Controllability results for cascade systems of m coupled parabolic PDEs by one control force

Transcription

1 Portugal. Math. (N.S.) Vol. xx, Fasc., x, xxx xxx Portugaliae Mathematica c European Mathematical Society Controllability results for cascade systems of m coupled parabolic PDEs by one control force Manuel González-Burgos, Luz de Teresa Abstract. In this paper we will analyze the controllability properties of a linear coupled parabolic system of m equations when a unique distributed control is exerted on the system. We will see that, when a cascade system is considered, we can prove a global Carleman inequality for the adjoint system which bounds the global integrals of the variable ϕ = (ϕ 1,..., ϕ m) in terms of a unique localized variable. As a consequence, we will obtain the null controllability property for the system with one control force. Mathematics Subject Classification (). 93B5, 93B7, 35K5. Keywords. Controllability, cascade parabolic systems. 1. Introduction Let Ω R N be a bounded connected open set with boundary Ω of class C. Let ω Ω be a nonempty open subset and assume T >. All along this work we will denote = Ω (, T ) and Σ = Ω (, T ). For m 1 given, we consider the parabolic linear system m m t y 1 L 1 y 1 + B 1j y j + a 1j y j = v1 ω in = Ω (, T ), m m t y L y + B j y j + a j y j = in, m m t y m L m y m + B mj y j + a mj y j = in, y i = on Σ = Ω (, T ), y i (x, ) = y,i (x) in Ω, 1 i m, Partially supported by D.G.I. (Spain), grant MTM Partially supported by grant 3746-E of CONACyT and IN166 of PAPIIT (Mexico).

2 M. González-Burgos, L. de Teresa where a ij = a ij (x, t) L (), B ij = B ij (x, t) L () N (1 i, j m), y,i L (Ω) (1 i m) and L k is, for every 1 k m, the self-adjoint second order operator N ( L k y(x, t) = i α k ij (x, t) j y(x, t) ), (1) ( i = / xi ) where i, α k ij W 1, (), α k ij(x, t) = α k ji(x, t) a.e. in, 1 i, j N, 1 k m, () and, for all k : 1 k m, the coefficients α k ij satisfy max i,j,k αk ij W 1, = M, N αij(x, k t)ξ i ξ j ã ξ, ξ R N, a.e. in, (3) i, for positive constants M and ã. Equivalently, the previous system can be written as { t y Ly + B y + Ay = Dv1 ω in, y = on Σ, y(x, ) = y (x) in Ω, (4) where L is the matrix operator given by L = diag (L 1,..., L m ), y = (y i ) 1 i m is the state, 1 ω is the characteristic function of the nonempty set ω and y = ( y i ) 1 i m, and where { y = (y,i ) 1 i m L (Ω) m, A(x, t) = (a ij (x, t)) 1 i,j m L () m, B(x, t) = (B ij (x, t)) 1 i,j m L () Nm and D e 1 = (1,,..., ) are given. Let us observe that, for each y L (Ω) m and v L (), system (4) admits a unique weak solution y L (, T ; H 1 (Ω) m ) C ([, T ]; L (Ω) m ). The main goal of this paper is to analyze the controllability properties of (4). It will be said that (4) is null controllable at time T if for every y L (Ω) m there exists a control v L () such that the solution y of (4) satisfies y i (, T ) = in Ω, i : 1 i m. (5) When m = 1 (one equation and one control force) the null controllability of parabolic problems has been studied by several authors (see for instance [18], [17], [4], [1],...). We also point out [14] and [15] where the null controllability of system (4) at time T was established for every T >, A L (), B L () N and ω Ω, using a global Carleman inequality for the solutions of the corresponding adjoint problem. There are few results on null controllability of system (4) when m > 1 and most of them are proved for m =. In [19] and [5] the authors consider a nonlinear system of two heat equations, one of them being forward and the other one

3 Controllability for parabolic cascade systems 3 backward in time, and show the null controllability of this system with sublinear nonlinearities ([19]) or slightly superlinear nonlinearities ([5]). In [1] and [], the authors give a null controllability result for a phase-field system and for reactiondiffusion systems (two nonlinear heat equations). The results in [1] and [] have been generalized in [1] (see also [11]) in two directions: on the one hand, there are not restrictions on the dimension N, and on the other hand, the authors consider nonlinearities which depend on the gradient of the state. Finally, in [8] the authors prove a result of local exact controllability to the trajectories for the Boussinesq system (N + 1 equations) when N (or N 1) distributed controls are exerted on the system. All previous works have a common point: they deal with cascade systems. In this work we want to generalize these works to the case of a general cascade system of m linear parabolic equations. To this end, we will suppose that A and B have the structure a 11 a 1 a a 1m a 1 a a 3... a m B 11 B 1... B 1m A = a 3 a a 3m B... B m, B = B... a m,m 1 a mm mm (6) with a ij L (), B ij L () N (1 i j m) and a i,i 1 L () ( i m), for an open set ω ω, satisfy a i,i 1 a > or a i,i 1 a > in ω (, T ), i : i m. (7) In order to study the null controllability of system (4), we will consider the corresponding adjoint problem which, under assumption (6) (cascade system), has the form i t ϕ i L 1 ϕ i [ (B ji ϕ j ) a ji ϕ j ] = a i+1,i ϕ i+1 in, (1 i m 1), (8) m t ϕ m L m ϕ m [ (B jm ϕ j ) a jm ϕ j ] = in, ϕ i = on Σ, ϕ i (x, T ) = ϕ,i (x) in Ω, 1 i m, where ϕ,i L (Ω) (1 i m). It is well known that the null controllability of system (4) (with L -controls) is equivalent to the existence of a constant C > such that the so-called observability inequality ϕ(, ) L (Ω) C m ϕ 1 (x, t) dx dt. (9) ω (,T )

4 4 M. González-Burgos, L. de Teresa holds for every solution ϕ = (ϕ 1,..., ϕ m ) to (8). Let us observe that in (9) we are estimating the L -norm of ϕ(, ) by means of the L -norm of the first component of ϕ localized in ω (, T ). We will prove inequality (9) as a consequence of a global Carleman inequality for the adjoint system (8). This inequality is established in our first main result: Theorem 1.1. Let us suppose that L k, A L () m and B L () Nm are given by (1) and (6) and satisfy (), (3) and (7). Let M = max i m a i,i 1. Then, there exist a positive function α C (Ω) (only depending on Ω and ω ), two positive constants C (only depending on Ω, ω, m, ã, M, a and M ) and σ = σ (Ω, ω, m, ã, M, M ) and l 3 (only depending on m) such that, for every ϕ L () m, the solution ϕ to (8) satisfies m I(3(m + 1 i), ϕ i ) C s l ω (,T ) [ s s = σ T + T + T 3(j i)+3 max i j ( a ij + B ij e sα γ(t) l ϕ 1, (1) 3(j i)+1 )]. In inequality (1), α(x, t), γ(t) and I(d, z) are given by: α(x, t) α (x)/t(t t), γ(t) (t(t t)) 1 and I(d, z) s d e sα γ(t) d z + s d e sα γ(t) d z. We will prove Theorem 1.1 from the corresponding global Carleman inequality satisfied by the solutions to the heat equation with a right hand side in the space L (, T ; H 1 (Ω)) (see [15]). To prove Theorem 1.1 we will follow the same argument given in [19] and [1] and which allows to show (1) when m =. As a consequence of Theorem 1.1 we will obtain the null controllability at time T of system (5). This is our second main result: Theorem 1.. Under assumptions of Theorem 1.1, given y L (Ω) m, there exists a control v L () such that Supp v ω [, T ] and the corresponding solution y to (4) satisfies (5). Moreover, the control v can be chosen such that v L () ec H m y,i L (Ω), (11) with C a positive constant, only depending on Ω, ω, ã, M, a and M, and H 1 + T + 1 ( T + max 3(j i)+3 3(j i)+1 a ij + B ij + T ( a ij + B ij ) ). i j Let us remark that, thanks to the cascade structure of the coupling matrices A and B (see (6) and (7)), we can control the system (4) (m functions) by means of one control force v L () (exerted in the right hand side of the first equation

5 Controllability for parabolic cascade systems 5 of the system). For general complete matrices, this is not possible, in general. Indeed, let us consider m = 3, L 1 = L = L 3 = (which, evidently satisfy () and (3)), B = and A L(R 3 ) a constant matrix given by A = Let λ 1 > be the first eigenvalue of in Ω with Dirichlet boundary conditions and let φ 1 be the associated eigenfunction with φ 1 L (Ω) = 1. If we now consider ϕ (x, t) = (, 1, 1) φ 1 (x) L (Ω) m in (8), it is not difficult to see that the corresponding solution to (8) is given by ϕ(x, t) = (, 1, 1) e (λ1+1)(t T ) φ 1 (x) which, evidently, does not satisfy inequality (1). In fact, the observability inequality (9) is also false and therefore, system (4) is not null controllable. The null controllability problem of system (4) for general coupling matrices A and B is nowadays widely open. Another important point to be underlined is that in our controllability problem the control is exerted on a little part ω of the set Ω (distributed control). The controllability properties of cascade systems like (4) can fail if boundary controls are considered instead of distributed controls. In [7] the boundary controllability of a cascade system of two parabolic equations is studied and it was found that even the boundary approximate controllability of the system is not in general true. To be precise, let us consider the controlled system (N = 1, m = ) t y Ly + Ay = in = (, 1) (, T ), y = Dv on {} (, T ), y = on {1} (, T ) y(, ) = y in (, 1), where v L (, T ) is the control, ( ) yxx Ly =, A = νy xx ( ), D = 1 ( ) 1 and ν > (which evidently satisfy (), (3) and (7)). Then, in [7] and by means of a simple counterexample, it is proved that this system is not approximately controllable if ν \ {1}. The rest of the work is organized as follows. In Section we will prove the Carleman inequality stated in Theorem 1.1. Theorem 1. will be proved in Section 3. Finally, we will devote Section 4 to give some remarks and additional results.. The global Carleman inequality. Proof of Theorem 1.1 We will devote this section to prove Theorem 1.1. To this end, we will suppose that the coupling matrices A L () m and B L () Nm are given by (6)

6 6 M. González-Burgos, L. de Teresa and satisfy (7). The basic tool we will use is a global Carleman inequality satisfied by the solutions to { t z L z = F in, z = on Σ, z(x, T ) = z (1) (x) in Ω, with z L (Ω) and F = F + N F i x i with F i L (), i =, 1,..., N, and L is given by N ( L y(x, t) = i α ij (x, t) j y(x, t) ) i, where the coefficients αij W 1, () (1 i, j N) satisfy αij (x, t) = α ji (x, t) a.e. in and max αij W 1, = M, i,j N αij(x, t)ξ i ξ j â ξ, ξ R N, a.e. in, i, for positive constants â and M. One has: Lemma.1. Let B Ω be a nonempty open subset and d R. Then, there exist a function β C (Ω) (only depending on Ω and B) and two positive constants C and σ (which only depend on Ω, B, â, M and d) such that, for every z L (Ω), the solution z to (1) satisfies s d e sβ γ(t) d z + s d e sβ γ(t) d z C (L B (d, z) N (13) + s d 3 e sβ γ(t) d 3 F + s d 1 e sβ γ(t) d 1 F i ), for all s s = σ (T +T ). In (13), L B (d, z) and the functions β and γ are given by L B (d, z) s d e sα γ(t) d z, β(x, t) = β (x), (x, t), t(t t) B (,T ) and γ(t) = (t(t t)) 1, t (, T ). The proof of this result can be found in [15] although the authors do not specify the way the constant s depends on T. This explicit dependence can be obtained arguing as in [9] (also see [6]). Proof of Theorem 1.1: Given ω ω, we choose ω 1 ω. Let α C (Ω) be the function provided by Lemma.1 and associated to Ω and B ω 1, and let α(x, t) the function given by α(x, t) = α (x)/t(t t). We will do the proof in two steps:

7 Controllability for parabolic cascade systems 7 Step 1. Let ϕ = (ϕ 1,..., ϕ m ) be the solution to (8) associated to ϕ L (Ω) m. We begin applying inequality (13) with B = ω 1 to each function ϕ i (1 i m) with L L i, d = 3(m + 1 i) and F i [ (B ji ϕ j ) a ji ϕ j ] a i+1,i ϕ i+1, if 1 i m 1, and F m [ (B jmϕ j ) a jm ϕ j ], for i = m, obtaining ( I(3(m + 1 i), ϕ i ) Ĉ + + i s 3(m i) a ji i s 3(m i)+ B ji L ω1 (3(m + 1 i), ϕ i ) + M I(3(m i), ϕ i+1 ) e sα γ(t) 3(m i) ϕ j e sα γ(t) 3(m i)+ ϕ j ), for every s ŝ = σ (T +T ), with Ĉ and σ two positive constant only depending on Ω, ω 1, ã, M (and m) (in the previous inequality we have taken ϕ i+1 when i = m). Now, it is not difficult to see that, for a new constant C (depending on Ω, ω 1, ã, M and M ), one has: ( m m I(3(m + 1 i), ϕ i ) C L ω1 (3(m + 1 i), ϕ i ) + m + m i s 3(m i) a ji i s 3(m i)+ B ji e sα γ(t) 3(m i) ϕ j e sα γ(t) 3(m i)+ ϕ j, s ŝ. Finally, we can get rid of the two last sums in the previous inequality if we take into account that t(t t) T /4 in (, T ) and we take ( )) s s = σ (T + T + T 3(j i)+3 3(j i)+1 max a ij + B ij, (14) i j with σ = σ (Ω, ω, ã, M, M ) >, obtaining, for a positive constants C 1 = C 1 (Ω, ω, ã, M, M ), ( m m ) I(3(m + 1 i), ϕ i ) C 1 L ω1 (3(m + 1 i), ϕ i ), s s. (15) Step. We will see that, thanks to assumptions (6) and (7), we can eliminate in (15) the local terms L ω1 (3(m + 1 i), ϕ i ) for i m. In order to carry this process out, we will need the following result:

8 8 M. González-Burgos, L. de Teresa Lemma.. Under assumptions of Theorem 1.1 and given l N, ε >, k {,..., m} and two open sets O and O 1 such that ω 1 O 1 O ω, there exist a positive constant C k (only depending on Ω, O, O 1, ã, M, a and M ) and l kj N, 1 j k 1 (only depending on l, m, k and j), such that, if ϕ is the solution to (8) associated to ϕ L () m and s s, one has L O1 (l, ϕ k ) ε [I(3(m + 1 k), ϕ k ) + I(3(m k), ϕ k+1 )] ( + C k ) L O (l kj, ϕ j ). ε (In this inequality we have taken ϕ k+1 when k = m). We will finish the proof of Theorem 1.1 showing that it is an easy consequence of Lemma.. To this end, we consider open sets Õi ω, with i m, such that ω 1 Õm Õm 1 Õ ω and we begin by applying formula (16) for O 1 = ω 1, O = Õm, k = m, l = 3 and ε = 1/C 1 (with C 1 the constant appearing in (15)). Thus, from (15), we obtain m ( m 1 I(3(m + 1 i), ϕ i ) C m L eom (l mi, ϕ i ) ) (16), (17) for all s s, with C m a new positive constant only depending on Ω, ω 1, Õm, ã, M, a and M. Observe that in (17) we have eliminated in the right hand side the term that depends on ϕ m. We can go on applying (16) for O 1 = Õm, O = Õm 1, k = m 1, l = l m,m 1 and ε = 1/ C m and eliminate in (17) the local term L eom (l m,m 1, ϕ m 1 ). By (a finite) iteration of this argument we obtain (1). Proof of Lemma.: For the proof of this result, we will reason out as in [19] and [1]. All along the proof, C will be a generic constant that may depend on Ω, O, O 1, ã, M, a and M, and also we will assume that s satisfies s s, with s given by (14). In particular, s C(T + T ) and then, for µ, ν Z, ν µ, and for every (x, t), one has { [sγ(t)] ν C [sγ(t)] µ, [s ν e sα γ(t) ν ] Cs ν+1 e sα γ(t) ν+1, t [s ν e sα γ(t) ν ] + N ij [sν e sα γ(t) ν ] (18) Cs ν+ e sα γ(t) ν+. i,j 1 Given ω 1 O 1 O ω, we consider ξ C (R N ) such that ξ 1 in R N, ξ 1 in O 1, Supp ξ O and ξ L (Ω) ξ 1/ and ξ L (Ω) N. (19) ξ 1/ Let us consider k {,..., m}. The coefficient a k, satisfies assumption (7) and, for simplicity, let us assume a k, a in ω (, T ). We fix l N and

9 Controllability for parabolic cascade systems 9 take u = s l e sα γ(t) l. We multiply the equation satisfied by ϕ by uξ ϕ k and integrate in. We get, a L O1 (l, ϕ k ) L(l, ϕ k ) = uξ ϕ k a j, ϕ j + where L(l, ϕ k ) = s l T T t ϕ + L ϕ, uξ ϕ k dt (B j, ϕ j ), uξ ϕ k dt = 4 K n, n=1 () e sα γ(t) l ξ a k, ϕ k and by means of, we are denoting the duality product between H 1 (Ω) and H 1 (Ω). Integrating by parts with respect to t and having in mind the equation satisfied by ϕ k (see (8)), we get K 1 = T = t ϕ, uξ ϕ k = u t ξ ϕ k ϕ + T uξ ϕ a k+1,k ϕ k+1 = L k ϕ k + 5 n=1 u t ξ ϕ k ϕ T t ϕ k, uξ ϕ k ( (B jk ϕ j ) a jk ϕ j ), uξ ϕ K (n) 1. Let us remark that, when k = m, in the previous equality the last term K (5) 1 does not appear. Taking into account (18) and (7), if s C(T + T ), we obtain K (1) 1 = with δ > to be chosen. u t ξ ϕ k ϕ Cs l+ δ L(l, ϕ k ) + C δ sl+4 δ L(l, ϕ k ) + C δ sl+4 e sα γ(t) l+ ξ ϕ k ϕ e sα γ(t) l+4 a 1 k, ξ ϕ e sα γ(t) l+4 ξ ϕ, Observe that, integrating by parts in K () 1, we get ( N K () 1 = ϕ ξ αij k i u j ϕ k + i,j 1 + uξ αij k i ϕ j ϕ k ). ϕ u α k ij i ξ j ϕ k

10 1 M. González-Burgos, L. de Teresa From (18) and (19), we also have: K () 1 Csl+1 e sα γ(t) l+1 ξ ϕ ϕ k + Cs l e sα γ(t) l ξ 1/ ϕ ϕ k + uξ ϕ ϕ k ε 1 s3(m k)+1 e sα γ(t) 3(m k)+1 ϕ k + C (s n+ e sα γ(t) n+ ξ ϕ ε + s n e sα γ(t) n ξ ϕ ), (1) with n = l 1 3(m k). For K (3) 1 we get: K (3) 1 = T (B jk ϕ j ), uξ ϕ dt + T (B kk ϕ k ), uξ ϕ dt = [ (uξ ) B jk ϕ ϕ j + uξ B jk ϕ ϕ j ] (uξ ) B kk ϕ ϕ k uξ B kk ϕ ϕ k = M 1 + M + M 3. Using (7), (18), (19) and taking into account that, if s C B kk T, then, B kk Csγ(t), it is not difficult to see that, for δ >, one has M + M 3 δ L(l, ϕ k ) + C δ + s l+1 (s l+3 e sα γ(t) l+1 ξ ϕ ). e sα γ(t) l+3 ϕ Now, using again (19) and (18), we get (n = l 1 3(m k)) M 1 C (s n e sα γ(t) n ξ ϕ ) + s n+ e sα γ(t) n+ ϕ + 1 s 3(m k)+1 e sα γ(t) 3(m k)+1 ξ B jk ϕ j.

11 Controllability for parabolic cascade systems 11 Since s CT 3(k j)+1 B jk, we also have B jk C ( s/t ) 3(k j)+1 C (sγ(t)) 3(k j)+1. This implies that, Summarizing, M 1 C (s n+ e sα γ(t) n+ ϕ ) + s n e sα γ(t) n ξ ϕ + C s 3(m j)+ K (3) 1 δ L(l, ϕ k ) + C δ + s l+1 (s l+3 e sα γ(t) 3(m j)+ ξ ϕ j. e sα γ(t) l+3 ϕ ) e sα γ(t) l+1 ξ ϕ + C (s n+ e sα γ(t) n+ ϕ ) + s n e sα γ(t) n ξ ϕ + C s 3(m j)+ e sα γ(t) 3(m j)+ ξ ϕ j. Now, reasoning as we did with K (3) 1, it is not difficult to deduce 1 = K (4) Cs n+ + 1 s 3(m k) 1 + δ L(l, ϕ k ) + C δ sl uξ ϕ a jk ϕ j uξ ϕ a kk ϕ k e sα γ(t) n+ ξ ϕ e sα γ(t) 3(m k) 1 a jk ξ ϕ j e sα γ(t) l a kk ξ ϕ, with δ >. Again, a jk C(sγ(t)) 3(k j)+3 for every s CT a ji 3(k j)+3.

12 1 M. González-Burgos, L. de Teresa This implies that K (4) 1 Csn+ e sα γ(t) n+ ξ ϕ + C s 3(m j)+ e sα γ(t) 3(m j)+ ξ ϕ j + δ L(l, ϕ k ) + C δ sl+3 e sα γ(t) l+3 ξ ϕ. As said above, if k = m then K (5) 1. If k m 1, we have, K (5) 1 = uξ ϕ a k+1,k ϕ k+1 ε 1 s3(m k) e sα γ(t) 3(m k) ϕ k+1 + 3M ε sn+1 e sα γ(t) n+1 ξ ϕ. Altogether we get K 1 3δ L(l, ϕ k ) + ε 1 [I(3(m + 1 k), ϕ k) + I(3(m k), ϕ k+1 )] + C(δ, ε)s J e sα γ(t) J ϕ + C(ε)s R e sα γ(t) R ξ ϕ k + C s 3(m j)+ e sα γ(t) 3(m j)+ ξ ϕ j with J = J(k) = max{l + 4, n +, 3(m k) + 5}, R = max{n, l + 1} and n = l 1 3(m k). Let us remark that in inequality () the positive constants C(δ, ε) and C(ε) are given by C(δ, ε) = C ( δ + ) ( ) 1 ε and C(ε) = C ε. Going back to the term K and integrating by parts, we get K = N i, ( ϕk α ij i (uξ ) j ϕ + u ξ α ) ij i ϕ k j ϕ On the other hand, if s C(T + T ), (see (18) and (19)), Therefore i (uξ ) u i ξ + Cs l+1 e sα γ(t) l+1 ξ Cs l+1 e sα γ(t) l+1 ξ 1/. K δ L(l, ϕ k ) + C δ sl+ e sα γ(t) l+ ϕ + ε 1 s3(m k)+1 e sα γ(t) 3(m k)+1 ϕ k + C ε sn e sα γ(t) n ξ ϕ. () (3)

13 Controllability for parabolic cascade systems 13 For the term K 3 we obtain, K 3 δ L(l, ϕ k ) + C δ s l e sα γ(t) l a j, ξ ϕ j, with δ >. Again, if s CT a j, K 3 δ L(l, ϕ k ) + C δ 3(k j) s l+3(k j), we can deduce We now have that T K 4 = (B j, ϕ j ), uξ ϕ k dt ( = (uξ ) B j, ϕ k ϕ j + e sα γ(t) l+3(k j) ξ ϕ j. (4) uξ ϕ k B j, ϕ j ) Proceeding as before and having in mind that s CT 3(k j) B j,, we obtain that K 4 δ L(l, ϕ k ) + C s l+3(k j) e sα γ(t) l+3(k j) ϕ j δ + ε 1 s3(m k)+1 e sα γ(t) 3(m k)+1 ϕ k (5) + C s l 3(m k+j+1) e sα γ(t) l 3(m k+j+1) ξ ϕ j, ε with δ > to be chosen. Coming back to (), putting together inequalities (), (3), (4) and (5), and choosing the δ = 1/1, we obtain 1 L(l, ϕ k ) ε 4 [I(3(m + 1 k), ϕ k) + I(3(m k), ϕ k+1 )] + C(ε) (s J e sα γ(t) J ϕ + s R e sα γ(t) R ξ ϕ k + s R jk e sα γ(t) R jk ϕ j, with k {,..., m}, C(ε) = C(1 + 1/ε) and J = max{l + 4, l 3(m k) + 1, 3(m k) + 5}, R = max{l + 1, l 3(m k) 1}, R jk = max{3(m j) +, l + 3(k j), l 3(m k + j + 1)}. (6)

14 14 M. González-Burgos, L. de Teresa Now we are interested in eliminating the term s R e sα γ(t) R ξ ϕ in inequality (6). So as to do that, we define ũ = s R e sα γ(t) R and we will use the equation satisfied by ϕ : L ϕ = a k, ϕ k + t ϕ a j, ϕ j + (B j, ϕ j ) Multiplying this equation by ũξ ϕ and integrating by parts in we get: N N ũξ α ij i ϕ i ϕ = α ij i (ũξ )ϕ j ϕ i, i, ũξ ϕ a k, ϕ k + T t ϕ a j, ϕ j + (B j, ϕ j ), ũξ ϕ dt = 5 H n. Next, we are going to estimate each term H i. Using (18) and (19) it is easy to deduce that, if s C(T + T ), i (ũξ ) Cs R+1 e sα γ(t) R+1 ξ 1/ and H 1 Cs R+1 e sα γ(t) R+1 ξ 1/ ϕ ϕ ã 4 ũξ ϕ + Cs R+ If we take δ >, we also have: H δ L(l, ϕ k ) + C δ s R l Proceeding as before and using (18), we get H 3 = 1 ũ t ξ ϕ Cs R+ n=1 e sα γ(t) R+ ϕ. (7) e sα γ(t) R l ξ ϕ. (8) e sα γ(t) R+ ξ ϕ. (9) 3( j)+3 In order to estimate H 4 we recall that if s CT a j, a j, C (sγ(t)) 3(k j). Thus, it is no difficult to check ũξ a, ϕ C e sα [sγ(t)] R+3/ ξ ϕ, ũξ a j, ϕ j ϕ C e sα [sγ(t)] R+3/ ξ ϕ + C e sα [sγ(t)] R 3/+3(k j) ξ ϕ j. then

15 Controllability for parabolic cascade systems 15 Therefore H 4 Cs R 3/+3(k j) e sα γ(t) R 3/+3(k j) ξ ϕ j. (3) Integrating by parts in H 5 and taking into account assumption (18) and (19), and that B j, C ( ) s 3(k j), T we can reason as before and obtain H 5 C s R 3/+3(k j) e sα γ(t) R 3/+3(k j) ϕ j + s R+3(k j) e sα γ(t) R+3(k j) ξ ϕ j + ã ũξ ϕ. 4 (31) Summarizing, taking into account (3), adding (7) (31) and using again (18), we deduce: ã +C ũξ ϕ Cs R+ + δ L(l, ϕ k ) + C δ s R l s R 3/+3(k j) e sα γ(t) R+ ϕ e sα γ(t) R l ξ ϕ e sα γ(t) R 3/+3(k j) ϕ j. Finally, if we now choose δ = ã /4C(ε), with C(ε) the constant appearing in (6), i.e., δ = eaε 4C(ε+1), and we combine (3) with (6), we obtain the proof of the result. (3) 3. Null controllability of system (4). Proof of Theorem 1. We will devote this section to show Theorem 1.. As said above, it is well known that Theorem 1. is equivalent to the observability inequality (9) satisfied by the solutions ϕ of the adjoint system (8) and, in fact, the constant e C H appearing in (11) coincides with C, the constant appearing in (9), (see for instance [9]). We establish this observability inequality in the next result: Proposition 3.1. Under the assumptions of Theorem 1.1, there exists a constant C > (which only depends on Ω, ω, ã, M, a and M ) such that, for every ϕ L (Ω) m, the corresponding solution ϕ to (8) satisfies ϕ(, ) L (Ω) ec H ϕ 1 (x, t) dx dt, (33) where H is given in Theorem 1.. ω (,T )

16 16 M. González-Burgos, L. de Teresa Proof: The proof of this result is standard (see e.g., [9], [6] and [1]) and it is a consequence of (1) and the energy inequality satisfied by the solutions to (8): ϕ(, t 1 ) L (Ω) m ec[1+max i j( a ij + B ij )](t t1) ϕ(, t ) L (Ω) m, where t 1 t T and C > is a new constant which depends on m, ã and M = max i m a i,i 1. If ϕ is a solution to (8), from the energy inequality, we deduce m ϕ i (, ) L (Ω) T ec[1+max i j( a ij + B ij )]T m 3T/4 On the other hand, taking into account (1) and (18), one has s 3 m 3T/4 T/4 Ω e sα γ(t) 3 ϕ i Cs l ω (,T ) T/4 Ω e sα γ(t) l ϕ 1, ϕ i. (34) for a new constant C = C(Ω, ω, a, M) > and for every s s. Secondly, it is not difficult to see ( s 3 e sα γ(t) s3 T 6 5 ) M s exp 3T 1 ( ) 3 ( l 5 ) M s 3 3 exp m 3T, (x, t) Ω (T/4, 3T /4), ( s l e sα γ(t) l s l l T l 3 ) ( ) l m s l exp T, (x, t), em if we choose s (l/8m )T, with m = min Ω α (x) > and M = max Ω α (x). Now, combining these three last inequalities, we readily deduce m 3T/4 T/4 Ω ϕ i Ce Cs/T ω (,T ) ϕ 1 [ for every s s 1 = σ 1 T + T + T 3(j i)+3 max i j ( a ij + B ij 3(j i)+1 )] and σ 1 = max{σ, (l/8m )}. Finally, by setting s = s 1 in the previous estimate and by recalling (34), we end the proof. 4. Further results and comments We will finalize this work doing some remarks and establishing some additional results.

17 Controllability for parabolic cascade systems It is possible to prove the null controllability of system (4) at time T if we replace (7) with the hypothesis a i,i 1 a > or a i,i 1 a > in ω (T, T 1 ), i : i m (35) with T < T 1 T. Indeed, let ŷ C ([, T ]; L (Ω) m ) be the solution to (4) for v. Hypothesis (35) allows us to prove the existence of a control v L (Ω (T, T 1 )) which drives system (4) (posed in the cylinder Ω (T, T 1 )) from the initial data ŷ(, T ) (at time T ) to the rest at time T 1. Now, if we take v in the set (, T ) (T 1, T ) and v v on the interval (T, T 1 ), we have that the solution y to (4) corresponding to the control v L () satisfies (5). Moreover, the control v can be chosen such that (11) holds with H given by ( ) H 1+T (j i)+3 3(j i)+1 +max a ij + B ij +T 1 ( a ij + B ij T 1 T ). i j On the other hand, let us assume that A and B are complete matrices which satisfy (35) and { aij in ω (T, T 1 ), i, j : i j +, B ij in ω (T, T 1 ), i, j : i j + 1, for a nonempty open subset ω ω and T, T 1 [, T ] with T < T 1. Then, it is also not difficult to check that Theorem 1. is still valid and Theorem 1.1 holds with m Ĩ(3(m + 1 i), ϕ i ) C s l e seα γ(t) l ϕ 1, ω (T,T 1) instead of (1). In the previous inequality α(x, t), γ(t) and Ĩ(d, z) are given by: α(x, t) = α (x)/(t T )(T 1 t), γ(t) = ((t T )(T 1 t)) 1 and Ĩ(d, z) s d e seα γ(t) d z + s d e seα γ(t) d z. Ω (T,T 1) Ω (T,T 1). As a direct consequence of the Carleman inequality (1) we obtain the unique continuation property: Under assumptions of Theorem 1.1, if ϕ C ([, T ]; L (Ω) m ) is solution to ( 8) and satisfies ϕ 1 (x, t) = in ω (, T ), then, ϕ in. It is well known that this unique continuation property for the adjoint problem (8) implies the approximate controllability property at time T of system (4). When m = and L 1 = L, i.e., for two equations, it is proved in [16], that the unique continuation property for the adjoint problem (8) is valid even when

18 18 M. González-Burgos, L. de Teresa a 1 = in ω but a 1 in ω an open subset of Ω. The problem remains open for L 1 L. 3. Theorems 1.1 and 1. can readily be generalized to the case where v = (v 1, v,..., v r ) (r control forces, r 1) and the coupling and control matrices A, B and D satisfies: B L () Nm is as in (6), A L () m is given by A 11 A 1 A 1r A A r A =......, A ii = A rr a i 11 a i 1 a i a i 1,s i a i 1 a a i 3... a i,s i a i 3 a i a i 3,s i a i s i,s i 1 a i s i,s i with s i N, r s i = m, a i j,j 1 satisfying (7) for every (i, j) (1 i r, j s i ), and D L(R r, R m ) such that D = ( e S1 e S e Sr ) with S i = 1+ i 1 s j, 1 i r (e j is the j-th element of the canonical basis of R m ). Observe that matrix A do not satisfy (7) for i = S,..., S r and even so, under the previous assumptions, a null controllability result for system (4) can be proved if we add a control in each equation where (7) does not hold. Indeed, let ϕ = (ϕ 1, ϕ,..., ϕ m ) be a solution to the adjoint problem (8). Thanks to the structure of the coupling matrices A and B, we can apply Lemma. to ϕ k, for every k S i with 1 i r, and obtain, from (15), m r I(3(m + 1 i), ϕ i ) C s li e sα γ(t) li ϕ Si, s s, ω (,T ) with s as in Theorem 1.1, C = C (Ω, ω, ã, M, a, M ) > and l i 3 (1 i r). Theorem 1. is a consequence of this last inequality. 4. Following the ideas of [4] (see Theorem 1.3.3, p. 156), from the Carleman inequality (1) we can prove the null controllability of system (4) with controls in L (). To be precise, it is possible to show Theorem 4.1. Under hypotheses of Theorem 1.1, there exists a control v L () satisfying v e C H m y,i L (Ω), such that Supp v ω [, T ] and the corresponding solution y to ( 4) satisfies ( 5). In this inequality, C is a positive constant only depending on Ω, ω, ã, M, a and M, and H is as in Theorem 1.. Sketch of the proof: From (1) and using the regularizing effect of problem (8), it is possible to prove the refined Carleman inequality for the solutions ϕ = (ϕ 1,..., ϕ m ) to (8) (valid for s s, with s given in Theorem 1.1): m (sγ(t)) K e sα ϕ + I(3(m + 1 i), ϕ i ) C 1 s l e sα γ(t) l ϕ 1, ω (,T )

19 Controllability for parabolic cascade systems 19 where C 1 = C 1 (Ω, ω, ã, M, a, M ) > and K = K(N, m) N are two new constants. A duality argument allows us to obtain the proof from this refined Carleman inequality. Theorem 4.1 is crucial in order to study the exact controllability to the trajectories of cascade nonlinear parabolic systems when the nonlinearities considered depend on the state and its gradient and have a superlinear growth at infinity (for the proof, see [13]). 5. When the coefficients a ij and B ij of system (4) are regular enough (for instance, if they are constants or only depend on t), it is possible to show Theorem 1. using the strategy of fictitious controls developed in [11] and [1]. Briefly, this technique consists in introducing a control function in each equation of our system (and, therefore, the null controllability of the system is a consequence of the Carleman inequality (15)) and, subsequently, eliminate the m 1 fictitious controls using the cascade structure of the system (see [1]). This strategy cannot be applied in the case of system (4) due to a lack of regularity of the coefficients. 6. Boundary controls: In view of known controllability results for a linear heat equation, it would be natural to wonder whether the null controllability result for system (4) remain valid when one considers one control force exerted on the boundary: y = e 1 v1 γ on Σ, where γ Ω is a relative open subset of Ω. Nevertheless, there exist negative results for some 1-d cascade linear coupled parabolic systems with m = (cf. [7]), which reveals the different nature of the controllability properties for a single heat equation and for coupled parabolic systems. 7. In the present work we have provided a sufficient condition on the matrices A, B and D which ensures the null controllability of system (4) at time T. Let us observe that when B and A is a constant matrix, under assumptions (6) and (7), the exact controllability of the ordinary differential system { y + Ay = Dv in [, T ], y() = y R N. holds with D e 1 since one has the so-called Kalman rank condition rank [ D AD A D... A m 1 D ] = m. Thus, it would be very interesting to try to generalize this condition to the case of coupled parabolic system like (4) and give a condition on the matrices A, B and D which is equivalent to the null controllability at time T of system (4). At the moment, the general problem is open but some results have been recently obtained in [3]. References [1] F. Ammar Khodja, A. Benabdallah, C. Dupaix, I. Kostin, Controllability to the trajectories of phase-field models by one control force. SIAM J. Control Optim., 4 (3), no. 5, pp

20 M. González-Burgos, L. de Teresa [] F. Ammar Khodja, A. Benabdallah, C. Dupaix, Null-controllability of some reactiondiffusion systems with one control force. J. Math. Anal. Appl. 3 (6), no., pp [3] F. Ammar Khodja, A. Benabdallah, C. Dupaix, M. González-Burgos, Controllability for a class of reaction-diffusion systems: generalized Kalman s condition. C. R. Math. Acad. Sci. Paris 345 (7), no. 1, [4] V. Barbu, Controllability of parabolic and Navier-Stokes equations. Sci. Math. Jpn. 56 (), no. 1, [5] O. Bodart, M. González-Burgos, R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity. Comm. PDE. 9 (4), no. 7 8, [6] A. Doubova, E. Fernández-Cara, M. González-Burgos, E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41 (), no. 3, [7] E. Fernández-Cara, M. González-Burgos, L. de Teresa Boundary controllability of parabolic coupled equations. Submitted. [8] E. Fernández-Cara, S. Guerrero, O.Yu. Imanuvilov, J.P. Puel, On the controllability of the N-dimensional Navier-Stokes and Boussinesq systems with N 1 scalar controls. C. R. Math. Acad. Sci. Paris, Ser. I 34 (5), no. 4, [9] E. Fernández-Cara, E. Zuazua, The cost of approximate controllability for heat equations: The linear case. Adv. Differential Equations 5 (), [1] A. Fursikov, O. Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes Ser. 34, Seoul National University, Korea, [11] M. González-Burgos, R. Pérez-García, Controllability of some coupled parabolic systems by one control force. C. R. Math. Acad. Sci. Paris, Ser. I 34 (5), no., [1] M. González-Burgos, R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force. Asymptotic Analysis 46 (6) [13] M. González-Burgos, R. Pérez-García, L. de Teresa, Controllability results for some nonlinear cascade parabolic systems. In preparation. [14] O. Yu. Imanuvilov, Controllability of parabolic equations, Mat. Sb. 186 (1995), [15] O. Yu. Imanuvilov, M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. Res. Inst. Math. Sci. 39 (3), no., [16] O. Kavian, L. de Teresa, Unique continuation principle for systems of parabolic equations. To appear in ESAIM:COCV. [17] G. Lebeau, L. Robbiano, Contrôle exact de l équation de la chaleur. Comm. PDE. (1995), no. 1, [18] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. (1978), no. 4, [19] L. de Teresa, Insensitizing controls for a semilinear heat equation. Comm. PDE. 5 (), no. 1, 39 7.

21 Controllability for parabolic cascade systems 1 Use \received{1 January, } before \begin{document} M. González-Burgos, Dpto. E.D.A.N., Universidad de Sevilla, Aptdo , Sevilla, Spain manoloburgos@us.es L. de Teresa, Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.U., 451 D.F., México deteresa@matem.unam.mx