint operator. Since L T ht =B ST t h for any h H and if A t denotes the adjoint operator in H of At, this last inequality can be written as T φt + ut C T ut, x dxdt, 9 Ω where. stands for the L -norm and φ, u is the solution of the adjoint problem: φ t = φ + aφ + du on Ω R + u t = u + cu + bφ on Ω R + φ δω = u δω = inr + φx, = φ, ux, = u. 1 Remark 1.1. We insist again here on the fact that the purpose of this article is to study the way to prove the observability estimate in the case of a reduced number of control forces. In particular we are not interested at this level by the localization in space of the control. However let us mention that if f = gχ ω in 5 then the corresponding observability estimate namely with ω instead of Ω in the right-hand-side of 9 can be deduced by using some suitable Carleman estimate see []. Let us show the observability inequality 9 for solutions of 1. Lemma 1.. For T>, assume that b δ> a.e in Q T. Then there exists a positive constant C T such that for all initial data φ,u L Ω L Ω the corresponding solution of 1 satisfies 9. Proof. 9 is an observability estimate. It shows the possibility of recovering the total L -norm at time T by observing only the L -norm of u from t =tot = T. The proof of this Lemma is based on the introduction of a suitable functional itself based on suitable multipliers. Let us introduce, for any solution φ, u of 1, the functional: Φφ, u =t 4 φ + λt u µt 3 φ, u, where λ, µ are positive constants witch will be chosen later.

5 43 F. AMMAR KHODJA ET AL. and One has d t 4 φ = 4t 3 φ +t 4 φ, φ t dt 4t 3 φ t 4 φ +t 4 a L Q T φ +t 4 d L Q T φ,u d t u dt = t u +t u, u t t u t u +t c L Q T u +t b L Q T φ, u, d dt t3 φ, u 3t φ, u } +t { 3 φ, u + a L Q T + c L Q T φ,u t 3 bφ, φ +t 3 d L Q T u. Thus, differentiating Φ with respect to time, we obtain: wherewehaveintroducedthefollowingnotations d dt Φu, v I 1 + I + I 3 I 1 = 4t 3 φ t 4 φ +t 4 a L Q T φ µt 3 bφ, φ, I = d L Q T t4 + µ a, c L Q T t3 + λ b L QT +3µ t φ,u +µ φ, u t 3, where a, c L Q T = a L Q T + c L Q T. I 3 = λt u λt u +λt c L Q T u + µt 3 d L Q T u. Now we estimate each of the three quantities I 1, I and I 3. Estimate of I 1. Recall that we have assumed that b δ> a.e. in Q T. Therefore, choosing µ = 8+4T a L Q T δ 11 leads to I 1 + T a L Q T t3 φ t 4 φ.

6 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE 431 Estimate of I. Using Cauchy-Schwarz s and Young s inequalities, we have: Estimate of I 3. I + T a L Q T t3 φ + t 4 φ + µ 4 t u [ + d L Q T t + µ a, c L Q T t ] t u +λ b L Q T +3µ +T a L Q T where I 3 P 1 t u λt u P 1 t =µ d L Q T t3 + λ c L Q T t +λt. In view of these last estimates, choosing λ µ 8,onegets d Φu, v P t u dt with t [ P t = P 1 t+ d + T a L L Q T Q T t + µ a L Q T + c L Q T t +λ b L Q T +3µ]. To end the proof, it remains to show that Φu, v is a quadratic definite form in L Ω for t = T. This is a direct consequence of the definition of Φ, with λ = µ. Actually, one has: Φu, vt Therefore, all solutions of 1 satisfy 9 with since P L O,T c 1T and 1 independent of T. 1 T + 1 λ 1 T 4 λt + λ T φt + ut. P L C T = O,T 1 T + 1 λ 1 T 4 λt + λ T T + λ T 4 λt + λ T C T 3, c T where C, c 1,c are positive constants Remark 1.3. The question of the optimality of the minimal norm steering control as T is by now wellknown. Let s briefly recall some background connected to this question. Consider an abstract and general

7 43 F. AMMAR KHODJA ET AL. control problem: Y t = AY + Bu, Y = Y H. where A, DA is an unbounded operator in a Hilbert space H and B, thecontrol operator, is defined from the control Hilbert space X in H. If the null controllability property holds for this abstract system, then for arbitrary T > andy H, thereexistsu L,T; X such that the corresponding solution satisfies Y T =. Then one can look for the control u which minimizes u L,T ;X. Let us denote it by u T and let the minimal energy function E min T defined by E min T := sup u T Y L,T ;X. Y =1 When dim H<, itiswellknownthate min T is exactly the best constant C T of the observability inequality and it has been proved see [11, 1] that C T T k 1 where k is the smallest integer such that A M n R etb M n,m R rg[b, AB,..., A k B]=n. If we consider the correspondence between our system and a finite dimensional one with: λ + a d A =,B =, d b λ + c 1 then a simple computation gives k =1andthenT k 1 = T 3. Therefore it is interesting to notice that our computations leads to the same behavior near T = since we have obtained C T CT Sufficient conditions. Null-controllability for general systems This section is devoted to the question of null-controllability for some general systems of parabolic type when using control forces acting on a single equation of the system. More precisely, let U and V be two real Hilbert spaces with scalar-product, U and, V respectively, and let U and V denote the corresponding norms. Let A and C be linear self-adjoint operators with the domains DA Uand DC V, respectively. We assume that these operators are positive definite. We also introduce two linear, possibly unbounded, densely defined operators B : DB V Uand D : DD U V. Assume also that the sets DA DD DB, DC DB DD are dense in U and V so that the matrix operators A D A = B C, Â = A B D C be defined on a common dense set F U V. Finally, suppose that for any u, v U V the inequality Bv,u U +Du, v V Au U + Cu V

8 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE 433 holds. Thus A is a densely defined dissipative operator. It is therefore closable [1], p. 15, Theorem 4.5. Let us retain the notation A for its closure. The domain of the adjoint operator A contains the dense set F and thus A is also dissipative. Therefore both A and A generate C semigroups of contractions [1], p. 15, Corollary 4.4. This allows us to consider the following initial value problem ψ t = A ψ + D w U w t = B ψ C w + f V 1 ψ = ψ, w = w, where f L R +, V is the control force. The aim of this section is to find conditions under which system 1 is L -null-controllable at any time T>, i.e., for any time T>and any initial data Y =ψ,w U Vthere exists a control force f L R +, V for which the corresponding solution ψ, w of 1 vanishes at time T. It is by now classical see for instance [13], Th..6, p. 13 that, if we denote by R : V U Vthe operator defined by Rf =, this property is f equivalent to the existence of a constant c T > such that for all ψ,w U V which reads for the solution u, v oftheadjointsystem, e A T T Y ct R e A t Y dt, ut U + vt V c T T u t = A u + Bv v t = Du C v u = u, v = v. vt V dt, 13 Throughout this section we agree to denote by c a constant depending only on the operators A, B, C, andd. The main result of this section reads as follows: Theorem.1. Let u,v DA. Assume that there exists a linear maybe unbounded but not necessarily closed operator M : DM V U such that for some p R the following inequalities are defined and hold for any u, v DA: 14 Bv,Mv U c Cv V 15 M u V c A p u U 16 A 1 p Bv U c Cv V 17 MDu,u U c A p u U 18 A 1+p Mv U c Cv V 19 A 1+p MC v U c Cv V.

9 434 F. AMMAR KHODJA ET AL. Assume also that there exists a positive constant γ such that for all ε>, A 1+p D v U ε Cv V + c ε γ v V 1 A 1+p Mv U ε Cv V + c ε γ v V for any u, v F. Then for any T>thesolution of problem 14 satisfies the estimate A p ut U + vt V ct γ 4 T If, in addition, there exists a constant q> such that the estimate vt V dt. ut U + vt V c1 + T q A p u U + v V 3 holds for any T>, then system 1 is L null-controllable at any time. Remark.. The assumptions of Theorem.1 may seem strange. Indeed, they naturally appear when we apply to the abstract system the same ideas which allowed us to treat the example in the previous section. Let us briefly explain these ideas. If we had to control system 1 by two functions one for each equation of the system, we should have to prove the estimate: T T ut U + vt V c T ut V dt + vt V dt, 4 for the solution u, v of the adjoint system 14. If this estimate is obtained, the difficulty now is to get rid of T ut V dt. Roughly speaking, it is the role of the second equation to allow this: thus we have to find a suitable multiplier the operator M of the theorem leading to an estimate of T ut V dt. If we multiply the second equation of the adjoint system 14 by M u, we are naturally led to the assumption 18 and the other assumptions are here to manage the bad terms appearing after this. For instance, if D was invertible with a bounded inverse and of the same order as B, the natural choice is M = D 1 and p = it corresponds to the situation in the example treated in the previous section. Based on these ideas, the functional we introduce allows both to prove 4 and to get rid of T ut V dt but, in doing this, the condition 3 it is a kind of parabolic property becomes necessary to end the proof. Remark.3. For another application of this process, the reader should refer to the third section system 41. Proof. For u and v satisfying 14, define the quadratic form Φ by Φu, v =t l+ A p u U + λtl v V µtl+1 Mv,u U, 5 where the positive parameters l, λ, andµ will be chosen later. The first step of the proof is to show that there exists a positive constant c such that d dt Φu, v c v V. Using 14 we deduce that: d t l+ A p u U = l +t l+1 A p u U dt +tl+ A p u, u t U = l +t l+1 A p u U tl+ A 1 p u U +t l+ Bv,A p u U, 6

10 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE 435 and d t l v V dt = lt l 1 v V +tl v, v t V = lt l 1 v V tl Cv V +tl Du, v V, 7 as well as d t l+1 Mv,u dt U = l +1t l Mv,u U + t l+1 Mv t,u U + t l+1 Mv,u t U = l +1t l Mv,u U + t l+1 MDu,u U t l+1 u, A M + MC v U + tl+1 Bv,Mv U. 8 Thus, differentiating Φ with respect to time, in view of 6 8 we obtain wherewehaveintroducedthefollowingnotations d dt Φu, v =I 1 + I + I 3, 9 I 1 = l +t l+1 A p u U tl+ A 1 p u U µtl+1 MDu,u U, = t l+ Bv,A p u U +λtl Du, v V µl +1t l Mv,u U + µt l+1 u, A M + MC v U, I 3 = λlt l 1 v V λtl Cv V µtl+1 Bv,Mv U. Now we estimate each of the three quantities I 1, I and I 3. Estimate of I 1. In view of 18, one can choose µ large enough so that I 1 t l+1 l + cµ A p u U tl+ A 1 p u U t l+ A 1 p u U. 3 Estimate of I. Using Cauchy-Schwarz s and then Young s inequality, we have I = t l+ A 1 p Bv,A 1 p u U +λt l A 1 p u, A 1+p D v U µl +1t l A 1+p Mv,A 1 p u U +µt l+1 A 1 p u, A 1+p M + A 1+p MC v U 1 tl+ A 1 p u U +tl+ A 1 p Bv U + 1 tl+ A 1 p u U +λ t l A 1+p D v U + 1 tl+ A 1 p u + 1 U µ l +1 t l A 1+p Mv U + 1 tl+ A 1 p u U + µ t l A 1+p Mv U + µ t l A 1+p MC v U,

11 436 F. AMMAR KHODJA ET AL. and therefore A I t l+ 1 p u U + A 1 p Bv U A +µ t l 1+p Mv U + A 1+p MC v U +t λ l A 1+p D v U µ l +1 A 1+p Mv. U Using 17 and 19, we have I t l+ A 1 p u U +ct l+ + µ t l Cv V +t l λ µ l +1 ε Cv V + cε γ v V, or, setting ε = t λ µ l +1 1 and l =+γ, 31 I t γ+4 A 1 p u U 3 +ct γ+4 +cµ t γ+ +t γ+ Cv V Estimate of I 3. To estimate I 3 we use 15 to obtain +4 λ µ γ +3 1+γ v V. I 3 λ + γt γ+1 v V + λt γ+ + cµt γ+3 Cv V 33 Now in view of 3, 3 and 33, which we substitute into 9 we have d Φu, v dt λt γ+ + cµt γ+3 +ct γ+4 +cµ t γ+ +t γ+ Cv V λ µ γ +3 1+γ + λ + γt γ+1 v V. Going back to 5, recalling 16 and making λ larger if necessary, we remark that Φu, v 1 Thus integration of 34 over [,T/] yields: t γ+4 A p u U + tγ+ v V. A p ut/ U + vt/ V ct T/ vt V dt, 35 with for T small: ct T γ 4 which already implies the null-controllability of system 1 in the case p. Otherwise use first 3 on the interval [T/,T] and then 35 to obtain ut U + vt V 1 + ct q A p ut/ U + vt/ V ct q γ 4 T/ vt V dt ct q γ 4 T vt V dt.

12 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE Some necessary conditions: a counterexample Here, we construct an example showing that for some coupling operator, the system cannot be null-controllable. Let A be an unbounded positive definite self-adjoint operator in a Hilbert space H with discrete spectrum {λ k } k N definitely, A could be viewed as with Dirichlet homogeneous boundary conditions. If we denote by {φ k } k the corresponding orthonormal basis of eigenfunctions, A may be written A = k λ k., φ k φ k. Therefore, if f is a real function defined on speca, we set fa := k fλ k., φ k φ k. Assume f : speca R satisfies < fs <s, s, and fs =os as s, so that the operator fa is relatively compact with respect to A. Consider the following control problem: given u,v H and a positive number T, find a function h L,T,H such that the solution of the Cauchy problem u t = Au + fav v t = Av + fau + h u = u, v = v 36 satisfies ut =andvt=. The desired h exists if and only if there exists a positive constant C T such that any solution u, v ofthe homogeneous problem u t = Au + fav v t = Av + fau u = u, v = v satisfies the estimate T ut + vt C T vt dt, 37 wherewedenoteby. the norm in H. The main result of this section is: Theorem.4. 1/- System 36 is null-controllable in time T if and only if the function s 3 f se Ts stays bounded away from zero as s. /- If fs decays polynomially, the system is null-controllable for any T >. 3/- For fs =e s we have the null-controllability result for any T>1, but not for smaller values of T. 4/- If the decay rate of fs is super-exponential, the system is never null-controllable. Remark.5. Note that if f satisfies one of the condition of this theorem, then fa is a compact operator as a limit of finite rank operators. From an heuristic point of view, this result gives some indications on the order of the coupling operators for the null-controllability property to hold. For instance, if the coupling operator is a partial differential operator of order or 1, or if it is of order A p for some real p>, then may be with some additional assumptions the

13 438 F. AMMAR KHODJA ET AL. null-controllability of the system should hold. If the coupling operator is too compact by this, we mean that its sequence of eigenvalues decays very fast to, then the null-controllability property will in general be false. Proof. In terms of quadratic forms, 37 reads as follows: T e LT C T e Lt P e Lt dt 38 where L and P are operators in H H given by A fa L = fa A, P = I. It is easy to see that the eigenvalues of the matrixl are λ ± A, where λ ± s =s ± fs, and that the orthogonal matrix V = diagonalizes L with So, it is easy to check that Thus: with Λ= λ+ A λ A e Λt = V e Lt V. = VLV. T e ΛT C T e Λt VPV e Λt dt 39 VPV = 1 I I I I For any real s we set ηs =e s 1/s. Computing explicitly the integral at the right-hand side of 39, one can rewrite the null-controllability condition in the form I C T. BA 4 where ηtλ + s η Tλ + s Tλ s Bs =. η Tλ + s Tλ s ηtλ s Denote by σs the smallest eigenvalue of the matrixbs. Clearly, using assumptions on fs det Bs trbs σs detbs trbs Since inequality 4 is equivalent to the positive definiteness of the operator σa, we arrive at the following assertion: system 36 is null-controllable if and only if the function detbs/trbs is bounded away from zero. To estimate det Bs/tr Bs first note that ln ηs = 1 s 1 e s +e s

14 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE 439 Thus the function η is log-convex and 1 s +1 ln ηs 1 s Applying the Taylor formula to ln ηs, for two real numbers s 1 and s we have ηs 1 ηs =η s 1 + s e ln η s s 1 s, where s [s 1,s ]. Replacing s 1 and s by λ + s andλ s, respectively, we obtain the following expression for the determinant of the matrix Bs: [ ] detbs =ηtλ + sηtλ s 1 e 4f st ψs, where 1 4T λ +s+1 ψs 1 4T λ s This, in particular, shows that the function det Bs/trBs is strictly positive for any value of s. Toverifyif it is bounded away from zero it suffices to study its asymptotic behavior as s.asiseasytosee, λ ± s = s1 + o1, ηλ ± s = ets o1,ψs = 1 + o1, Ts 4Ts 1 e 4f st ψs = f s s 1 + o1, and therefore Finally we obtain detbs = f se 4Ts 4T s o1, trbs = ets 1 + o1. Ts det Bs trbs = f se Ts 4Ts o1. 3. Application to the null-controllability of power-like systems 3.1. Preliminaries Let Ω be a bounded domain in R n with C boundary and let ω be any subdomain of Ω with characteristic function χ ω.fort>, we set Q T =,T Ω and we consider the system of parabolic equations: ϕ t = Lϕ + L α w in,t Ω w t = L α ϕ L β w + χ ω u in,t Ω 41 ϕ = ϕ,w = w in Ω, where L is the realization of the Dirichlet Laplacian operator on Ω: L =, DL =H Ω H 1 Ω,

15 44 F. AMMAR KHODJA ET AL. α R, β R +, ϕ, w L Ω and u L Q T. We then set H = L Ω L Ω and we define the linear and unbounded operator: L L α L = L α L β with domain DL =DL DL α DL α DL β. We also denote by <.,.>and. the scalar product and associated norm in H and by.,. and. the scalar product and associated norm in L Ω. Introducing µ k k N and ϕ k k N the sequences of eigenvalues and normalized eigenfunctions of L, the eigenvalues and corresponding normalized in H eigenfunctions of L are given by: λ ± k = µ k + µ β k µ k µ β k +4µ α k 4 Φ ± k = To simplify the notations, we set: 1 µ α k + λ ± k + µ k µ α k λ ± k + µ k ϕ k, 43 λ k = Φ k = { λ + k if k 1 λ k if k 1, { Φ + k if k 1 Φ k if k 1. Clearly Φ k k Z is an orthonormal family in H. Actually, it is total i.e. span {Φ k k Z } = H. We can now introduce the self-adjoint extension of L for which we keep the same notation with respect to its base of eigenfunctions according to: L = k Z λ k <.,Φ k > Φ k 44 DL = { Y H, } λ k Y k < 45 k Z where Y k :=< Y,Φ k >. A necessary and sufficient condition for L to generate a C -semigroup e Lt is then that: C >, λ k C, k Z, and this last condition is, from the expression of λ + k see 4, equivalent to: α β Together with L defined by 44 45, we introduce the control operator B : L Ω H by setting Bu = χ ω u. Using the notation Y =ϕ,w, system 41 can be rewritten as: { Y = LY + Bu Y = Y H. 47

16 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE 441 We will now prove the null-controllability of 47 namely that, for any Y H, there exists a control u L Q T such that the solution Y of 47 satisfies Y T =. Noting that B =,χ ω, this property amounts to prove ϕ that any solution Y = of the adjoint problem w { Y = LY Y = Y, 48 satisfies the estimate: T ϕt + wt C T ω wt, x dxdt, 49 For a single parabolic equation, this property is well-known and has been obtained in particular by Lebeau-Robbiano [6] and Fursikov-Imanuvilov [7]. In our case two more difficulties occur. The first one is due to the fact that the control force only acts on a single equation of this system and the second one is that this control force is localized in a subdomain of Ω. To prove 49, we will find in a first step the set of α and β for which this inequality is valid if ω =Ω. This result is obtained as a direct consequence of Theorem.1 in the particular case of system 47. We will restrict ourselves to the case α <β+1. 5 For these values of α and β, we then adapt the method developed in [6] to prove Null controllability when ω =Ω The main result of this section is: Theorem 3.1. Assume that β>and α <β+1. Then for all ϕ,w L Ω and all time T > there exists a constant C T > such that the corresponding solution of 48 satisfies T ϕt + wt C T wt, x dxdt. 51 Proof. We prove Theorem 3.1 by showing that the assumptions 15 3 of Theorem.1 hold. We first concentrate on 15. Therefore we have to find p R and M : L Ω L Ω such that DM DL p and such that the estimates 15 are satisfied. We seek M as a power of L namely M := L α p. Estimates 15 and 18 are then automatically satisfied while the estimates 16 give S α p p for 16 1 p + α β for 17 1+p α p β for 19 1+p α p + β β for 1+p + α< β for 1 1+p α p< β for Ω with for 1 and for L 1+p L 1+p +α w α p w ε L β c w + ε γ w 5 ε L β c w + ε γ w. 53

17 44 F. AMMAR KHODJA ET AL. In order to find γ, remark that the first inequality of system S leadsto L 1+p α p w = L 1 p α w c L 1+p +α w, so that 5 implies 53. If 1+p + α, then 5 holds with γ =. Otherwise, recall that see [1] Th. 6.1, p. 73 for a positive selfadjoint operator Γ, for any r, 1 there exists C> such that for any ρ>and x DΓ Setting ε = ρ r 1, this last inequality writes: Γ r x Cρ r x + ρ r 1 Γx. Γ r x C ε r 1 r x + ε Γx. Applying this with Γ = L β/, r = 1+p + α, we get 5 with: β γ = r 1 r = max Finally we have to find p R such that system S holds. This is equivalent to: {, } 1+α + p. 54 β +1 α p max α 1 β, β 1 α <p<1+β α 55 and it is now easily seen that the assumptions β> and α<1+β imply the existence of such a real p. For example, one can verify that p := max{, 1 β α}, 56 is convenient. Thus the assumptions 15 of Theorem.1 are satisfied and this implies the existence of a constant c independent of T such that the solution of 48 satisfies L p ϕt + wt ct γ 4 T wτ dτ. It remains to verify that Assumption 3 also holds for the solution of system 48. In this case, Assumption 3 becomes: there exists q> such that the solution of system 48 satisfies L ϕt + wt c1 + T q p ϕ + w. This estimate is easily proved by writing down the solution of 48 in the basis Φ k k Z. Indeed, we have: Y t = k Z e λ kt Y, Φ k Φ k.

18 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE 443 Thus, using the properties of the Φ k and setting Y =ϕ,w Y t = k Z e λ kt Y, Φ k C e λ kt k Z µ α k + µ α k λ k + µ ϕ,ϕ k + λ k + µ k w,ϕ k k C k Z [ µ p k e λ kt µ p k ϕ,ϕ k + w,ϕ k ]. 57 Studying the asymptotic behavior of λ k yields that there exist constants C>andδ> such that λ k Cµ δ k for all k Z. On the other hand, it is a matter of computation to check that for all t>: with q = p δ µ p k e Cµδ k t C t q and C is a constant independent of k and t. Thus: which is the desired estimate. Y t k Z C t q µ p k < ϕ,ϕ k > + < w,ϕ k > C1 + 1 t q L p ϕ + w Thus the conclusion of Theorem.1 follows: T ϕt + wt C T where the constant C T is estimated by γ beingdefinedin Null controllability when ω Ω Ω wt, x dxdt, C T = ct p γ 4, 58 This part is devoted to the proof of the main result of this section. Remark 3.. The abstract result does not apply in this situation because the control force is constrained to be in a strict subspace of L Q T the abstract result gives a control function in the whole space without more precision. Theorem 3.3. Assume that β>and α <β+1. Then for any Y H and any T >, thereexistsa control u L Q T such that the solution Y of 47 satisfies Y T =. Proof. We give it for completeness because it is essentially the same than the one given by G. Lebeau and L. Robbiano [6] in the case of a single equation. Indeed, the original point here is Lemma 3.4 which itself uses the results of the previous section. Following [6], we fix δ,t/ and ρ,b/nwhereb =min1,β 1 and n is the space dimension. For l 1weset σ l = l, T l = A ρl A > is chosen such that l 1 T l = T δ.

19 444 F. AMMAR KHODJA ET AL. Then we introduce, for any t>, the operator: L t Y,u=e tl Y + so Y t =L t Y,u is the unique solution of 47. We define the sequence a l l by setting t e t sl Busds, a = δ a l = a l 1 +T l, l 1, and for any initial data Y H let u l = K T,σl Y be a control such that: π σl L T Y,u l =, where π σl is the projection operator on the subspace H σl =span { Φ ± k, 1 k σ l} of H. Of course, the control u l needs not to be unique but we will see below that, first it exists and second that it can be chosen as the one which minimizes a given cost functional. We are now in position to construct a sequence of states: Y H Y 1 = e δl Y 59 Z l = L Tl Y l,k Tl,σ l Y l 6 Y l+1 = e TlL Z l. 61 We first explain briefly this process: until time δ = a the system is freely evolving and dissipates without control u = in 47. On the time interval a,a +T 1, we first introduce a control u 1 which will drive Y 1 to a function of Hσ 1 in time T 1 and we then let again system 47 evolves freely with u =fromt = a + T 1 to t = a +T 1. We obtain the sequence Y l by repeating this construction on a l 1,a l. The claim of the Theorem holds true if we are able to prove that lim l Y l =. Of course, this construction needs to be justified by proving that, at any step, the control u l = K Tl,σ l Y l which drives the initial data Y l to a function of Hσ l in time T l exists. For the solution of system 47 the following result holds: Lemma 3.4. Let T >. For any l 1 and any Y H, there exists at least a control u = u T,l., Y L,T; U such that π l Y T = π l L T Y,u = 6 where π l is the projection operator on H l = span { Φ ± k, 1 k l}. This control can be chosen such that: u L,T ;U ct p 4 γ e C µl Y. 63 Proof of the Lemma. We fix arbitrary l 1andY H. Equality 6 can be rewritten as T π l e T L Y + π l e T sl Busds =. 64 If we denote by D T the operator defined from the space L,T; U inh by D T u = T e T sl Busds,

20 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE 445 the first part of Lemma 3.4 is equivalent to R π l e T L R π l D T. 65 This last inclusion holds if and only if see for instance [13] Th.., p. 8 there exists C = C T,l > such that e T L π l Y C DT π l Y L,T ;U Y H. H Since DT ht =B e T tl h for any h H and L = L, this last inequality can be written as T ϕt + wt C T wt, x dxdt, 66 ϕ for any solution Y = of System 48 with Y w H l. So we now prove 66. From Theorem 3.1, we have, with p and γ satisfying 55 and 54: T ϕt + wt ct p 4 γ wt, x dxdt, 67 ϕ for any solution Y = of System 48 with Y w H. Now since Y H l, we deduce from the invariance of the eigenspaces of L under e tl, Y t H l for any t,t] and thus: Y t,. = l k=1 From this last equality and 43, we deduce that: ω { } e λ+ k t Y, Φ + k Φ+ k +eλ k t Y, Φ k Φ k. Ω wt,. = = l k=1 { } e λ+ k t Y, Φ + k c+,k +eλ k t Y, Φ k c,k ϕ k l a k tϕ k, k=1 where c ±,k = λ ± k + µ k µ α k + λ ± k + µ, k and ϕ k is a normalized eigenfunction of the Dirichlet Laplacian. Now, the observability estimates on the eigenfunctions of the Laplacian due to G. Lebeau and L. Robbiano [6] assert that there exist constants C 1,C > such that: l a k t C 1 e C µl l a k tϕ k x dx, k=1 ω k=1 or, equivalently: wt, x dx C 1 e C µl wt, x dx. 68 Ω ω

21 446 F. AMMAR KHODJA ET AL. Estimates 68 and 67 yield 66 with C T = ct p 4 γ e C µl. 69 This proves the existence of the desired control. That u canbechoseninsuchawaythat63issatisfiedisclassical and we recall for completeness how this can be deduced from 66 with C T given by 69. Let Y H. For all ε>, we define the quadratic convex and continuous functional J ε,l : H l R by ψ where X = v J ε,l X = 1 is the solution of: T vt, x dxdt + Y,X + ε X, ω Clearly, Zt, x =XT t, x is the solution of X = LX, XT =X H l. 7 and applying 66 with C T given by 69 yields This implies that for any δ> J ε,l X Z = LZ, Z = X, T X C T and then for sufficiently small values of δ we obtain T C T ω ω vt t, x dxdt v t, x dxdt. 1 δc T v t, x dxdt C δ Y + ε X, ω J ε,l X ε X C δ Y. It follows that J ε,l is a strictly convex, continuous and coercive functional and, thus, achieves a unique minimum at X,ε H l. Let X ψε ε = be the solution of 7 associated with the initial data v X,ε and let Y ε the ε solution of 47 with u := v ε.itsatisfies Besides, one has: Y ε T Hl ε. 71 J ε,l X,ε J ε,l =,

22 NULL-CONTROLLABILITY OF SOME SYSTEMS OF PARABOLIC TYPE BY ONE CONTROL FORCE 447 and thus T ω v ε t, x dxdt Y, X,ε Y X,ε T 1/ C T v ε t, x dxdt Y. ω From this last estimate, we deduces the boundedness of { v ε } ε> in L [,T] ω. This allows us to let ε going toin 47withu := v ε. From 71 we deduce that the weak limit Y of at least a subsequence of {Y ε } ε> satisfies π l Y T =, and from the previous estimate, that the weak limit u of a subsequence of { v ε } ε> satisfies 63. This ends the proofofthelemma. We now come back to the proof of Theorem 3.3. From 61, since Z l Hσ l we get Y l+1 = e T l L Z l = = k σ l +1 k σ l +1 { e λ+ k T l Z l, Φ + k Φ+ k +eλ k T l Z l, Φ k Φ k } e λ+ k T l 1/ Zl, Φ + k +e λ k T l Zl, Φ k e λ+ σ l +1 T l Z l. 7 From 6, using that e Lt is a semigroup of contractions together with estimate 63, we obtain that: Z l = L Tl Y l,k Tl,σ l Y l al 1 e T l L +T l Y l + e Tl sl Bu l sds a l 1 Y l + u l L a l 1,a l 1 +T l ω Combining 7 and 73 yields Y l + ct p 4 γ l e C µ σl Yl Y l+1 e λ+ σ l +1 T l 1+ ct p 4 γ l e C µ σl Y l ct p 4 γ l e C µ σl Y l. 74

23 448 F. AMMAR KHODJA ET AL. Now, recall that σ l = l,t l = A ρl = Aσ ρ l and that from Weyl s formula, µ σl CΩ σ l /n. On the other hand, direct computations yield λ + σ l +1 µθ σ l +1 Cσθ/n l, with θ =min1,β. Thus λ + σ l +1 T l Cσ θ n ρ l. For simplicity we introduce the notation Then a> and one has: C µσl λ + σ l +1 T l a := Since by assumption ρ, θ 1 n, it appears that p 4 γ C σ1/n l σ θ n ρ l = C 1 σ θ 1 n ρ l C T a l e λ+ σ l +1 T l+c µσl Cσ aρ l e Cσ θ ρ n l. Coming back to 74, we get Y l+1 Cσ aρ l e Cσ θ ρ n l Y l C l aρll+1 e C θ n ρ l Y1 C l aρll+1 e C θ n ρ l Y, which shows that and Theorem 3.3 is proved. lim l Y l+1 =, References [1] F. Ammar-Khodja, A. Benabdallah, C. Dupaix and I. Kostine, Controllability to the trajectories of phase-field models by one control force. SIAM J. Control. Opt [] F. Ammar-Khodja, A. Benabdallah and C. Dupaix, Controllability of some reaction-diffusion models by one control force. To appear. [3] S. Anita and V. Barbu, Local exact controllability of a reaction-diffusion system. Diff. Integral Equ [4] V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Optim [5] V. Barbu, Local controllability of the phase field system. Nonlinear Analysis [6] G. Lebeau and L. Robbiano, Contrôle exact de l équation de la chaleur. Comm. Partial Diff. Equ [7] A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations. Seoul National University, Korea. Lect. Notes Ser [8] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré, Anal. Non Linéaire [9] O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural ceva, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, AMS [1] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag New York [11] T.I. Seidman, How fast are violent controls? Math. Control Signals Syst [1] T.I. Seidman and J. Yong, How fast are violent controls, II? Math Control Signals Syst [13] J. Zabczyk, Mathematical Control Theory: An Introduction. Birkhäuser 199.