CONTROL AND STABILIZATION OF A FAMILY OF BOUSSINESQ SYSTEMS

Transcription

1 DISCRETE AND CONTINUOUS doi:1.3934/dcds DYNAMICAL SYSTEMS Volume 4, Number, June 9 pp CONTROL AND STABILIZATION OF A FAMILY OF BOUSSINESQ SYSTEMS Sorin Micu Facultatea de Matematica si Informatica Universitatea din Craiova, 585, Romania Jaime H. Ortega Depto. Ingeniería Matemática and Centro de Modelamiento Matemático UMI 87 CNRS-UdeChile Universidad de Chile Casilla 17-3, Correo 3, Santiago, Chile Blanco Encalada 1, Casilla 17-3, Santiago, Chile Lionel Rosier Institut Élie Cartan UMR 75 UHP/CNRS/INRIA, B.P. 39 F-5456 Vandœuvre-lès-Nancy Cedex, France Bing-Yu Zhang Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 451-5, USA Communicated by Jerry Bona) Abstract. This paper studies the internal controllability and stabilizability of a family of Boussinesq systems recently proposed by J. L. Bona, M. Chen and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. The space of the controllable data for the associated linear system is determined for all values of the four parameters. As an application of this newly established exact controllability, some simple feedback controls are constructed such that the resulting closed-loop systems are exponentially stable. When the parameters are all different from zero, the local exact controllability and stabilizability of the nonlinear system are also established. 1. Introduction. Considered in this paper is a family of Boussinesq systems η t + w x + ηw) x + aw xxx bη txx =, 1.1) w t + η x + ww x + cη xxx dw txx =, Mathematics Subject Classification. Primary: 93B5, 93D15; Secondary: 35Q53. Key words and phrases. Boussinesq system, smoothing effect, exact controllability, stabilizability. 73

2 74 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG which were recently proposed by Bona, Chen and Saut [5, 6] to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. The systems may also arise, for example, when modeling the propagation of longcrested waves on large lakes or on the oceans. Contrary to the classical Korteweg-de Vries equation which assumes that the waves travel only in one direction, system 1.1) is free of the presumption of unidirectionality and may have a wider range of applicability. In 1.1), η is the elevation from the equilibrium position and w = w θ is the horizontal velocity in the flow at height θh, where h is the undisturbed depth of the liquid. The parameters a, b, c and d in 1.1) are not completely independent. They are, in fact, required to fulfill the relations a + b = 1 θ 1 3 ), c + d = 1 1 θ ), 1.) where θ [, 1] specifies which horizontal velocity the variable w represents cf. [5]). Consequently, a + b + c + d = 1 3. A detailed analysis of this family of Boussinesq systems when posed on the whole real axis R has been conducted by Bona, Chen and Saut [5, 6], in which they have established, in particular, various well-posedness results for the initial value problem IVP) for system 1.1). In this paper, we will focus on the system 1.1) posed on a finite interval, L) with periodic boundary conditions imposed on η and w. As in [5], we will restrict our attention to system 1.1) with its parameters a, b, c, d satisfying one of the assumptions below. C1. b, d, a, c ; C. b, d, a = c >. As it has been proved in [5], the IVP of the linear systems associated to 1.1) when posed on R is well-posed if either C1 or C is satisfied. Since the length of the interval is irrelevant, we will assume L = π through all our study. In this new setting, the well-posedness of 1.1) and their associated homogeneous and nonhomogeneous linear systems will be studied. However, the main task of this paper is to study control and stabilization problems for system 1.1) by means of some localized control actions. More precisely, we will consider the following nonhomogeneous systems η t + w x + ηw) x + aw xxx bη txx = f for x, π), t, T), 1.3) w t + η x + ww x + cη xxx dw txx = g for x, π), t, T) with the periodic boundary conditions r η x t, ) = r η r x t, π) for t, T), r r r, q w x t, ) = q w q x t, π) for t, T), q q q. 1.4) The number of boundary conditions depends on the values of the parameters. For instance, if a = b = then r =, if a = and b then r = 1 and if a then r =. The values of q depend on the parameters c and d in a similar way. The forcing functions f and g, which will be considered as control inputs, are assumed

3 BOUSSINESQ SYSTEMS 75 to be supported in ω, a nonempty open subinterval of, π). We will be mainly interested in the following two problems for system 1.3)-1.4). Problem 1 Exact controllability): Given T >, the initial state η, w ) and the terminal state η 1, w 1 ) in an appropriate space, can one find controls f and g in a suitable space such that 1.3) admits a solution ηx, t), wx, t)) satisfying the boundary conditions 1.4) and ηx, ), wx, )) = η x), w x)), ηx, T), wx, T)) = η 1 x), w 1 x))? Problem Stabilizability): Can one find some linear) feedback controls f = K 1 η, w), g = K η, w) such that the resulting closed-loop system η t + w x + ηw) x + aw xxx bη txx = K 1 η, w), x, π), t, T), w t + η x + ww x + cη xxx dw txx = K η, w), x, π), t, T) is stabilized, i.e., its solution η, w) tends to zero in an appropriate space as t? Control and stabilization of the dispersive wave equations have been studied intensively in the last decade. In particular, for the Korteweg-de Vries KdV) equation, the study of control and stabilization problems began with the works of Russell [36] and Zhang [43]. In [37, 38], Russell and Zhang considered the KdV equation posed on the finite interval, π) with periodic boundary conditions and with localized control action. They showed in [37] that the associated linearized system is exactly controllable and exponentially stabilizable. Aided by then the newly discovered Bourgain smoothing property [8] for the KdV equation, those results were extended to the nonlinear) KdV equation in [38] assuming its solutions have small amplitude. In [31], Rosier studied the boundary controllability of the KdV equation posed on the finite interval, L) with the Dirichlet type boundary conditions: u t + u x + uu x + u xxx =, x, L), t, 1.5) u, t) = h 1 t), ul, t) = h t), u x L, t) = h 3 t), t, where the boundary value functions h j t), j = 1,, 3 are considered as control inputs. Using only a single control input h t) letting h 1 t) = h 3 t) ) and assuming that the length L of the domain, L) does not belong to the set { } k + kl + l K := π ; k, l N, 1.6) 3 Rosier showed that the associated linear system is exactly controllable in the space L, L) using the Hilbert Uniqueness Method HUM), and that the nonlinear system is locally exactly controllable in the space L, L). Interestingly, while Rosier showed in [31] that the linear system associated to 1.5) is not exactly controllable if L K, it has been proved recently by Coron and Crépeau [1] cf. also Crépeau [11]) that the nonlinear system 1.5) is locally exactly controllable in the space L, L) for some values of L in the set K. Using all the three boundary control inputs and a very different approach, Zhang [46] showed that 1.5) is locally controllable in the space H s, L) for any s

4 76 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG without any restriction on L. It should be pointed out that the exact controllability result presented in [46] is different from that in [31]; system 1.5) is shown to be exactly controllable in a neighborhood of any given smooth solution of the KdV equation in [46] while 1.5) is demonstrated in [31] to be exactly controllable in a neighborhood of the zero solution of the KdV equation. To stabilize system 1.5), the simplest feedback control law to use is h 1 t) = h t) = h 3 t), t. 1.7) With the help of an observability inequality established by Rosier in [31], Perla Menzala, Vasconcellos, and Zuazua [8] showed, assuming L / K, that the linear system associated to 1.5) satisfying 1.7) is exponentially stable and that small amplitude solutions of the nonlinear system 1.5) satisfying 1.7) decay exponentially as t. In order to stabilize large amplitude solutions and to remove the length restriction on L, Perla Menzala, Vasconcellos, and Zuazua [8] introduced an extra localized feedback control to 1.5) resulting in the following closed-loop system: u t + u x + uu x + ax)u + u xxx =, x, L), t, 1.8) u, t) =, ul, t) =, u x L, t) =, t, where a = ax) is a nonnegative smooth function supported in Ω, a subdomain of [, L]. Under the assumption Ω, L Ω, 1.9) they showed that all solutions of 1.8) decay exponentially to as t. In addition, they conjectured that the assumption 1.9) could be removed. This is indeed the case; the conjecture has been confirmed recently by Pazoto [9] see also Rosier and Zhang [34]). The stability result of Perla Menzala, Vasconcellos, and Zuazua has been extended by Rosier and Zhang [34] to the generalized KdV equation u t + u x + u p u x + ax)u + u xxx =, x, L), t, 1.1) u, t) =, ul, t) =, u x L, t) =, t with 1 p < 4. Many other dispersive equations have also been studied for their control and stabilization problems the reader is referred to [4, 1, 19,, 3, 5, 6, 3, 33, 38, 41, 44, 45, 47] and the references therein). Among them, the Benjamin-Bona- Mahony BBM) equation w t + w x + ww x w xxt = deserves a special attention. As an alternative of the KdV equation, the BBM equation is also considered as a model in nonlinear studies whenever one wishes to include and balance a weak nonlinearity and weak dispersive effects [3, 7]. However, while its mathematical theory such as the well-posedness, stability analysis and long time behavior is easier to establish than that of the KdV equation, its control theory seems harder to study. In contrast to some significant progress made for the KdV equation, the nonlinear) BBM is still not known to possess any controllability. An exception is the boundary control of the linearized BBM equation w t + w x w txx =, x, 1), t, 1.11) u, t) =, u1, t) = ht) t,

5 BOUSSINESQ SYSTEMS 77 which has been shown by Micu [5] to have poor controllability; the system is not even spectrally controllable though it possesses approximate boundary controllability cf. [43, 5]). It suggests that the BBM equation may not be exactly boundary controllable. Depending on the values of its parameters, system 1.3) couples two equations that may be of KdV or BBM types. It is therefore interesting to see to which extent the controllability properties of each equation type are maintained and/or improved. Our strategy of study of 1.3) is to consider first the associated linear systems and prove the corresponding controllability and stabilizability properties. This linear problem is interesting by itself. As we have pointed out earlier, depending on the parameters a, b, c and d, system 1.3) may couple two KdV-type equations, two BBM-type equations, or one KdV and one BBM-type equations. It is not unusual that in the first case KdV case) some good controllability properties are proved whereas in the second case BBM case) there are no such controllability properties. What happens in the last case is a priori less clear. We will however prove that the system is controllable in that case. Different approaches will be used to establish the exact controllability depending on whether we employ a single control input or two control inputs. If only a single control action is used, the exact controllability will be established via the Hilbert Uniqueness Method HUM) cf. []). If both control actions are used, a stronger exact controllability result will be obtained by using the classical moment method cf. [35]). As an application of the established exact controllability, some simple feedback controls are constructed such that the resulting closed-loop systems are shown to be exponential stable. The contraction mapping principle will be used to extend the results obtained for the associated linear systems to the nonlinear systems. However, only the case when all the parameters of the system are different from is considered here. Remark that 1.1) may be written in the following equivalent form η t a b w x a b ) I b x ) 1 p w x = I b x ) 1 p ηw) x), w t c d η x c ) I d d x) 1 p η x = I d x) 1 p ww x ) 1.1) for x, π), t, T). In 1.1), a very important smoothing effect on the nonlinear term may be noted if b and d are different from zero, which allows us to apply the contraction mapping principle. If α >, the operator I α x ) 1 p from 1.1) is defined in the following way: v αv xx = ϕ in, π), I α x ) 1 p ϕ = v v) = vπ), v x ) = v x π). Since for any ϕ L, π), the elliptic equation from the right has a unique solution I α x) 1 p v H p, π) = {v H, π) v) = vπ), vx ) = v x π)}, is a well-defined, compact operator in L, π). The rest of the paper is organized as follows. Section is devoted to the study of the associated linear systems and is split into three subsections. The well-posedness problem will be discussed in subsection.1. Control and stabilization problems for the linear systems with a single control action will be addressed in subsection.

6 78 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG while the same problems for the linear systems with two control actions will be investigated in subsection.3. The results obtained in section will be extended to the nonlinear systems with nonzero parameters in section 3. The paper is finished with section 4 which provides some concluding remarks and some open problems for further study.. Linear systems. In this section we study the initial-boundary-value problem IBVP) for the linear system associated to 1.3), namely η t + w x + aw xxx bη txx = f for x, π), t, T),.1) w t + η x + cη xxx dw txx = g for x, π), t, T) with the periodic boundary conditions r η x t, ) = r η r x t, π) for t, T), r r r,.) q w x t, ) = q w q x t, π) for t, T), q q q and the initial condition η, x) = η x), w, x) = w x) for x, π)..3) Its well-posedness in a suitable classical Banach space will be investigated in subsection.1. Then, in subsections. and.3, considering f and g as control inputs acting only on a subdomain ω of, π), we will study its control and stabilization problems. In particular, exact controllability and stabilizability will be established in subsection. for system.1) with a single control while subsection.3 will be devoted to study system.1) with two control inputs for its control and stabilization problems..1. Well-posedness. We first introduce a few notations. Given any v L, π) and k Z, we denote by v k the k Fourier coefficient of v, v k = 1 π π vx)e ikx dx and for any m N we define the space { Hp m, π) = v L, π) v = v k e ikx, } v k 1 + k ) m < k Z k Z which is a Hilbert space with respect to the inner product v, h) m = k Z v k ĥ k 1 + k ) m..4) The norm corresponding to.4) is denoted by m. It may be seen that Hp {v m, π) = H m, π) r } v x r ) = r v x r π), r m 1, where H m, π) stands for the classical Sobolev space of exponent m in, π). We also consider the closed subspace { H,p m, π) = v Hp m, π) v = } v k e ikx, v =. k Z

7 BOUSSINESQ SYSTEMS 79 We may extend the definition of Hp m, π) to the case m = s, a real number, by setting { Hp, s π) = v = } v k e ikx H s, π) v k 1 + k ) s <, k Z where the convergence of the series is in the Sobolev space H s, π). For any real number s, Hp, s π) may also be seen as a Hilbert space with respect to the inner product defined by.4) with m replaced by s. In particular, for any v Hp s, π), v s = k Z k Z v k 1 + k ) s )1. The following map is a duality product between Hp, s π) and Hp s, π) for any s f, g s = k Z f k ĝ k, f H s p, π), g H s p, π)..5) It follows that Hp s, π) is the topological dual of Hs p, π). Assume that the initial data in.3) and the forcing terms in.1) are given by η, w ) = η k, ŵ ) k e ikx, f, g)t) = fk t), ĝ k t)) e ikx. k Z k Z At least formally, the solution of.1) may be written as η, w)t, x) = k Z η k t), ŵ k t))e ikx where η k t), ŵ k t)) fulfill η k ) t + ikŵ k iak 3 ŵ k + bk η k ) t = f k, t, T), ŵ k ) t + ik η k ick 3 η k + dk ŵ k ) t = ĝ k, t, T), η k ) = η k, ŵ k) = ŵk. System.6) may be solved explicitly. If we set ω 1 = ω 1 k) = 1 ak 1 + bk, ω = ω k) = 1 ck 1 + dk, Ak) = it is easy to see that.6) is equivalent to η k + ikak) η k = ŵ k t ŵ k Hence, the solution of.6) is given by η k t) = e iktak) η k + ŵ k ŵ k 1 1+bk f k 1 1+dk ĝ k t, η k e ikt s)ak) ŵ k ω 1 ω ) = η k 1 1+bk f k s) 1 1+dk ĝ k s) ŵ k,..6) ds..7)

8 8 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG Note that the eigenvalues of the matrix Ak) are ±σk), with σk) = ω 1 k)ω k), and are always real. By a standard diagonalizing procedure we get that cos[kσk)t] i ω1 ω sin[kσk)t] e iktak) = i ω ω 1 sin[kσk)t] Consequently, the solution of.6) is given by η k t) = cos[kσk)t] η k i ω1 ω sin[kσk)t]ŵk cos[kσk)t]. t { } cos[kσk)t s)] ω1 sin[kσk)t s)] 1 + bk fk s) i ω 1 + dk ĝ k s) ds, ŵ k t) = i ω ω 1 sin[kσk)t] η k + cos[kσk)t)]ŵ k + t { } ω sin[kσk)t s)] cos[kσk)t s)] i ω bk fk s) dk ĝ k s) ds..8) Remark.1. The eigenvalues of system.1) are given by λ k = ikσk), k Z..9) Note that not all the eigenvalues in.9) are different. If we count only the distinct eigenvalues, we obtain the sequence λ k ) k I, where I Z has the property that λ k1 λ k for any k 1, k I. For each k 1 Z set Ik 1 ) = {k Z : kσk) = k 1 σk 1 )} and Ik 1 ) = mk 1 ). We have the following properties of mk 1 ): mk 1 ) 6. This is a consequence of the fact that mk 1 ) is less than the number of entire roots of the equation xσx) = α, where α is an arbitrary real number. The roots of this equation are also roots of a polynomial of degree less or equal to 6. If the sequence of eigenvalues tends to infinity, there exists k1 N such that mk 1 ) = 1 for all k 1 > k1. This is a consequence of the fact that the function xσx) is strictly increasing for x large enough. The number of the eigenfunctions corresponding to an eigenvalue λ k1 is mk 1 ) for any k 1 I. These eigenfunctions read then e ikx, σk) ) e ikx, e ikx, σk) ) e ikx, k Ik 1 ). ω 1 ω 1 On the other hand, the zero eigenvalue has multiplicity two, unless there exists some k Z \ {} such that a = c = 1 in which case it is of multiplicity six with k associated eigenfunctions 1, ),, 1), e ik x, ),, e ik ) x, e ik x, ),, e ik ) x.

9 BOUSSINESQ SYSTEMS 81 For each s R, we define the space V s = { η, w) Hp s, π) Hs p, π) η, w) V := s η s + Hw s < } where s is the Hp, s π) norm and the operator H is defined in the following way ) H v k e ikx = ω1 v k e ikx. ω k Z k Z V s is a Hilbert space with respect to the inner product f 1, f ), g 1, g )) V s = f 1, g 1 ) s + Hf, Hg ) s. The space V s depends on the values of the parameters a, b, c and d and more precisely on the value of ω1 ω. Let us introduce the number l Z with the property that ω1 ω C k l as k.1) where C is a positive constant not depending on k. We have that V s = Hp, s π) Hp s+l, π). Note that, depending on the parameters a, b, c and d, we may have an asymmetry of regularity of the two components of the space V s. Note also that, in any case, V s H s, π) H s, π). The following result uses formula.8) and introduces a C group associated with system.1). Theorem.. The family of linear operators St)) t R defined by St)η, w )ηt), wt)) = k Z η k t), ŵ k t))e ikx,.11) where the Fourier coefficients of ηt), wt)) are obtained from the ones of η, w ) by η k t) = cos[kσk)t] η k i ω1 ω sin[kσk)t]ŵ k, ŵ k t) = i ω ω 1 sin[kσk)t] η k + cos[kσk)t]ŵ k,.1) is a group of isometries in V s, for any s R. Proof. First, let us prove that St) is a well-defined linear and continuous operator for any t R. If η, w ) = k Z η k, ŵ k )eikx V s, then we claim that the series k Z η kt), ŵ k t))e ikx converges in C[, ), V s ). This is equivalent to say that the sequence P = η k t), ŵ k t)) e ikx k N is a Cauchy sequence in C[, ), V s ). N 1

10 8 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG It is clear that P C[, ), V s ) and that sup η k t), ŵ k t)) e ikx t [, ) = sup = N< k N+p t [, ) N< k N+p N< k N+p V s η k t) + ω 1 ω ŵ k t) η k + ω 1 ω ŵ k ) 1 + k ) s. ) 1 + k ) s Thus P is a Cauchy sequence in C[, ), V s ) since η, w ) V s. Hence the operator St) is well-defined in V s and S )η, w ) C[, ), V s ). Moreover, since η k t), ŵ k t))e ikx = η k + ω ) 1 ŵk ω 1 + k ) s, k N V s k N St)) t R is a family of linear and continuous operators which are also isometries. It is easy to see that S) = I, St) Ss) = St + s) for any t, s R and, in addition, St)η, w ) η, w ) V s = k Z = 4 k Z cos[kσk)t] 1) + sin [kσk)t] ) [ η k + ω ] 1 ŵk ω 1 + k ) s [ ] [ kσk)t sin η k + ω ] 1 ŵ ω k 1 + k ) s. Consequently lim t St)η, w ) = η, w ) in V s and the proof is complete. Another important parameter in our analysis will be the number e Z defined in the following way ω1 ω C k e as k.13) where C is a positive constant not depending on k. Theorem.3. The infinitesimal generator of the group St)) t R is the unbounded operator DA), A) in V s where DA) = V s+1+max{ 1, e}) and a b w x 1 + a ) I b x b ) 1 p w x Aη, w) = c d η x 1 + c ), η, w) DA)..14) I d x d ) 1 p η x Proof. We show that St)η, w) η, w) lim = Aη, w),.15) t t if and only if η, w) V s+1+max{ 1, e}). This is equivalent to show that the derivative in zero of the series k Z η kt), ŵ k t))e ikx, where η k t), ŵ k t)) is given by.1), is convergent to Aη, w) in V s if and only if η, w) V s+1+max{ 1, e}).

11 BOUSSINESQ SYSTEMS 83 If we denote by S N t) = k N η k t), ŵ k t))e ikx a partial sum of the series, a straightforward computation which takes into account.1) shows that [S N ] t ) = AS N )). Both terms are convergent in V s when N tends to infinity if η, w) V s+1+max{ 1, e}) and [ ] η k t), ŵ k t))e ikx ) = Aη, w). k Z On the other hand, the derivative of the series k Z η kt), ŵ k t))e ikx is convergent only if η, w) V s+1+max{ 1, e}). Indeed, this may be easily seen if we compute [S N+p ] t ) [S N ] t ) V = ) k σ k) η s k + ω 1 k) ω k) ŵ k 1 + k ) s N< k N+p which is convergent if and only if η, w) V s+1+max{ 1, e}). System.1)-.3) may be written equivalently in the following form ) ) ) ) ) η η f η η t) = A + w w g, ) = w w t t.16) where f = 1 f k Z 1+bk k e ikx and g = 1 k Z 1+dk ĝ k e ikx. The following existence and uniqueness result for the solutions of.1)-.3) is a direct consequence of the general theory for evolution equations associated with a group of isometries see e.g. [9]). Theorem.4. Let T > and s R be given. If η, w ) V s and f, g ) L 1, T; V s ), then.16) admits a unique solution η, w) C 1 [, T], V s 1+max{ 1, e}) ) C[, T], V s ). Moreover, there exists a positive constant C > depending only on s such that η, w) C[,T];V s ) C [ f, g ) L 1,T;V s ) + η, w ) V s]..17) Remark.5. System.1) has the following important regularity property: if b and d then f, g) L 1, T; V s ) implies f, g ) L 1, T; V s+ ). Hence, if η, w ) V s+ and f, g) L 1, T; V s ), then η, w) C[, T]; V s+ ). Remark.6. If we define V s, = {η, w) V s : η = ŵ = }, V s, = {η, w) V s : η = }, V, s = {η, w) V s : ŵ = }, we obtain that V, s, V, s and V, s are all closed subspaces of V s. The group St)) t R is well-defined in those spaces. Hence, for instance, if we consider that in Theorem.4 f = ĝ =, then the corresponding solution of.1) is well defined in V, s. Remark that the elements of V, s have the property that the mean of η and w are

12 84 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG zero. This is a quantity which is conserved in system.1) if f and g also have zero mean. To end this subsection we consider the following backward IBVP of the homogeneous adjoint system of.1): ξ t + u x + cu xxx bξ txx = for x, π), t, T), u t + ξ x + aξ xxx du txx = for x, π), t, T), r ξ x t, ) = r ξ r x t, π) for t, T), r r r,.18) q u x t, ) = q u q x t, π) for t, T), q q q, ξt, x) = ξ T x), ut, x) = u T x) for x, π). Letting t = T t, x = π x, one can easily see that.18) is.1)-.3) with f and g being zero and a and c exchanged. Let ω 1 = ω 1 k) = 1 ck 1 + bk, ω = ω k) = 1 ak 1 + dk,.19) As before, for each s R, we define the space { Ṽ s = ξ, u) Hp, s π) Hp s, π) ξ, u) V s := ξ s + Hu s < where s is the Hp, s π) norm and the operator H is defined in the following way ) H v k e ikx = ω 1 v k e ikx. ω k Z k Z Introduce the number l Z with the property that ω 1 ω C k l as k where C is a positive constant not depending on k. We have that Ṽ s = H s p, π) Hs+ l p, π). Theorem.7. Let T > and s R be given. If ξ T, u T ) Ṽ s, then.18) admits a unique solution ξ, u) in C 1 [, T], Ṽ s 1+max{ 1, e}) ) C[, T], Ṽ s ). Moreover, there exists a positive constant C > depending only on s such that ξ, u) C[,T]; Ṽ s ) C ξ, u ) Ṽ s..) } If ξ T, u T ) = k Z ξt k, û T k ) e ikx, then the solution of.18) may be written as ξ, u)t, x) = ξ k t), û k t))e ikx k Z

13 BOUSSINESQ SYSTEMS 85 where ) ξ k = 1 ω ξt k + 1 ω û T ikσk)t t) k e û k = 1 ) + 1 ω ξt k 1 ω û T ikσk)t t) k e ) ω ω ξt 1 k + û T ikσk)t t) k e + 1 ) ω ω ξt 1 k + û T k e ikσk)t t)..1) This formula will be very useful for the controllability problem, considered in the next section... Linear systems with a single control input. In this subsection we study the control and stabilization of the following system with a single control input: η t + w x + aw xxx bη txx = Qh for x, π), t, T), w t + η x + cη xxx dw txx = for x, π), t, T), r η x t, ) = r η r x t, π) for t, T), r r r,.) q w x q t, ) = q w x q t, π) for t, T), q q, η, x) = η x), w, x) = w x) for x, π), where the operator Q is defined by [Qh]x, t) = qx)hx, t), and q L, π) is a given non-negative function supported in ω and such that qx) > C on a nonempty) open set ω ω, C > being some constant. First, we look for a time T > and a space V to be made precise latter on) with the property that for η, w ) V there exists h L, T), π)) such that the corresponding solution η, w) of.) satisfies ηt, ) = wt, ) =..3) Property.3) represents a null-controllability. Since system.) is conservative and time reversible, this property is equivalent to the exact controllability. Hence, any initial state in V may be led to any final state from V in time T. The controllability properties of.) will depend on the values of the coefficients a, b, c and d. As we shall see latter on, a whole bunch of situations may occur, from non-spectral controllability to exact controllability in any time. This controllability problem will be solved in this subsection by using the Hilbert Uniqueness Method HUM) introduced by J.-L. Lions see []). The following duality product will play an important role. ϕ, ζ), ψ, z) D = k Z ϕ k ψk 1 + bk ) + ζ ) k ẑ k 1 + dk ).4)

14 86 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG for any ϕ, ζ) V, ψ, z) Ṽ,. The space V is defined in the following way V, if b =, V =.5) V, if b. Notice that ϕ, ζ) V if and only if the sum in.4) is well defined and its absolute value is less than C ψ, z) Ṽ C being some constant) for any ψ, z) Ṽ,. Moreover, the duality product.4) is found to be π ϕx)ψx)dx + ζ, I d x)z l if b =, ϕ, ζ), ψ, z) D = π I b x)ϕx)ψx)dx + ζ, z +l if b, d =, π I b x)ϕx)ψx)dx + I d x)ζ, z l if b, d, where, l is the duality product defined by.5). The following proposition presents an equivalent condition for the controllability of.). Proposition.8. Given η, w ) V, there exists h L, π), T)) such that the corresponding solution η, w) of.) satisfies.3) if and only if there exists h L, π), T)) such that η, w ), ξ), u)) D + T π for any ξ T, u T ) Ṽ,, where ξ, u) is the solution of.18). [Qh]x, t)ξx, t)dxdt =.6) Proof. Note that it is sufficient to prove.6) for regular data. Multiplying the first and the second equation in.) by ξ and u respectively, integrating in time and space over the domain, T), π) and adding the relations we obtain that T π T π + = π [Qh]x, t)ξx, t)dxdt = T π [w t + η x + cη xxx dw txx ] udxdt [ η bηxx )ξ + w dw xx )u ] dx = ηt), wt)), ξ T, u T ) D η, w ), ξ), u)) D T [η t + w x + aw xxx bη txx ] ξdxdt from which.6) follows immediately. A standard argument shows that the variational equality.6) has a solution if and only if there exists a constant C > such that the following observation inequality holds true for any ξ T, u T ) Ṽ, T π ξ), u)) C [Qξ]x, t)ξx, t)dxdt, Ṽ

15 BOUSSINESQ SYSTEMS 87 where ξ, u) is the solution of.18) with final data ξ T, u T ). Thus, because of the definition of the operator Q, it suffices to show T ξ), u)) C ξx, t) dxdt..7) Ṽ ω We show that.7) holds true for certain values of the parameters a, b, c and d by using the Fourier expansion of the solutions of.18). Theorem.9 Observability). Suppose that neither of the following two situations occur S1) b, d and ac =, S) a = c = and b + d. Then there exist a time T > and a constant C > such that, for any ξ T, u T ) the corresponding solution ξ, u) of.18) satisfies the inequality.7). Ṽ, Proof. Let ξ T, u T ) Ṽ, be of the form ξ T, u T ) = k Z ξ T k, û T k )e ikx. The corresponding solution of.18) is given by ξ, u) = k Z ξ k, û k )e ikx with ) ξ k = 1 ω ξt k + 1 ω û T ikσk)t t) k e û k = 1 ) + 1 ω ξt k 1 ω û T k e ikσk)t t), ) ω ω ξt 1 k + û T ikσk)t t) k e + 1 ) ω ω ξt 1 k + û T k e ikσk)t t)..8) From the fact that a group of isometries is associated to.18), it follows that ξ, u)) = ξt Ṽ k + ω ) 1 û T k = ξ T, u T )..9) ω Ṽ k Z

16 88 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG On the other hand, T = 1 4 ω ξx, t) dxdt = 1 4 ω + T + ξ Tk ω 1 û T k ω k 1 I k Ik 1) T ω ) [ ξ Tk + ω 1 û T k e ikσk)t ω k Z )e ikσk)t ] e ikx ) [ ξ Tk + ω 1 û T k e ikx ω dxdt ξ T k ω 1 û T k )e ]e ikx ik1σk1)t dxdt. ω Remark that, if a, b, c and d do not fulfill S1) or S), then there exists a number γ, ] such that lim inf k λ k+1 λ k = γ >. Moreover, the elements of the sequence e ) λ kt are all different. By using a k I generalization of Ingham s inequality see [13] and []) we deduce that, for any T > π γ, there exists a constant C > such that T [ ) ξt, x) dxdt C ξ T ω 1 k + û T k e ikx ω ω ω + k 1 I k Ik 1) ] ξ k T ω 1 û T k )e ikx dx. ω Note that, in the hypothesis of the theorem, lim k λ k =. From Remark.1 we obtain that Ik 1 ) = mk 1 ) = 1 for k 1 large enough. For the remaining finite number of valued k 1 I we use the fact that ω is a nonempty open interval and Ik 1 ) 6 to deduce that there exists a constant C >, independent of k 1, such that ) ) ] [ ξ Tk ω + ω 1 û T k e ikx + ξ T k ω ω 1 û T k e ikx dx ω k Ik 1) C k Ik 1) k ξ T + ω 1 û T k ω + k ξ T ω 1 û T k. ω This may also be seen as a consequence of the same generalized Ingham s inequality used before. It follows that there exists a constant C > such that T ξx, t) dxdt C ξt k + ω ) 1 ût ω ω k..3) k Z Hence, from.9) and.3) it follows that.7) holds and the proof ends.

17 BOUSSINESQ SYSTEMS 89 Remark.1. If either S1) or S) holds, the family of eigenvalues λ k ) k Z is bounded. It is well known that in this case Ingham s inequality does not hold and the system is not controllable. In [5] the BBM equation is studied in some details. The corresponding spectrum is equally bounded in that case and the equation is not controllable. Note that the non-controllability of.) corresponds exactly to the cases in which two BBM-type equations are coupled. Remark.11. If S3) b d = and ac, or S4) b = d =, a + c and a = or c =.31) we have that lim inf k λ k+1 λ k ) = and inequality.7) holds for any T >. Hence, the system is controllable in any positive time. In the remaining cases the controllability time should be large enough. The following controllability result follows from Proposition.8 and Theorem.9, Theorem.1 Exact controllability). Suppose that neither of the following two situations occur S1) b, d and ac =, S) a = c = and b + d. Then there exists a time T > such that for given η, w ) V, η T, w T ) V, one can find a control input h L, T; L, π)) such that.) admits a unique solution satisfying η, w) C[, T]; V) η, ), w, )) = η ), w )), η, T), w, T) = η T ), w T )) in V. Moreover, there exists a constant C > such that h L,T;L,π)) C η, w ) V + η T, w T ) V ). Remark.13. The details of the controllability results are provided in Table 1.

18 9 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG No. b d a c Controllable space Control time 1 V, = Hp H,p bd T > π ac No No 3 No No 4 No No 5 V, = L H,p 1 Arbitrarily small 6 No No 7 V, = L L T > π d c 8 V, = L H,p T > π d a 9 V, = Hp H,p 1 Arbitrarily small 1 No No 11 V, = Hp L T > π b c 1 V, = Hp H,p T > π b a 13 V, = L L T > π 14 V, = L L Arbitrarily small 15 V, = L H 1,p Arbitrarily small 16 V, = L H,p 1 Arbitrarily small Table 1. Controllability results for the linear system. Remark.14. One may consider the control system.) but with a control input acting on the second equation: η t + w x + aw xxx bη txx = for x, π), t, T), w t + η x + cη xxx dw txx = Qh for x, π), t, T), r η x t, ) = r η r x t, π) for t, T), r r r,.3) q w x q t, ) = q w x q t, π) for t, T), q q, η, x) = η x), w, x) = w x) for x, π). Note that if we let a c, b d, w η, then the system.3) becomes.). Thus one may produce the table of controllability for the system.3) from Table 1 by interchanging a and c, b and d as well as the order of the product spaces. Next we turn to the stabilization problem for system.). We seek a linear feedback control law such that the resulting closed-loop system is exponentially stable. Note that.) can be written in the form: η k ) t + ikŵ k iak 3 ŵ k + bk η k ) t = q k, t, T) ŵ k ) t + ik η k ick 3 η k + dk ŵ k ) t =, t, T).33) η k ) = η k, ŵ k) = ŵk for < k <, where the q k s denote the Fourier coefficients of Qh. Multiplying both sides of the first equation in.33) by η k and the second equation by ŵ k if a = c or by 1 ak 1 ck if a <, c <, and then adding the resulting

19 BOUSSINESQ SYSTEMS 91 first equation to the conjugate of the resulting second equation, we obtain d dt 1 + bk ) η k + ω ) 1k) ω k) ŵ k = Req k η k ) for < k <. Thus, if we define π I E[η, w]t) = b x) 1/ ηx, t) + I b x) 1/ Hwx, t) ) dx,.34) then d ) dt E[η, w]t) = Re π [Qh]x, t)ηx, t)dx. Thus, if one chooses the feedback control law then the resulting closed loop system η t + w x + aw xxx bη txx = Qη hx, t) = ηx, t), for x, π), t, T), w t + η x + cη xxx dw txx = for x, π), t, T), r η x r t, ) = r η x r t, π) for t, T), r r,.35) q w x q t, ) = q w x q t, π) for t, T), q q, η, x) = η x), w, x) = w x) for x, π) has the property that d E[η, w]t) = dt π [Qη]x, t)ηx, t)dx C ηx, t) dx ω for any t, where C > is a constant independent of η and w. Our main concern is whether the solution η, w) tends to zero as t and if it does, how fast it decays? Let us define the spaced U b,d by Hp 1 H,p 1 if b > and d > ; and the norm Ub,d by Hp 1 L if b > and d = ; U b,d = L H,p 1 if b = and d > ; L L if b = and d = η, w) U b,d = π η + b η x + w + d w x ) dx. Theorem.15. Assume that i) there exist constants c 1 > and c > such that for any u, v) U b,d ; c 1 E 1 u, v) u, v) Ub,d c E 1 u, v)

20 9 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG ii) the following open loop control system the adjoint system of.)) χ t + ζ x + cζ xxx bχ txx = Qh for x, π), t, T), ζ t + χ x + aχ xxx dζ txx = for x, π), t, T), r χ x r t, ) = r χ x r t, π) for t, T), r r,.36) q ζ x t, ) = q ζ q x t, π) for t, T), q q q is exactly controllable in the space U b,d in the time interval, T) for some T > with control input h chosen in the space L, π), T)). Then for any η, w ) U b,d, the closed-loop system.35) admits a unique solution η, w) CR + ; U b,d ). Moreover, there exist two constants δ > and C > such that for any t. η, t), w, t)) Ub,d C η, w ) Ub,d e δt Proof. Let η, w) be the solution.35) with initial value η, w ). By assumption ii), there exists a control h in the space L, π), T)) such that the system.36) admits a solution χ, ζ) C[, T]; U b,d ) satisfying χ, ζ) =, ) at t =, χ, ζ) = η, w) at t = T. Moreover, there exist positive constants C 1 and C such that h L,π),T)) C 1 η, T), w, T)) Ub,d and sup χ, t), ζ, t)) Ub,d C η, T), w, T)) Ub,d. t T Multiplying the first and the second equation in.36) by η and w respectively, integrating in time and space and adding the relations we obtain that = η, T), w, T)) U bd T π ηx, t)[qh]x, t)dxdt T π [Qη]x, t)χx, t)dxdt = T π [Qη]x, t)hx, t) χx, t))dxdt [Qη] L,π),T)) h χ L,π),T)) C 3 T π [Qη]x, t)ηx, t)dxdt ) 1 η, T), w, T)) Ub,d

21 BOUSSINESQ SYSTEMS 93 where C 3 > is a constant. Consequently, T π [Qη]x, t)ηx, t)dxdt C 3 η, T), w, T)) U b,d Since T π E[η, w]t) E[η, w]) = [Qη]x, t)ηx, t)dxdt, there exists a constant C > by assumption i) such that Thus E[η, w]t) E[η, w]) CE[η, w]t). E[η, w]t) 1 E[η, w]). 1 + C The theorem follows then consequently. The proof is complete. Remark.16. If a = c or a < and c <, then Assumption i) of Theorem.15 is satisfied. Assumption ii) of Theorem.15 is also satisfied if, in addition, b =. Remark.17. A similar theorem holds if the feedback control is acting on the second equation instead of the first one. The assumptions of Theorem.15 are quite restrictive. According to Table 1, the theorem may be applied only in the cases of No. 5, 13 and 14. In particular, it cannot be applied for the case where both b and d are not zero. This is mainly caused by the simple feedback control law we have used. If some more complicated linear feedback control laws are used, we will have much stronger stabilizability results. To see this, we rewrite system.) as an abstract control system in the Hilbert space V : d dt η = A η + B h, η) = η.37) with ) ) ) η η η =, η w = w, h1 h =, B h h = I b x) 1 Qh 1 and the operator A is as given in.14), which is an infinitesimal generator of a C group St) in the space V and A = A. Note that B is a bounded linear operator from the space L, π) to the space V. The following theorem is derived from Theorem.1 and a classical principle exact controllability implies exponential stabilizability for conservative control systems [4, 4]. Theorem.18. Assume that the assumptions of Theorem.1 are satisfied. Then i) there exist a T > and a δ > such that for any η V. T B S t) η L,π) dt δ η V.38)

22 94 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG ii) For any given α >, there exists an operator K LV, L, π)) such that if one chooses h = K η in.37), then the resulting closed-loop system d dt η = A η + BK η, η) = η.39) has the property that its solution satisfies ηt) Y M η V e αt.4) for any t where M is a constant independent of η. Remark.19. According to Slemrod [4], one can choose with D T,α η = T K = B D 1 T,α e αt S t)bb S t) η dt. In addition, if one simply chooses K = B, then there exists a ν > such that estimate.4) holds with α replaced by ν..3. Linear systems with two control inputs. In this subsection we study control and stabilization of the following linear system with two control inputs. η t + w x + aw xxx bη txx = f for x, π), t, T), w t + η x + cη xxx dw txx = g, for x, π), t, T), r η x r t, ) = r η x r t, π), for t, T), r r,.41) q w x q t, ) = q w x q t, π), for t, T), q q, η, x) = η x), w, x) = w x), x, π). We assume throughout this subsection that In consideration of practical applications, we require that [f, t)] = π a = c, b = d..4) fx, t)dx =, [g, t)] = π gx, t)dx =. With this restriction on f and g, any smooth solution ηx, t), wx, t)) has the property that d dt π ηx, t)dx =, d dt π wx, t)dx = for any t. Thus the quantities [η, t)] and [w, t)], the mean values of η and w, are conserved.

23 BOUSSINESQ SYSTEMS 95 A more interesting case is obtained if some a priori restrictions are imposed on the applied control fx, t) and gx, t). Let us suppose that ρx) is a nonnegative periodic function of period π supported in ω such that [ρ] = π ρx)dx = 1. For any function h = hx, t), we define the control operator G by π ) [Gh]x, t) = ρx) hx, t) ρy)hy, t)dy..43) Our control inputs f and g in.41) will take the form: fx, t) = [Gh 1 ]x, t), gx, t) = [Gh ]x, t).44) and.41) becomes η t + w x + aw xxx bη txx = Gh 1 for x, π), t, T) w t + η x + aη xxx bw txx = Gh for x, π), t, T) r η x t, ) = r η r x t, π) for t, T), r r r.45) q w x q t, ) = q w x q t, π) for t, T), q q η, x) = η x), w, x) = w x) for x, π) with h 1 x, t) and h x, t) as our new control inputs. Note that π [Gh]x, t)dx = π ρx)hx, t)dx π ρx)dx π ρy)hy, t)dy = because of the restriction on ρ. The mean values of both Gh 1 and Gh are zero. The next two technique lemmas are useful in discussing control properties of system.45). Lemma.. Let s R be given. There exists a constant C depending only on s such that Gh H s p C h H s p.46) for any h Hp s. In addition, π for any ξ L, π). Proof. If s, [Gξ]x)ξx)dx = π [Gh]x) = ρx) hx) π ρx) ξx) ρy)ξy)dy dx.47) π ) ρy)hy)dy. Because ρ = ρx) is assumed to be smooth, it is easy to see that the estimate.46) holds. If s <, for any h Hp s, we define Gh by [Gh]x) = ρx)hx) ρx) ρ, h s, where, s is defined in.5). For any h H s p, ρh Hs p and ρ ρ, h s H s p, it is straightforward to verify that.46) holds.

24 96 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG and To see that.47) is true, note that π ρx) = π 1 π ρy)dy = 1 ρy)ξy)dy ξx) π ) π ρx)dx π ρy)ξy)dy ) ρy)ξy)dy dx =..47) follows then by a direct computation. Lemma.1. Let φ j x) = e ijx for j = ±1, ±, and m j,k = 1 π π G[φ j ]x)φ k x)dx, j, k = ±1, ±,. In addition, for any given finite sequence of nonzero integers k j, j = 1,, 3, n, let m k1,k 1 m k1,k n m k,k 1 m k,k n A n =. m kn,k 1 m kn,k n Then i) there exists a constant µ > such that m k,k µ for any k = ±1, ±, ;.48) ii) A n is an invertible n n hermitian matrix. Proof. Note that m k,k = 1 π π for any k, and G[φ k ]x)φ k x)dx = 1 π lim m k,k = 1 k π π ρx)dx π ρx)dx. π ) ρx)φ k x)dx >. Estimate.48) follows consequently. To see that ii) is true, let Σ n be the space spanned by φ kj, j = 1,,, n. In addition, let p j be the projection of Gφ kj ) onto the space Σ n, i.e., n p j = m kj,k l φ kl l=1 for j = 1,,, n. It suffices to show that p j, j = 1,,, n is a linearly independent set in the space Σ n. Assume that there exist scalars λ j, j = 1,,, n such that λ 1 p 1 x) + λ p x) + + λ n p n x).

25 BOUSSINESQ SYSTEMS 97 Then, by the definition of p j, n λ j m kj,k l φ kl = j,l=1 n n G l=1 j=1 λ j φ kj,φ kl φ kl. Here < f, g > stands for the inner product of f and g in the space L, π). Since φ kl, l = 1,,, n is a basis of Σ n, n G λ j φ kj,φ kl = j=1 j=1 for l = 1,,, n. As a result, n n G λ j φ kj, λ l φ kl =, which implies that and therefore The proof is complete. l=1 n λ l φ kl =, l=1 λ l =, l = 1,,, n. Now we return to the study of the controllability of system.45). Consider the change of variables η = v + u and w = v u. In terms of these new variables, the equations in.45) become ) ) ) v v H1 Gf t + B = ) u u H 1 Gg ) with f = h 1 + h )/, g = h 1 h )/, where B is the skew-adjoint operator with symbol ) σk) ik, σk) = 1 ak σk) 1 + bk and the symbol of H 1 is 1 1+bk. We take f x, t) in.49) to have the following form f x, t) = f j q j t)φ j x) j=.49) where φ j x) = e ijx, j =, ±1, ±., f j and q j t) are to be determined later. Then dtˆv d 1 kt) + ikσk)ˆv k t) = 1 + bk f j q j t)m j,k where m j,k = 1 π π j= Gφ j )x)φ k x)dx.

26 98 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG Thus, or ˆv k T) e ikσk)t ˆv k ) = ˆv k T)e ikσk)t ˆv k ) = bk bk j= j= T f j m j,k e ikσk)t τ) q j τ)dτ T f j m j,k e ikσk)τ q j τ)dτ. Let p k t) = e λ kt = e ikσk)t. Recall that I is a subset of Z such that λ k1 λ k for any k 1, k I with k 1 k. If a, then kσk) ± as k ± and there exists an integer k > such that k I if k > k. Thus there are only finite many integers in I, saying, k j, j = 1,,, n, such that one can find another integer k k j with λ k = λ kj. Let Then I j = {k Z, k k j, λ k = λ kj }, j = 1,,, n. Z = I I 1 I I n. Note that each I j contains at most two integers. Without loss of generality, we may assume that I j = {k j,1, k j, } j = 1,,, n and rewrite k j as k j,. Moreover, P {p k k I} forms a Riesz basis for its closed span, P T, in L, T) if T > π γ where γ := lim inf k λ k+1 λ k ). We let L {q j j I} be the unique dual Riesz basis for P in P T, i.e., the functions in L are the unique elements of P T such that In addition, we choose T q j t)p k t)dt = δ kj, j, k I. q k = q kj if k I j. For such choice of q j t), < j <, we have then, for any k N, ˆv k T)e ikσk)t ˆv k ) = bk f km k,k, if k k j,l, l =, 1,, j = 1,,, n; ˆv kj, T)e ikj,σkj,)t ˆv kj, ) = 1 1+bk j, l= f k j,l m k j,l,k j,,.5) ˆv kj,1 T)e ikj,1σkj,1)t ˆv kj,1 ) = 1 1+bkj,1 l= f k m j,l k j,l,k j,1,.51) ˆv kj, T)e ikj,σkj,)t ˆv kj, ) = 1 1+bkj, l= f k j,l m kj,l,k j, if k = k j,l for j = 1,,, n and l =, 1,. According to Lemma.1, for given initial state v and terminal state v 1 with zero mean, system.5)-.51) admits a unique solution f, f, f 1, f 1, f, ). In particular, f k = 1 + bk m k,k v 1 k eikσk)t v k ), if k k.

27 BOUSSINESQ SYSTEMS 99 Similarly, for given initial state u and terminal state u 1 with zero mean, choose g x, t) = g j q j t)φ j x) j= with g k = 1 + ) dk û 1 m ke ikσk)t û k, if k k. k,k Then the system.49) admits a unique solution vx, t), ux, t)) satisfying vx, ), ux, )) = v x), u x)), vx, T), ux, T)) = v 1 x), u 1 x)) for x, π). This analysis leads to the following controllability result for the system.49). Proposition.. Assume that the parameter a and T > π γ. Let s, s R be given and let n 1 = if b and n 1 = if b =. Then, for any given initial state v, u ) H,p s H,p s and the terminal state v 1, u 1 ) H,p s H,p, s there exist f L, T; H s n1,p ) and g L, T; H s n 1,p ) such that the system.49) admits a unique solution v, u) C[, T]; H,p s Hs,p ) satisfying vx, ), ux, )) = v x), u x)) and vx, T), ux, T)) = v 1 x), u 1 x)). Moreover, there exists a constant C > depending only on T, s and s such that C f L,T;H s n 1,p ) + g L,T;H s n 1 v, u ) H s,p H + v 1, u 1 ),p s H s,p H,p s,p ) Consequently we have the following exact controllability result for the original system.45). Theorem.3. Assume that the parameter a and T > π γ. Let s R be given. Then, for any η, w ) and η 1, w 1 ) belonging to the space H,p s Hs,p, there exist h 1, h L, T; H s n1,p ) such that the system.45) admits a unique solution η, w) C[, T]; H,p s H,p) s satisfying η, T) = η 1 ), w, T) = w 1 ) in H s,p. Remark.4. The following remarks are in order. i) The choice of the control inputs in Proposition. and Theorem.3 are based on the moment method instead of the HUM. For this approach, it is necessary to apply controls on both equations. ii) In contrast to using only one control input, the advantage of using two control inputs is that the system.45) is exactly controllable in H,p s Hs,p for any s R. Next we turn to the stabilization issue. Again, we rewrite system.45) as an abstract control system in the space V s,: d dt η = A 1 η + B 1 h, η) = η.5) ).

28 3 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG where A 1 = A with a = c and b = d, and B 1 h = I b x) 1 Gh 1. I b x) 1 Gh The following stabilization result using two feedback controls will then be useful. Theorem.5. Assume that the parameter a. Let s R be given. For any given α >, there exists an operator K 1 LV, s, V s n1, ) such that if one chooses h = K1 η in.5), then the resulting closed-loop system has the property that its solution satisfies d dt η = A 1 η + B 1 K 1 η, η) = η.53) for any t where M is a constant independent of η. ηt) V s M η V se αt.54) Proof. The proof is the same as that of Theorem Nonlinear systems. In this section we consider the nonlinear) Boussinesq systems posed on the finite interval, π) η t + w x + ηw) x + aw xxx bη txx = f for x, π), t, T), 3.1) w t + η x + ww x + cη xxx dw txx = g for x, π), t, T) with the periodic boundary conditions r η x t, ) = r η r x t, π) for t, T), r r r, and the initial condition q w x q t, ) = q w x q t, π) for t, T), q q, 3.) η, x) = η x), w, x) = w x), x, π). 3.3) Its well-posedness in a suitable Sobolev space will be studied in subsection 3.1. Its controllability and stabilizability will be investigated in subsection 3. and subsection 3.3, respectively Well-posedness. The following lemma, whose proof can be found in [6], is needed in this study. Lemma 3.1. Let s 1 be given. There exists a constant C > depending only on s such that for any f, g H s+1 p. fg H s p C f H s+1 g p H s+1 p

29 BOUSSINESQ SYSTEMS 31 We first consider the case when both parameters b and d in 3.1) are positive. Such systems are called weakly dispersive systems in [6]. Recall that l is an integer such that hk) = ω 1 k)/ω k)) 1/ C k l, as k and V s = H s p Hs+l p. Theorem 3.. Assume b > and d >. Let T > and s be given such that s if l and s 1 if l = 1. Then there exists a constant r > such that for any η, w ) V s and any f, g) L 1, T; V s ) satisfying η, w ) V s r and f, g) L1,T;V s ) r, 3.4) the system 3.1) admits a unique solution η, w) C[, T]; V s ) satisfying the boundary condition 3.) and the initial condition 3.3). Moreover, the corresponding solution map is locally Lipschitz continuous and η, w)t) V s C η, w ) V s 3.5) for any t T where C is a constant depending only on s, but independent of T and r. Remark 3.3. It is also true that for any r > there exists a T > depending only on s and r such that for η, w ) V s satisfying 3.4), the system 3.1)-3.3) admits a unique solution η, w) C[, T]; V s ) satisfying the estimate 3.5). Proof. Rewrite 3.1)-3.3) in its integral form: η, w)t) = St)η, w ) + t St τ)f, g)τ)dτ t St τ)ηw) x, ww x )τ)dτ. This suggests us, for given η, w ) V s and f, g) L 1, T; V s ), to define a map by Γu, v)t) = St)η, w ) + Γ : C[, T]; V s ) C[, T]; V s ) t St τ)f, g)τ)dτ t St τ)uv) x, vv x )τ)dτ 3.6) for any u, v) C[, T]; V s ). According to Theorem.4, there exist constants C 1 > and C > independent of T such that Γu, v)t) V s C 1 η, w ) ) V s + f, g) L 1,T;V s ) T +C 1 uv) x, vv x )τ) V s dτ C 1 η, w ) T ) V s + f, g) L 1,T;V s ) + C1 uv, v /)τ) V s 1dτ C 1 η, w ) V s + f, g) L 1,T;V )) s + C1 C T sup u, v)t) V s t T

30 3 SORIN MICU, JAIME H. ORTEGA, LIONEL ROSER AND BING-YU ZHANG for any t T. Let r > and R > be chosen according to R = C 1 r, 4C 1C rt = 1, 3.7) then, if η, w ) V s + f, g) L1,T;V s ) r and u, v) C[,T];V s ) R, we have that Γu, v)t) V s C 1 r + 4C 1 r C 1 C T = 3 C 1r < R. Thus Γ maps the ball B, R) in the space C[, T]; V s ) into itself. Moreover, for any u, v) B, R) and any µ, ν) B, R), T Γu, v)t) Γµ, ν)t) V s C 1 uv) x µν) x, vv x νν x )τ) V s dτ T C 1 uv µν, 1 v 1 ν )τ) V s 1dτ C 1 C T u, v) C[,T];V s ) + µ, ν) C[,T];V s )) u µ, v ν) C[,T];V s ) 4C 1 C rt u µ, v ν) C[,T];V s ) 1 u µ, v ν) C[,T];V s ). Thus, Γ is a contraction mapping on the ball B, R) in the space C[, T]; V s ). Its fixed point η, w) is thus the desired solution of 3.1)-3.3). The proof is complete. Next we consider the system 3.1) with b = d =, a, c. 3.8) In view of the constraints C1) and C), the only admissible case is when a = c >. The system 3.1) then takes the form η t + w x + ηw) x + aw xxx = f, 3.9) w t + η x + ww x + aη xxx = g, for x, t), π), T). Introducing v and u by η = v + u and w = v u, we obtained the equivalent system v t + v x + av xxx + 1 [v u)v + u)] x + 1 v u)v u) x = 1 f + g), u t u x au xxx + 1 [v u)v + u)] x 1 v u)v u) x = 1 f g), 3.1) for x, t), π), T). This is a system of two linear KdV-equations coupled through nonlinear terms. One can apply the theory developed by Kato [15] for the scalar KdV to obtain the following local well-posedness result. Theorem 3.4. Let s > 3 and T > be given and assume that b = d =, a = c >.