A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems
Abstract
We present a generalization of the Kalman rank condition to the case of non linear parabolic systems with constant coefficients and diagonalizable diffusion matrix. To reach the result, we are led to prove a global Carleman estimate for the solutions of a scalar 2nd−order parabolic equation and deduce from it an observability inequality for our adjoint system.
Full text
A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems. Ammar-Khodja Farid∗ , Benabdallah Assia† , Dupaix C´edric‡ , Gonz´alez-Burgos Manuel§ January 26, 2009 Abstract We present a generalization of the Kalman rank condition to the case of n×nlinear parabolic systems with constant coefficients and diagonalizable diffusion matrix. To reach the result, we are led to prove a global Carleman estimate for the solutions of a scalar 2n−order parabolic equation and deduce from it an observability inequality for our adjoint system. AMS 2000 subject classification: 93B05, 93B07, 35K05, 35K55, 35R30. Keywords: Kalman condition, control, observability, Carleman estimates, reaction diffusion systems. 1 Introduction. The main result Controllability of linear ordinary differential systems is well-known. In particular we have at our disposal the famous Kalman rank condition (see for example [17, Chapter 2, p. 35]), that is to say, if n, m ∈Nwith n, m ≥1 and A∈L(Rn) and B∈L(Rm,Rn), then the linear ordinary differential system Y′=AY +Bu is controllable at time T > 0if and only if (1.1) rank [A|B] = rank An−1B|An−2B| · · · | B=n. The main goal of our work is to give an extension of this algebraic condition to a class of parabolic systems of partial differential equations where the controls ∗Laboratoire de Math´ematiques UMR 6623, Universit´e de Franche-Comt´e, 16, route de Gray, 25030 Besan¸con cedex, France. E-mail: [email protected]. †CMI-LATP, UMR 6632, Universit´e de Provence, Technopˆole Chˆateau-Gombert, 39, rue F. Joliot-Curie, 13453 Marseille cedex 13, France. E-mail: [email protected] ‡Laboratoire de Math´ematiques UMR 6623, Universit´e de Franche-Comt´e, 16, route de Gray, 25030 Besan¸con cedex, France. E-mail: [email protected]. §Dpto, E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. E-mail: [email protected]. Supported by D.G.E.S. (Spain), grant MTM2006-07932. 1
act on a subdomain of the domain where the parabolic system is posed. To this end, let Ω ⊂Rdbe a bounded open set with a C2-boundary ∂Ω, let ω⊂Ω be a nonempty open subset and assume T > 0. Let us consider the second order elliptic self adjoint operator given by (1.2) R= N X i,j=1 ∂i(rij(x)∂j) + c(x) with (rij ∈W1,∞(Ω), c ∈L∞(Ω) (1 ≤i, j ≤d), rij(x) = rji(x) a.e. in Ω, and the coefficients rij satisfy the uniform elliptic condition (1.3) d X i,j=1 rij(x)ξiξj≥a0|ξ|2,∀ξ∈Rd,a.e. in Ω, for a positive constant a0. For n, m ∈N∗and T > 0, we consider the following n×nparabolic system (1.4) (∂ty= (DR +A)y+Bv1ωin ΩT= Ω ×(0, T ), y= 0 on ΣT=∂Ω×(0, T ), y(·,0) = y0(·) in Ω, where D∈L(Rn), A= (aij)1≤i,j≤n∈L(Rn) and B∈L(Rm,Rn) are given, v∈L2(ΩT)m(ΩT= Ω ×(0, T )) and y0= (y0,i)1≤i≤n∈L2(Ω)n. In (1.4), y= (yi)1≤i≤nis the state variable while 1ωdenotes the characteristic function of the open subset ω. All along this work we will assume that the diffusion matrix Dis diagonalizable with positive real eigenvalues, i.e., for J= diag (di)n×nwith d1, d2, ..., dn> 0, one has (1.5) D=P−1JP, with P∈L(Rn),det P6= 0. Let us observe that, thanks to the assumptions on the operator Rand the diffusion matrix D, for every y0∈L2(Ω)nand v∈L2(ΩT) system (1.4) possesses a unique solution y∈L2(0, T ;H1 0(Ω)n)∩C0([0, T ]; L2(Ω)n). The exact controllability to zero of system (1.4) reads as follows: For given T > 0and y0∈L2(Ω)ncan we find v∈L2(ΩT)msuch that the corresponding solution yof (1.4) satisfies y(T) = 0 a.e. in Ω?In the linear case, this property is equivalent to the exact controllability to the trajectories, that is to say, to the following property: for every trajectory y∗∈C0([0, T ]; L2(Ω)n) of (1.4) (i.e., a solution to (1.4) corresponding to v∗∈L2(ΩT)mand y∗ 0∈L2(Ω)n) and y0∈L2(Ω)n, there exists a control v∈L2(ΩT)msuch that the corresponding solution yto (1.4) satisfies y(·, T ) = y∗(·, T ) in Ω. 2
Finally, it will be said that (1.4) is approximately controllable in L2(Ω)nat time Tif, for any y0, yd∈L2(Ω)nand any ε > 0, there exists a control function v∈L2(ΩT)msuch that the solution y∈C0([0, T ]; L2(Ω)n) to (1.4) satisfies ||y(·, T )−yd||L2(Ω)n≤ε. Let us remark that in the scalar case, i.e., when n= 1, the controllability properties of system (1.4) are well known (see for instance [18], [11] and [10]). To be precise, under the previous assumptions and for every ω⊂Ω and T > 0, if B6= 0, system (1.4) is exactly controllable to the trajectories, null controllable and approximately controllable in L2(Ω) at time T > 0. Therefore we will concentrate on the study of the controllability properties of system (1.4) in the non scalar case and all along this paper we will assume that n≥2 and m≥1. The study of the controllability properties of system (1.4) is a mathematical interesting problem. Moreover, systems as (1.4) arise, for example, in the modeling of biological phenomena as the growth of tumors. In this context, the control vrepresents a therapy which only acts on some of the variables of the systems. There are few results providing sufficient conditions for the exact controllability to the trajectories of system (1.4) (when n > 1 and 1 ≤m < n) and most of them are proved for n= 2 and B= (1,0)∗: see, for instance, [22], [1], [4], [12] and [14] (also see [9] and [7] for similar controllability results of cascade system of parabolic-hyperbolic equations). The most general result in this direction seems to be the one in [13] where the authors study a cascade parabolic system of nequations. Finally we also point out [3] where the authors provide a sufficient condition on the coupling matrices for obtaining the null controllability on the time interval (T0, T1), with 0 ≤T0< T1≤T, of the non-autonomous parabolic linear system: (∂ty=L(t)y+A(t)y+B(t)v1ωin Ω ×(T0, T1), y= 0 on ∂Ω×(T0, T1), where A(·)∈Cn−1([0, T ]; L(Rn)), B(·)∈Cn([0, T ]; L(Rm,Rn)) and L(t) is a scalar time-dependent elliptic second order operator given by L(t)y(x, t) = − N X i,j=1 ∂ ∂xiaij (x, t)∂y ∂xj (x, t)+ N X i=1 bi(x, t)∂y ∂xi (x, t) +d(x, t)y(x, t), where aij ∈W1,∞(ΩT), bi, d ∈L∞(ΩT), aij(x, t) = aji(x, t) a.e. in ΩT(1 ≤ i, j ≤N), and N X i,j=1 aij(x, t)ξiξj≥ea0|ξ|2,∀ξ∈RN,a.e. in ΩT, for a positive constant ea0. 3
For obtaining a necessary and sufficient condition for the controllability of system (1.4), let us introduce the Kalman operator. We denote by Lthe operator given by L:= DR +A. Thanks to (1.5), we deduce D(L) = D(R)n= H2(Ω) ∩H1 0(Ω)n. Then the Kalman operator associated with (L, B) is the matrix operator (K:= [L|B] : D(K)⊂L2(Ω)nm →L2(Ω)n,with D(K) := {u∈L2(Ω)nm :Ku∈L2(Ω)n}, where [L|B] = Ln−1B|Ln−2B|... |LB |B. Our main result is the following: Theorem 1.1. Let us assume that Dsatisfies (1.5). Then, system (1.4) (with Rgiven by (1.2)) is exactly controllable to trajectories at any time Tif and only if the Kalman operator Ksatisfies (1.6) Ker (K∗) = {0}. As said above the Kalman rank condition (1.1) is a necessary and sufficient condition for the controllability of linear ordinary differential systems (see [24]). The first attempt to extend this condition to the case of an infinite dimensional systems are described and discussed in [23]. The closest result to Theorem 1.1 we know is due to H. Leiva [19] which gives a necessary and sufficient condition for the approximate controllability of (1.4) in the case in which Dis a diagonalizable matrix whose eigenvalues have positive real part, A≡0 and B(x)∈L2(Ω; L(Rm;Rn)). Thanks to (1.2), it is well known that the operator −Rwith Dirichlet boundary conditions admits a sequence of eigenvalues 0< λ1< λ2≤... ≤λp≤λp+1 ≤... , λp→ ∞, such that the associated sequence of normalized eigenfunctions (φp)p∈N∗is an orthonormal basis of L2(Ω). With the previous notations, we will see that, in fact, condition (1.6) can be easily checked since it is equivalent to the algebraic condition (1.7) rank [(−λpD+A)|B] = n, ∀p≥1 (see Proposition 2.2; we have used the notation [(−λpD+A)|B] = [(−λpD+A)n−1B|(−λpD+A)n−2B| · · · | B]). Remark 1.1. It is interesting to observe that system (1.4) can be exactly controlled to the trajectories with one control force even if A≡0. Indeed, let us assume that D= diag (di)n×n, with di>0 for every 1 ≤i≤n, and 4
B= (b1, ..., bn)∗∈Rn. Then, in order to apply Theorem 1.1 we should check condition (1.6), i.e., condition (1.7). So, [(−λpD+A)|B] = (−λpd1)n−1b1(−λpd1)n−2b1··· b1 (−λpd2)n−1b2(−λpd2)n−2b2··· b2 . . .. . ..... . . (−λpdn)n−1bn(−λpdn)n−2bn··· bn ∈L(Rn), and (1.7) holds if and only if bi6= 0 for every iand the diffusion coefficients di are distinct. Remark 1.2. Let us remark that the controllability properties of system (1.4) when the diffusion matrix Dgiven by D=β0Id, with β0>0, have been already studied in [3]. To be precise, in [3] it is proved that system (1.4) (with D=β0Id) is exactly controllable to the trajectories if and only if rank [A|B] = n. On the other hand, for D=β0Id, it is not difficult to see that condition (1.7) is independent of the eigenvalues λpand is equivalent to rank [A|B] = n. As a consequence, in this work we recover the results on controllability of system (1.4) proven in [3] in the case D=β0Id, with β0>0, A∈L(Rn) and B∈L(Rm;Rn). It is also interesting to point out that the techniques developed in this work are completely different from those used in [3]. In order to study the exact controllability to the trajectories of system (1.4), we will consider the corresponding adjoint problem (1.8) (−∂tϕ= (D∗R+A∗)ϕin ΩT, ϕ= 0 on ΣT, ϕ(x, T ) = ϕ0in Ω, where ϕ0∈L2(Ω)n. It is by now well known that the exact controllability to the trajectories of system (1.4) is equivalent to the existence of a positive constant Csuch that, for every ϕ0∈L2(Ω)n, the solution ϕ∈C0([0, T ]; L2(Ω)n) to the adjoint system (1.8) satisfies the observability inequality: (1.9) ||ϕ(·,0)||2 L2(Ω)n≤CZZωT |B∗ϕ(x, t)|2, where we have introduced the notation ωT=ω×(0, T ). The observability inequality (1.9) will be deduced from an appropriate global Carleman inequality satisfied by the solutions to the adjoint problem (1.8) (see Theorem 1.3). In order to state this Carleman inequality we will reason as follows: firstly we will show a global Carleman inequality (see Theorem 1.2) which bounds a weighted global integral of K∗ϕby means of a weighted local integral of B∗ϕ(ϕis the solution to the adjoint problem (1.8) corresponding to ϕ0). Secondly, and using condition (1.6), we will be able to deduce from the previous inequality the appropriate global Carleman inequality for the solutions to the adjoint problem and, in particular, (1.9). Our second result reads as follows: 5
Theorem 1.2. Let us assume that Dsatisfies (1.5). Then, given τ∈Rand k≥(n−1)(2n−1),there exist a positive function β0∈C2(Ω) (only depending on Ωand ω), r=r(n)∈Nand two positive constants Cand σ(only depending on Ω,ω,n,(rij )1≤i,j≤N,D,A,kand τ) such that for every ϕ0∈L2(Ω)nthe corresponding solution ϕto (1.8) satisfies (1.10) ZT 0 (sρ)τe −2sM0 t(T−t)||RkK∗ϕ||2 L2(Ω)nm ≤CZZωT (sρ)τ+K+re−2sη |B∗ϕ|2, for every s≥σT+T2+T2||c||2/3 ∞. In (1.10), ρ,η,M0and Kare respectively given by: ρ(t) = t−1(T−t)−1,η(x, t) = β0(x)/t(T−t),M0= maxΩβ0 and K= 4k+n−4. Remark 1.3. Let us observe that if ϕ0∈L2(Ω)n, the corresponding solution ϕto (1.8) satisfies ϕ∈C∞((0, T ]; D(Rk)n), for every k≥1. Indeed, Ais a constant matrix and the operator D∗R+A∗with domain D(D∗R) = H2(Ω) ∩H1 0(Ω)n, is the generator of an analytic semigroup on L2(Ω)n(see Section 2 and Subsection 3.2. Finally, in our third result we state a global Carleman inequality for the solutions to (1.8). At this point we use condition (1.6) in an essential way: Theorem 1.3. In addition to the assumptions in Theorem 1.2, we assume condition (1.6). Then, given τ∈Rand k≥(n−1)(2n−1),there exist two positive constants Cand σ(only depending on Ω,ω,n,(rij )1≤i,j≤N,D,A,k and τ) such that for every ϕ0∈L2(Ω)nthe corresponding solution ϕto (1.8) satisfies (1.11) ZZΩT (sρ)τe −2sM0 t(T−t)|Rk−(n−1)(2n−1)ϕ|2≤CZZωT (sρ)τ+K+re−2sη |B∗ϕ|2, for every s≥σT+T2+T2||c||2/3 ∞. In (1.11), ρ,η,M0,Kand r(n)are as in Theorem 1.2. The plan of the paper is the following: In section 2, we will address some properties of the unbounded Kalman operator Kand prove the continuity (in a suitable sense) of its inverse, when it exists. Then, in section 3, a proof of Theorem 1.2 is given using a global Carleman estimate for a scalar parabolic equation of order 2n(see Theorem 3.2). In section 4 we give some comments and open problems. The results that we present in this work have been announced in [2]. 2 Properties of the Kalman operator Define D= ∞ \ p=0 D(Rp). 6
It is well-known that Dis dense in D(Rp) for every p≥0 and straightforward that Dnm ⊂D(K).Thus, D(K) = L2(Ω)nm and K∗is well-defined from D(K∗)⊂L2(Ω)ninto L2(Ω)nm. The formal adjoint operator of K, again denoted by K∗,is given by K∗= B∗(L∗)n−1 . . . B∗L∗ B∗ , and it coincides with the adjoint operator of Kon Dn. This section will be devoted to the proof of the following result which contains two crucial properties of the Kalman operator K: Theorem 2.1. There exists a positive constant Csuch that: 1. Continuity of Kand K∗:If u∈D(Rn−1)nm (resp. u∈D(Rn−1)n) then Ku∈L2(Ω)n(resp. K∗u∈L2(Ω)nm) and kKuk2 L2(Ω)n≤CRn−1u2 L2(Ω)nm , (resp. kK∗uk2 L2(Ω)nm ≤CRn−1u2 L2(Ω)n). 2. Invertibility of K∗and continuity of the inverse: Assume that condition (1.6) is fulfilled and let k∈Nwith k≥(2n−1)(n−1). Then, for every ϕ∈L2(Ω)nsatisfying K∗ϕ∈D(Rk)nm, one has ϕ∈ D(Rk−(2n−1)(n−1))nand (2.12) Rk−(2n−1)(n−1)ϕ 2 L2(Ω)n≤CRkK∗ϕ2 L2(Ω)nm . Remark 2.1. Let us remark that, in general, the operator K(resp. K∗) is not onto, even if Ker K={0}. Indeed, assume that n≥2, m= 1 and take, for instance, D= diag (di)n×n(d1, d2, ..., dn>0) and B≡e1. We consider f≡(Ae1)g∈L2(Ω)nwith g∈H1 0(Ω) such that Rg 6∈ L2(Ω). If we pose the equation x∈D(K) such that Kx=f, we readily see that it admits the solution x= (0, ..., 0, x1, x0)twith x1=g∈ L2(Ω) and x0=−d1Rx16∈ L2(Ω): Kx=LBx1+Bx0≡(d1Rx1+x0)e1+ (Ae1)x1≡f. Thus, the previous equation has no solution in D(K) and Kis not onto. Before proving Theorem 2.1, let us introduce some notations. We recall that (λp)p∈N∗is the sequence of eigenvalues of the operator −Rwith Dirichlet 7
boundary conditions and (φp)p∈N∗the associated sequence of eigenfunctions. For any j, p ∈N∗, we consider the projection operator Pj p: Ψ = (Ψk)1≤k≤j∈L2(Ω)j→Pj p(Ψ) = ((Ψk, φp))1≤k≤j∈Rj, where (·,·) stands for the scalar product in L2(Ω). All along this paper we denote by |·|the euclidian norm in Rj. Thus, if j∈N∗, a characterization of Djis: Dj= Ψ = X p≥1 Ψpφp: Ψp∈Rjand X p≥1 λ2m p|Ψp|2<∞,∀m≥0 . For p∈N∗, let us denote Lp=−λpD+A∈L(Rn) and Kp= [Lp|B] = [Ln−1 pB|Ln−2 pB| · · · | B]∈L(Rnm,Rn). It is not difficult to show the equalities (L(bφp) = (Lpb)φp,∀b∈Rn,∀p≥1 and K(bφp) = (Kpb)φp,∀b∈Rnm,∀p≥1. From these identities and taking into account that Land Kare closed unbounded operators, a direct computation gives Ly =X p≥1 LpPn p(y)φp,∀y∈D(L), Ku=X p≥1 KpPnm p(u)φp,∀u∈D(K), and then, D(K) =u∈L2(Ω)nm :X p≥1KpPnm p(u)2<∞. In the same way, we also have: K∗=X p≥1 K∗ pPn p(·)φp,K∗ p= [Lp|B]∗ D(K∗) =ϕ∈L2(Ω)n:X p≥1K∗ pPn p(ϕ)2<∞. Finally, we define the operator KK∗:D(KK∗)⊂L2(Ω)n→L2(Ω)n, with domain D(KK∗) = {ϕ∈L2(Ω)n:K∗ϕ∈D(K),KK∗ϕ∈L2(Ω)n}. 8
Note that KK∗is a closed unbounded operator and, again, a simple calculation provides the equalities: KK∗ϕ=X p≥1 KpK∗ pPn p(ϕ)φp, D(KK∗) = ϕ∈L2(Ω)n:X p≥1K∗ pPn p(ϕ)2,X p≥1KpK∗ pPn p(ϕ)2<∞ . With these notations, it is easy to show that: Proposition 2.2. The following conditions are equivalent: 1. Ker (K∗) = {0}. 2. Ker (KK∗) = {0}. 3. det KpK∗ p6= 0 for every p≥1. We are now ready to prove Theorem 2.1. Proof of Theorem 2.1. 1. From the expression of Kp(resp. K∗ p), it is not difficult to prove that for some positive constant we have (2.13) |Kpb| ≤ Cλn−1 p|b|,∀b∈Rnm and p≥1, (resp. |K∗ pb| ≤ Cλn−1 p|b|, for every b∈Rnand p≥1). Now, if u∈D(Rn−1)nm, one has kKuk2 L2(Ω)n=X p≥1KpPnm p(u)2≤CX p≥1 λ2(n−1) pPnm p(u)2 =CRn−1u2 L2(Ω)nm . The proof for K∗can be obtained in a similar way. 2. Let us consider ϕ∈L2(Ω)nwith K∗ϕ∈D(Rr)nm and let us set u≡K∗ϕ∈ D(Rr)nm. This last equality can be equivalently written as K∗ pPn p(ϕ) = Pnm p(u),∀p≥1. Applying Kp,we get: (2.14) KpK∗ pPn p(ϕ) = KpPnm p(u),∀p≥1. Let us remark that the elements of the matrix [−λD +A|B] [−λD +A|B]∗ are polynomials of degree 2(n−1). Therefore, we can write [−λD +A|B] [−λD +A|B]∗= [M1(λ),··· , Mn(λ)] , 9
for all s≥σ1T+T2+T2||c||2/3 ∞, which in view of (3.19) implies (3.25) I(τ0+ 3(n−1), φ) + n X i=2 I(τ0+ 3(n−i), Pi−1...P1φ) ≤CZZωT (sρ)τ0+re−2sη |φ|2+ZZΩT (sρ)τ0e−2sη |F(φ)|2. Note, at this level, that we have got rid of the local terms in the right-hand- side of (3.21) but the left-hand-side of (3.25) does not contains enough terms to absorb the term corresponding to F(φ). So let Π denote then any permutation of {1,2,...,n}and consider, instead of (3.19), the new change of variable ψ1=φ ψi=PΠ(i−1)ψi−1=∂t+dΠ(i−1)Rψi−1,2≤i≤n. Then system (3.20) becomes ∂t+dΠ(1)Rψ1=ψ2in ΩT, ∂t+dΠ(2)Rψ2=ψ3in ΩT, . . . ∂t+dΠ(n)∆ψn=F(φ) in ΩT, ψi= 0 on ΣT and the same procedure as above leads to a similar estimate as (3.25) which reads then I(τ0+ 3(n−1), φ) + n X i=2 I(τ0+ 3(n−i), PΠ(i−1) ...PΠ(1)φ) ≤CZZωT (sρ)τ0+re−2sη |φ|2+ZZΩT (sρ)τ0e−2sη |F(φ)|2, for all s≥σ1T+T2+T2||c||2/3 ∞, where Cis a new positive constant. Thus, considering all such possible permutations with associated change of variable, we finally obtain (3.26) I(τ0+ 3(n−1), φ) + n X i=1 I(τ0+ 3(n−2), Piφ) + n−1 X p=2 X 1≤i1<···<ip≤n I(τ0+ 3(n−p−1), Pip...Pi1φ) ≤CZZωT (sρ)τ0+re−2sη |φ|2+ZZΩT (sρ)τ0e−2η|F(φ)|2, 16
where s≥σ1T+T2+T2||c||2/3 ∞and Cis a new positive constant. Now, we are going to prove that the term corresponding to F(φ) can be absorbed by the left-hand-side of (3.26). Using the definition of F(φ) (see (3.18)) we have that: (3.27) ZZΩT (sρ)τ0e−2sη |F(φ)|2≤CZZΩT (sρ)τ0e−2sη|φ|2+ n X i=1 |Piφ|2 + n−1 X p=2 X 1≤i1<···<ip≤nPi1...Pipφ2, where Cis a new positive constant (observe that Calso depends on Athrough the coefficients αi1,...,ip,αi,α). Finally, choosing s≥σ2T+T2+T2||c||2/3 ∞, with σ2depending on Ω, ω, n, (rij )1≤i,j≤N,D,Aand τ0, we deduce C(sρ)τ0≤1 2(sρ)τ0+3(n−p),∀p: 0 ≤p≤n−1, and (3.22) from (3.26) and (3.27). Remark 3.1. In view of steps 1 and 2, instead of (3.16), one can consider the following system (P(∂t,∇)φ=fin ΩT, Rkφ= 0 on ΣT,∀k≥0, with f∈L2(ΩT). It is a direct application to deduce: I(τ0+ 3(n−1), φ) + n X i=1 I(τ+ 3(n−2), Piφ) + n−1 X p=2 X 1≤i1<···<ip≤n I(τ+ 3(n−p−1), Pip. . . Pi1φ) ≤CZZωT (sρ)τ0+re−2sη |φ|2+ZZΩT (sρ)τ0e−2sη |f|2, for every s≥σ2T+T2+T2||c||2/3 ∞. Step 3: It is interesting to remark that, thanks to the regularity assumptions imposed on φ, if 1 ≤i≤k1and 1 ≤j≤k2,Ri∂j tφalso fulfills equation (3.16) and the two previous steps developed for φcan be repeated for Ri∂j tφ. Therefore, from (3.22), if we take τ∈Rwe deduce the existence of two positive constants Cτand στ, which depend on Ω, ω,n, (rij )1≤i,j≤N,D,Aand τ, such that I(τ, Ri∂j tφ)≤CτZZωT (sρ)τ+re−2sη Ri∂j tφ 2, 17
for every s≥στT+T2+T2||c||2/3 ∞(we recall that I(τ, φ) is defined in the statement of Theorem 3.2). On the other hand, it is easy to check that for any ψ∈L2(0, T ;D(R)) with ∂tψ∈L2(ΩT) ZZωT (sρ)τ+re−2sη |Rψ|2+|∂tψ|2≤I(τ+ 4, ψ)≤I(τ+ 4, ψ), (I(·,·) is given by (3.15)). Thus, if s≥στT+T2+T2||c||2/3 ∞, one has (I(τ, Ri∂j tφ)≤CτI(τ+ 4, Ri−1∂j tφ), I(τ, Ri∂j tφ)≤CτI(τ+ 4, Ri∂j−1 tφ). Now, if we successively apply these last inequalities for 1 ≤i≤k1and 1 ≤ j≤k2we deduce the existence of two positive constants e Cτand eστ(depending on Ω, ω,n, (rij)1≤i,j≤N,D,A,τ,k1and k2) for which I(τ, Ri∂j tφ)≤e CτI(τ+ 4(i+j), φ) holds for every s≥eστT+T2+T2||c||2/3 ∞and for every i, j with 0 ≤i≤k1 and 0 ≤j≤k2. Finally, choosing τ=τ0−4(i+j) and taking into account inequality (3.22), we obtain (3.17). Remark 3.2. As far as the Carleman estimate is concerned, it is enough to assume that the diffusion coefficients satisfy di6= 0 for 1 ≤i≤n. 3.2 Proof of Theorem 1.2. The Carleman inequality for the adjoint problem We will devote this section to show the Carleman estimate for the adjoint problem (1.8) stated in Theorem 1.2. Let us recall that Dstands for the space D=T∞ p=0 D(Rp), a space which is dense in L2(Ω). The starting point relies on the following observation. Proposition 3.3. Let us consider ϕ0∈Dnand let ϕ= (ϕ1,··· , ϕn)∗be the corresponding solution of problem (1.8). Then, ϕ∈Ck([0, T ]; D(Rp)n)for every k, p ≥0, and, for every i(with 1≤i≤n)ϕisolves (3.16). Proof. Assume ϕ0∈Dnand let ϕ= (ϕ1,··· , ϕn)∗be the corresponding solution of problem (1.8). Setting P∗ψ(x, t) = ϕ(x, T −t) for (x, t)∈ΩT,ψ solves problem: (∂tψ= (JR + (P∗)−1A∗P∗)ψin ΩT, ψ= 0 on ΣT, ψ(x, 0) = (P∗)−1ϕ0in Ω, (see (1.5) for the definition of Jand P). The operator JR with domain D(JR) = H2(Ω) ∩H1 0(Ω)n, is dissipative and self adjoint on the space L2(Ω)n: it is then 18
the generator of an analytic semigroup. Since (P∗)−1A∗P∗is a bounded linear operator on L2(Ω)n,by the perturbation theory of analytic semigroups (see [21, Corollary 2.2., p. 81]), the operator JR+(P∗)−1A∗P∗is also the generator of an analytic semigroup. Since D((JR + (P∗)−1A∗P∗)p) = D(Rp)n, the regularity part follows from the semigroup theory. Let Q(η, ξ) = ηI +D∗ξ+A∗for (η, ξ)∈R2and set Q(∂t,∇) = ∂t+ D∗R+A∗:= (Qij(∂t,∇))1≤i,j≤n. As operators acting on C∞([0, T ]; Dn), all the entries Qij(∂t,∇) live this space invariant and commute between them. So, as it is done in [15, I.VI.4, p. 144], in the system Q(∂t,∇)ϕ= 0,we carry out the (Gauss) algebraic elimination process as if the operators were constants. With P(∂t,∇) = det Q(∂t,∇),the conclusion follows. Proof of Theorem 1.2: Firstly, we will assume that ϕ0∈Dn. Let ϕbe the solution to (1.8) corresponding to ϕ0. We can then apply Proposition 3.3 and deduce that ϕ∈Cℓ([0, T ]; D(Rp)n), for every ℓ, p ≥0, and ϕisatisfies (3.16) for every i, 1 ≤i≤n. Also, (B∗ϕ)jlies in Cℓ([0, T ]; D(Rp)n) and solves (3.16), for all j(with 1 ≤j≤m). Thus Theorem 3.2 can be applied to φ= (B∗ϕ)i(1 ≤i≤m) with k1=k, k2=n−1 and τ0=τ−4k−n+ 4 deducing the existence of two positive constants e Cand eσ(only depending on Ω, ω,n, (rij)1≤i,j≤N,D,A,kand τ) such that k X l=0 n−1 X j=0 I(τ0−4(l+j), Rl∂j t(B∗ϕ)i)≤e CZZωT (sρ)τ0+re−2sη |B∗ϕ|2 for every s≥es:= eσT+T2+T2||c||2/3 ∞. Observe that thanks to (3.24), if s≥eswe can affirm (sρ(t))τ≤C(sρ(t))τ0−4(l+j)+3n,∀t∈(0, T ), for every l, j with 0 ≤l≤k, 0 ≤j≤n−1. The definition of I(see the statement of Theorem 3.2) and Kleads immediately to (3.28) n−1 X j=0 ZZΩT (sρ)τe−2sη Rk∂j t(B∗ϕ)i 2≤e CZZωT (sρ)τ+K+re−2sη |B∗ϕ|2, for every s≥eσT+T2+T2||c||2/3 ∞and i= 1,··· , m. In (3.28), r=r(n) is as in the statement of Theorem 3.2 and K= 4k+n−4. On the other hand, using (1.8) and the expression of K∗we get: K∗ϕ(·, t)≡(−1)n−1∂n−1 tB∗ϕ, (−1)n−2∂n−2 tB∗ϕ, ··· ,−∂tB∗ϕ, B∗ϕ∗(·, t) in [0, T ). Now, if we set M0= maxΩβ0and use (3.28) we infer (1.10) for every s≥eσT+T2+T2||c||2/3 ∞. This concludes the proof in the case ϕ0∈Dn. The general case can be easily obtained from a density argument. Indeed, if ϕ0∈L2(Ω)nthere exists a Cauchy sequence {ϕℓ 0}ℓ≥1⊂Dnsuch that ϕℓ 0→ϕ0 19
in L2(Ω)n. If ϕℓand ϕare, respectively, the solution to (1.8) corresponding to ϕℓ 0and ϕ0, we have ϕℓ→ϕin L2(Q)nand RkK∗ϕℓ→RkK∗ϕin D′(ΩT)nfor every k≥0. Observe that (ϕℓ, ϕℓ 0) satisfies inequality (1.11) for every ℓ≥1. Also, inequality (1.11) implies that {RkK∗ϕℓ}ℓ≥1is a Cauchy sequence in the weighted space L2((sρ)τ/2e −sm0 t(T−t); ΩT). These previous considerations permit to pass to the limit in the Carleman inequality (1.11) satisfied by (ϕℓ, ϕℓ 0) and obtain the result in the general case. This ends the proof. We end the section with the proof of Theorem 1.3. As said above, at this point we will use the assumption (1.6). Proof of Theorem 1.3: It is a direct consequence of the assumption on K∗: taking account (2.12) and (1.10), if k≥(2n−1)(n−1) we deduce the existence of a new positive constant Cfor which (1.11) holds for every s≥eσT+T2+T2||c||2/3 ∞. This finalizes the proof. 4 The Kalman condition. Proof of Theorem 1.1 We will devote this section to the proof of Theorem 1.1. Proof of Theorem 1.1: Let us begin by proving the necessary part of Theorem 1.1: It is well-known that the null-controllability is equivalent to the observability estimate (1.9) for any solution of the adjoint problem (1.8). Suppose that Ker (K∗)6={0}. From Proposition 2.2 it follows that there exists p0∈N∗such that rank Kp0= rank [(−λp0D+A)|B]< n. Therefore, thanks to Kalman’s rank condition applied to the ordinary differential system y′= (−λp0D+A)y+Bv we infer that this system is not controllable. Thus, there exists a non zero solution zp0(t)∈Rnto the associated adjoint system −z′= (−λp0D∗+A∗)zin (0, T ), satisfying B∗zp0(t) = 0 for every t∈[0, T ]. Then, letting ϕ0=zp0(T)φp0, where φp0is the normalized eigenfunction associated with λp0, it is easy to check that the function ϕ(t, x) = zp0(t)φp0(x) is the solution of (1.8) corresponding to ϕ0. It is non zero and satisfies B∗ϕ(x, t) = 0 in ΩT. Obviously this solution does not satisfy the observability inequality (1.9), and thus (1.4) is not null controllable. Remark 4.1. Observe that, if condition (1.6) is not satisfied, then system (1.4) is not null controllable even if ω= Ω. We turn now to the sufficient part of Theorem 1.1. As said above, the exact controllability to the trajectories of system (1.4) is equivalent to the null controllability of this system. Therefore, it is enough to show the observability inequality (1.9) for the solutions of the adjoint problem (1.8). Following [10], we show inequality (1.9) combining the global Carleman inequality (1.11) and the energy inequality satisfied by the solutions to the adjoint problem. 20
Let ϕ∈L2(0, T ;H1 0(Ω)n) be the solution to (1.8) corresponding to ϕ0∈ L2(Ω)n. Thanks to the hypothesis (1.6), we can apply Theorem 1.3 with τ= 0 and k= (n−1)(2n−1) obtaining Z3T/4 T/4ZΩ e −2sM0 t(T−t)|ϕ|2≤CZZωT (sρ)le−2sη |B∗ϕ|2, for every s≥σT+T2+T2||c||2/3 ∞, with l= 4(n−1)(2n−1) + n+rand r as in Theorem 1.2. It can be easily verified that e −2sM0 t(T−t)≥exp −25sM0/(3T2),∀t∈(T/4,3T/4), and, if we take s≥(l/8m0)T2, also (sρ)le−2sη ≤sl22lT−2lexp −23m0s/T 2≤l 2em0l ,∀(x, t)∈ΩT, where m0= minΩβ0. From this three last inequalities, we readily deduce (4.29) Z3T/4 T/4ZΩ |ϕ|2≤CeCs/T 2ZZωT |B∗ϕ|2, for every s≥σ1T+T2+T2||c||2/3 ∞and σ1= max{σ, (l/8m0)}(Cis a positive constant depending on Ω, ω,n, (rij )1≤i,j≤N,Dand A). On the other hand, taking into account the assumptions imposed to the operator Rand the matrix D(see (1.2) and (1.5)), it is not difficult to get the following energy inequality satisfied by ϕ d dt eC(1+||c||∞)t||ϕ(·, t)||2 L2(Ω)n≥0,∀t∈(0, T ), with Ca positive constant depending on Dand A. From this last inequality we also obtain ||ϕ(·,0)||2 L2(Ω)n≤eC(1+||c||∞)T/4||ϕ(·, T/4)||2 L2(Ω)n ≤2 TeC(1+||c||∞)3T/4Z3T/4 T/4ZΩ |ϕ|2. This last inequality together with (4.29) (with s=σ1T+T2+T2||c||2/3 ∞) imply the observability inequality for the solutions to the adjoint problem (1.8): (4.30) ||ϕ(·,0)||2 L2(Ω)n≤eC(1+1/T +||c||2/3 ∞+T+T||c||∞)ZZωT |B∗ϕ(x, t)|2, with C > 0 a constant which depends on Ω, ω,n, (rij )1≤i,j≤N,Dand A. This finalizes the sufficient part and Theorem 1.1. 21
5 Some additional comments, results and open problems 1. Approximate controllability. As a consequence of Theorem 1.1, we can prove that condition (1.6) also is a necessary and sufficient condition for the approximate controllability of system (1.4). In fact, one has Theorem 5.1. Let us assume that Dsatisfies (1.5). Then, system (1.4) (with Rgiven by (1.2)) is approximately controllable at time Tif and only if the Kalman operator Ksatisfies (1.6) Proof: The sufficient part can be proved taking into account the global Carleman inequality (1.11). Indeed, it is well known that system (1.4) is approximately controllable at time Tif and only if the adjoint problem (1.8) satisfies the following unique continuation property: “If ϕ∈C0([0, T ]; L2(Ω)n)is a solution to (1.8) and B∗(t)ϕ≡0in ω×(0, T ), then ϕ≡0in Q.” This property is a direct consequence of (1.11). Finally, the necessary condition can be deduced arguing as in the proof of Theorem (1.1). 2. When condition (1.6) is fulfilled, it is also possible to compute the cost of the exact controllability to the trajectories of system (1.4). As a consequence of the observability inequality (4.30) and following the ideas of [10] we can prove: Theorem 5.2. Let us assume that Ris given by (1.2) and Dsatisfies (1.5). Let y∗∈L2(0, T ;H1 0(Ω)n)∩C0([0, T ]; L2(Ω)n)be a trajectory of system (1.4) and let us fix y0∈L2(Ω)n. Then, if (1.6) holds, there exists v∈L2(ΩT)m such that the solution to (1.4) satisfies y(·, T ) = y∗(·, T )in Ω. Moreover, for a positive constant C(which only depends on Ω,ω,n,(rij )1≤i,j≤N,Dand A) one has ||v||2 L2(Ω)m≤exp C1 + 1/T +||c||2/3 ∞+T+T||c||∞||y0−y∗(·,0)||2 L2(Ω)n. 3. It is possible to give a result on exact controllability to trajectories of system (1.4) when condition (1.6) is not fulfilled but there exists bp∈N∗such that rank [(−λbpD+A)|B] = n. To be precise, let us consider (J={p∈N∗: rank [(−λpD+A)|B] = n}and X= span {φp:p∈J} ⊂ L2(Ω). Thanks to Remark 2.2 we can conclude that Jis an infinite set whence we deduce that Xis an infinite-dimensional closed subspace of L2(Ω). On the other hand, let us fix y0∈L2(Ω)nand a trajectory y∗∈L2(0, T ;H1 0(Ω)n)∩ C0([0, T ]; L2(Ω)n) of system (1.4). Thus, one has: 22
Theorem 5.3. Under the previous assumptions, if y0−y∗(·,0) ∈Xn, then there exists v∈L2(ΩT)msuch that the solution yto system (1.4) satisfies y(·, T ) = y∗(·, T )in Ω. Proof: This result is a consequence of the following observability inequality for the solutions ϕof the adjoint problem (1.8) associated to initial data ϕ0∈Xn: (5.31) ||ϕ(·,0)||2 L2(Ω)n≤CZZωT |B∗ϕ(x, t)|2, where Cis a positive constant. So, let us prove this inequality. Firstly, observe that thanks to the previous assumptions we can repeat the proof of Theorem 2.1 and infer the existence of a constant C > 0 such that for every ψ∈Xnsatisfying K∗ψ∈D(Rk)nm (k≥(2n−1)(n−1)) one has ψ∈D(Rk−(2n−1)(n−1))nand Rk−(2n−1)(n−1)ψ 2 L2(Ω)n≤CRkK∗ψ2 L2(Ω)nm . Secondly, we readily obtain that if ϕ0∈Xnthen the solution ϕto (1.8) satisfies ϕ(·, t)∈Xnfor every t∈[0, T ]. Taking into account the two previous considerations and inequality (1.10) (with τ= 0 and k= (2n−1)(n−1)), we get ZZΩT e −2sM0 t(T−t)|ϕ|2≤CZZωT (sρ)K+re−2sη |B∗ϕ|2, for every s≥σT+T2+T2||c||2/3 ∞(M0,ρ,K,rand ηare as in the statement of Theorem 1.2). Finally, The observability inequality (5.31) can be obtained from the previous Carleman inequality reasoning as in the proof of Theorem 1.1. This ends the proof. 4. Theorems 1.1, 1.2 and 1.3 are still valid if in (1.4) we consider Neuman boundary conditions instead of Dirichlet boundary conditions. 5. In our analysis we have strongly used the structure of the diffusion matrix D. It would be very interesting to extend the results of this work to the case in which Dis a general positive definite matrix. Observe that, in this case, even the case B≡Id ∈L(Rn) (i.e., a distributed control in each equation of the system) is still open. 6. The non autonomous problem. To our knowledge, the null controllability problem of (1.4) when the operators and the coupling matrices Aand Bare time-dependent is still open. We will address in details this question in the case D=Id in a forthcoming paper (see [3]). On the other hand, the case in which Aand Bdepend on xseems to be much more complicated. A necessary and sufficient condition for the exact controllability to the trajectories of system (1.4) is open. 23
7. Boundary controls. In view of known controllability results for a linear heat equation, it would be natural to wonder whether the controllability result for system (1.4) remains valid when one considers boundary controls exerted on a relative open subset γof the boundary ∂Ω. Nevertheless, there exist negative results for some 1-d cascade linear coupled parabolic systems with n= 2 which are null controllable in (0, T ) when we apply a distributed control e1v1ωand they are not if we take y=e1v1γon ∂Ω×(0, T ) as boundary control (cf. [8]). These counterexamples reveal the different nature of the controllability properties for a single heat equation and for coupled parabolic systems. 8. Extension to non smooth diffusion operators. Inspection of the proof of Theorem 1.1 shows that the operator Rhas to satisfy only two conditions •Rmust be self-adjoint and uniformly elliptic as (1.3), •Rmust satisfy a Carleman estimate as in Theorem 3.1. According to [6], [5] and [20], our result extends to operators Ras (1.2) with non smooth coefficients rij (see [6], [5] and [20] for exact assumptions). References [1] F. Ammar-Khodja, A. Benabdallah and C. Dupaix,Null controllability of some reaction-diffusion systems with one control force, J. Math. Anal. Appl. 320 (2006), no. 2, 928–943. [2] F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. Gonz´ alez- Burgos,Controllability for a class of reaction-diffusion systems: the generalized Kalman’s condition, C. R. Acad. Sci. Paris, Ser. I 345 (2007), no. 10, 543-548. [3] F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. Gonz´ alez- Burgos,A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, in preparation. [4] F. Ammar-Khodja, A. Benabdallah, C. Dupaix and I. Kostine, Null controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var. 11 (2005), no. 3, 426–448. [5] A. Benabdallah, Y. Dermenjian, J. Le Rousseau,Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math. Anal. Appl. 336 (2007), no. 2, 865-887. [6] A. Doubova, A. Osses, and J.-P. Puel,Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim. Calc. Var. 8 (2002), 621–661. 24
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