On the exact controllability and the stabilization for the Benney-Luke equation

In this work we consider the exact controllability and the stabilization for the generalized Benney-Luke equation \begin{document}$\begin{equation} u_{tt}-u_{xx}+a u_{xxxx}-bu_{xxtt}+ p u_t u_{x}^{p-1}u_{xx} + 2 u_x^{p}u_{xt} = f, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\end{equation}$ \end{document} on a periodic domain \begin{document}$ S $\end{document} (the unit circle on the plane) with internal control \begin{document}$ f $\end{document} supported on an arbitrary sub-domain of \begin{document}$ S $\end{document} . We establish that the model is exactly controllable in a Sobolev type space when the whole \begin{document}$ S $\end{document} is the support of \begin{document}$ f $\end{document} , without any assumption on the size of the initial and final states, and that the model is local exactly controllable when the support of \begin{document}$ f $\end{document} is a proper subdomain of \begin{document}$ S $\end{document} , assuming that initial and terminal states are small. Moreover, in the case that the initial data is small and \begin{document}$ f $\end{document} is a special internal linear feedback, the solution of the model must have uniform exponential decay to a constant state.

on a periodic domain S (the unit circle on the plane) with internal control f supported on an arbitrary sub-domain of S. We establish that the model is exactly controllable in a Sobolev type space when the whole S is the support of f , without any assumption on the size of the initial and final states, and that the model is local exactly controllable when the support of f is a proper subdomain of S, assuming that initial and terminal states are small. Moreover, in the case that the initial data is small and f is a special internal linear feedback, the solution of the model must have uniform exponential decay to a constant state.
1. Introduction. In this paper, we consider the controllability and the stabilization problems associated with the forced Benney-Luke equation in which u = u(x, t) with x, t ∈ R and the subscripts denote the corresponding partial derivatives. For p = 1 and f = 0, this equation, known as the Benney-Luke equation, is a formally valid approximation for describing small-amplitude of the isotropic type for two-dimensional water waves (see [12,2]), where the parameters a, b > 0 are such that a − b = σ − 1 3 , with σ being associated with the surface tension (the Bond number). It is worth to mention that the Benney-Luke equation is an approximation formally valid for describing two-way water wave propagation, in contrast to one-way equations such as the KdV, or BBM equations.
For instance, D. Russell in [15], L. Rosier in [13], and D. Russel and B. Zhang in [17] showed the exact boundary controllability of linear and nonlinear Korteweg-de Vries equation on bounded domains with various boundary conditions. In particular, L. Rosier in [13] derived the exact boundary controllability for the nonlinear KdV equation on bounded domains, when the initial and terminal states are sufficiently small (see also the work by C. Laurent and L. Rosier [8]).
B. Zhang in [20] considered a distributed control f for the generalized Boussinesq equation on the periodic domain S, showing that the system is exactly controllable for either the control f acting on the whole domain S, or the control f acting only on a sub-domain of S, imposing a smallness condition on the initial and terminal states in the later case (see also the work [10]).
M. Chapouly in [6] considered both the global exact controllability to the trajectories (one boundary left control) and the global exact controllability of a nonlinear Korteweg-de Vries equation in a bounded interval. In this work, there were introduced, in addition to the boundary value at the left endpoint, a control in the boundary value at the right endpoint and an internal control in the right member of the equation, assumed both to be x-independent, proving the global exact controllability to the trajectories, for any positive time T . It was also obtained the global exact controllability, for any positive time, by introducing a fourth control on the first derivative at the right endpoint.
E. Cerpa and E. Crépeau in [4] established the controllability of the improved Boussinesq equation posed either on a bounded or periodic domain. For the bounded domain [0, 1] with boundary control, they showed that the linearized equation is not spectrally controllable, and so not null controllable, either. However, they proved an approximate controllability result.
J. Amara and H. Bouzidi in [1] considered the exact boundary controllability for a Boussinesq equation with variable physical parameters, when the control is acting at the end x = l of the interval [0, l]. They established that the linearized problem is exactly controllable in any time T > 0, using a spectral analysis together with the moment method. They also established the local exact controllability for the nonlinear problem by fixed point argument.
C. Laurent, F. Linares and L. Rosier in [11] and F. Linares and L. Rosier in [9] considered the control and stabilization problem for the Benjamin-Ono equation in L 2 (T), where T = R/(2πZ). In the latter work, authors proved a controllability result in L 2 (T) that allows to get the global controllability in large time.
Our goal in this work is to study the generalized Benney-Luke equation from a control point of view with a forcing term f = f (x, t) added to the equation as a control input, which is assumed to be supported in a given open set ω ⊆ S. In the controllability theory the following problems are essential: Exact Control Problem: Given T > 0, an initial state (u 0 , v 0 ) t and a terminal state (u T , v T ) t in an appropriate space, can one determine an appropriate control input f so that the model (2) has a solution u satisfying that u Stabilization Problem: Can one find a feedback law f = Γ(u x , u t ) so that the resulting closed-loop system We point out that we follow the classical approach used by D. Russell and B. Zhang in the case of the KdV equation (see [17]), and by B. Zhang in the case of the good Boussinesq equation (see [20]).
Hereafter, for s ∈ R, we define the space V s+1 given by In the case that the support of f is acting on the whole S, we get an exact controllability result for the forced Benney-Luke equation (2) with a distributed control f having the general form with g being a smooth function defined on S having S g(x) dx = 1, to guarantee that f (·, t) has mean zero on S.
Theorem 1.1. (Exact controllability) Let T > 0 and s ≥ 2 be given. Suppose that the function g smooth on S is such that there exists a control function f ∈ L 2 ((0, T ), H s−2 (S)) such that the initial value problem associated with the nonlinear Benney-Luke equation (2) has a solution satisfying that In the case that the support of f is acting on a proper subdomain ω ⊂ S, we get an exact controllability result for the forced Benney-Luke equation (2) with a distributed control f having the general form (3), assuming that the initial and terminal states are small. and there exists a control function f = Lh with h ∈ L 2 ((0, T ), H s−2 (S)) such that the initial value problem associated with the Benney-Luke equation (2) has a solution u ∈ C((0, T ), V s+1 ) C 1 ((0, T ), H s (S)) satisfying that We obtain previous result due to the exact controllability result for the system associated to the forced Benney-Luke equation with a distributed control f of the form (3) acting on the whole domain S or a subdomain of S. Hereafter, we set where B −1 is the inverse for the operator B = I − b∂ 2 x with b > 0. Regarding the stabilization issue, we get the following result.
the forced generalized Benney-Luke equation where C 1 (·, ·) is a positive function.
The paper is organized as follows. In section 2, we consider the well-posedness of the initial value problem for the forced generalized Benney-Luke model on the periodic domain S, where the forcing term f ≡ f (x, t) is known as the control input. In section 3, we perform the spectral analysis for the operator x for a, b > 0 are defined using Fourier series. For the linear operator M, we prove the existence of a discrete spectral decomposition since the eigenvectors form a Riesz basis of the space H s , for s ≥ 0. In Section 4, we establish linear exact controllability by showing the existence of a bounded linear operator Ψ from the initial/end state pairs (q 0 , r 0 ), (q T , r T ) in the space H s , to the corresponding control h in the space L 2 ((0, T ), H s−2 ). In section 5, we prove the nonlinear controllability, which will be global in the case |g(x)| ≥ β > 0 on S and local for general g, by imposing smallness of the initial and terminal states in the later case. In section 6, we establish the stabilization result for the Benney-Luke with a forcing feedback depending on (u x , u t ).

2.
Preliminares. Before we go further, we consider the Benney-Luke equation on the periodic domain S, the unit circle in the plane, consisting with the bounded domain (−π, π) with periodic boundary conditions. For s ∈ R, the periodic Sobolev space H s (S) is defined by where C ∞ per (R) denotes the space of periodic functions on R with period 2π and ψ n denotes the n-Fourier coefficient with respect to the spatial variable x. We also define the Sobolev spaceḢ which means that ψ has average zero on S, On the other hand, for any d > 0, we define the operators D = I − d∂ 2 x and D −1 , respectively by We clearly have that D : H s (S) → H s−2 (S) and D −1 : H s (S) → H s+2 (S) with s ∈ R. Moreover, we also have the n-Fourier symbols for the operators ∂ x and B −1 A are given respectively by (∂ x ) n = in, (B −1 A) n = 1 + an 2 1 + bn 2 := Λ 2 (n).
In particular, we also have that n-Fourier symbol for the operator M is given by We begin this section by rewriting the Benney-Luke equation (2) (f = 0) as a first order equation. To do this, we consider the following variables In this case, we see formally that and that the Benney-Luke equation (2) can be expressed as So, we have that Thus, the Benney-Luke equation (2) can be written as the first order system or equivalent to the first order system where U = (q, r) t , M and G are given by with G i for i = 1, 2 being Remark 1. Before we go further, if F = (0, F 2 ) t is such that F 2 has average zero on S, then we want to discuss the relationship between the Boussinesq type system and the forced generalized Benney-Luke equation (2). If we assume that q ∈ H s (S) has zero average in S (q 0 = 0), then we have that the function u = ∂ −1 x q ∈ V s+1 is such that q = u x , where the operator ∂ −1 x : H s (S) → H s+1 (S) is defined as We first note that the quantity is conserved in time for classical solutions and even for mild solutions, as long as the solution exists. So, if we consider the Cauchy problem associated with the system in the variable (q, r) with the initial data q 0 ∈ H s (S) with mean zero property Then, as long as the solution exists for t, we have that S q(x, t) dx = 0, meaning that q(·, t) ∈ H s (S) and has the mean zero property, as the solution exists for t. In this case, the function u( From this we see that u solves the forced Benney-Luke equation (2) for a special f . In fact, On the other hand, we also have that In other words, we have that u satisfies formally the forced generalized Benney-Luke equation Due to this fact, we focus our attention in the exact controllability and the stabilization associated with the system in the variable (q, r).
where S n is defined using Fourier series by where λ n = isign(n) n 2 Λ 2 (n) and Λ 2 (n) = 1+an 2 1+bn 2 . In other words, S n is the exponential matrix of e tMn , with M n being the n-Fourier coefficient of M. Moreover, we have the following estimate, Theorem 2.1. Let T > 0 and s ≥ 0 be given. Then there is a constant C > 0 such that for any given U ∈ H s , and for any given Proof. We see directly that for any (q, r) ∈ R 2 , showing the estimate (14). From this estimate, we see that

JOSÉ R. QUINTERO AND ALEX M. MONTES
In order to perform the nonlinear estimates in the case of the forced system (10), we combine the Sobolev Multiplication Law (see [18]) and a result by D. Roumégoux ([14]).
Now, we are able to get the nonlinear estimates. We set for y > 0 the following notation: η p (y) = y 2p + y p .
Moreover, for U ∈ H s , we have that Proof. We must recall that H s (S) is an algebra for s > 1 2 . Let U i = (q i , r i ) t ∈ H s with i = 1, 2. Then, we see using the estimate (17) in Lemma 2.3 that there is a positive constant C 1 (p, b) such that On the other hand, we set We see from the same approach that . Now, we perform the estimate for G 4 for p > 1, since p = 1 is similar.
where C l = C l (p, b) for l = 3, 4 and we are using the Sobolev Multiplication Law with t = s, s 1 = s and s 2 = s + 2, and that Now, using Hölder inequality with α = p+1 p−1 for the product q 2 p−1 s q 1 p+1 s and α = 2 for the product q 2 p s q 1 p s , we conclude that In other words, we have shown that The second inequality follows by taking U 1 = U and U 2 = 0.
From the classical semigroup theory, we are able to establish the local wellposedness associated with the system (10) in the space H s . For this, using the Banach fixed point Theorem, we will show the existence of a solution for the integral equation Before we go further, for T > 0 and s ≥ 0, we define the space Theorem 2.5. Let s, p be as in Lemma 2.4 and T > 0 be given. For any given U 0 ∈ H s and for any given F = (0, F 2 ) t with F 2 ∈ L 1 ((0, T ), H s (S)), there exists T * > 0, depending only on ||U 0 || Hs and ||F 2 || L 1 ((0,T ),H s (S)) , such that initial value problem has a unique solution U ∈ Z s T * and the corresponding solution map Using the semigroup S(t), we have that the integral form associated with (20) can be written as So, we consider the mapping Φ defined as From previous results, we conclude that and also that sup If we choose R > 0, and T * > 0 such that From these choices, we conclude for any U ∈ X R T * that From the same type of computations, we have that meaning that Φ is a contraction on X R T * , and so, the Contraction Mapping Theorem guaranties the existence and uniqueness of a local solution to the initial value problem (20) in the space Z s T * .
On the other hand, we also have that which implies that lim k→±∞ (e 1,k , e 2,k ) = 1 1 ±i ∓i and also that lim In other words, {η 0,1 , η 0,2 , η 1,k , η 2,k : k ∈ Z * } forms a Riesz bases for H s , and also that {ν 0,1 , ν 0,2 , ν 1,k , ν 2,k : k ∈ Z * } forms an orthonormal bases for H s with Moreover, we also have for j = 1, 2 that ν So, we have the following result, Theorem 3.1. Let λ k and φ j,k for j = 1, 2 be given by b) The set {φ j,k : k ∈ Z} forms an orthogonal bases for H s such that for any U ∈ H s , we have the Fourier expansion 4. Linear exact controllability. In this section, we establish the linear exact controllability associated with the initial value problem (20) in the case G(U ) ≡ 0 and F = (0, B −1 f ). In this case for s ≥ 2 and f ∈ L 1 ((0, T ), H s−2 (S)), the solution U of the first order linear system is given by where α j,n and β j,n (τ ) for j = 1, 2 are given by α j,n = U 0 , φ j,n Hs , , with φ (l) denoting the l component of φ, ·, · s denoting the inner product in H s (S) and ·, · s−2,s+2 denoting the pairing between H s−2 (S) and H s+2 (S). In the case f (x, t) = (Lh)(x, t) defined in (3), we have that .
Before we go further, we observe that P = {e λ k t : k ∈ Z} is a Riesz basis for its closed span P T generated in L 2 (0, T ), with a unique dual Riesz basis given by We first assume that f has the form (3) with h given by the expansion From this representation, using the notation ρ j,n,k = L(B −1 φ 2,0 ) e −iky dy(ρ j,n,0 ) where ·, · l denotes the inner product in H l (S) for l ≥ 0. Now, for n ∈ Z * , we have that 1,n ) + c 2,n L(B −1 φ 2,n ) e −iky dy.
So, we also have that Putting these estimates, we get that We note for l ∈ Z that where we are using that We also have that k-Fourier coefficient for L(B −1 e il· ) is On the other hand, we have that Now, we expand L(B −1 φ r,l ) with r = 1, 2 as with the k-coefficient given by From previous calculations, we see that Before we go further, we see the following estimate for s ≥ 2 and l, k ∈ R (1 + |k + l|) 2(s−2) ≤ (1 + |k|) 2(s−2) (1 + |l|) 2(s−2) .
From this estimate, we conclude for s ≥ 2 that In a similar fashion, we have that From (26) and using the expansion (32), we see that h(x, t) = l∈Z * k∈Z q l (t) (c 1,l α 1,l,k + c 2,l α 2,l,k )) e ikx .
If we assume that l∈Z (1 + |l|) 2(s−4) (|c 1,l | 2 + |c 2,l | 2 ) is finite, then we have that h ∈ L 2 ([0, T ]; H s−2 (S)), according with previous computations. In fact, Now, we define the space Y s ⊂ X s = H s × H s by where U (1) denotes the first component of U . such that for any (U 0 , U T ) ∈ Y s , the solution U (t) of the initial value problem associated with the linear system for initial data U 0 and F = (0, B −1 LΨ T (U 0 , U T )) t , satisfies that Moreover, we also have that Proof. Let U 0 and U T be having the following decompositions We know that So, in each node, we have for j = 1, 2 and n ∈ Z * that α j,0 + T 0 β j,0 (τ ) dτ = γ j,0 α j,n + T 0 e −λnτ β j,n (τ ) dτ = γ j,n e −λnT . Now, we suppose the Ψ T (U 0 , U T ) has the form (26). In other words, From the computations above, we are able to characterize the coefficients c 0 , c 1,l and c 2,l later on in such a way that the series (26) converges appropriately. In fact, for n = 0, we have from (27) that and for l ∈ Z * , we have from (28) that c 1,l and c 2,l must satisfy the linear system r,l ) are linear independent for any l ∈ Z * . On the other hand, we have that which implies that Then we have that ∆ l = a 11 a 21 a 12 a 22 1,l ), L(φ Moreover, using that ν r,l (x) = b r,l e ilx and the estimate (24), we have that So, we conclude that c 1,l and c 2,l are uniquely determine by Using the estimates of E. Cerpa and I. Rivas in [5] for the Boussinesq equation and B. Zhang in [20] for the generalized the Boussinesq equation, we also have for any l ∈ Z that∆ l ≥ C 0 > 0, and so we get that |ã ij | 2 |α 1,l | 2 + |α 2,l | 2 + |γ 1,l | 2 + |γ 1,l | 2 (1 + bl 2 ) −8∆2 l ≤ C 1 (1 + |l|) 8 |α 1,l | 2 + |α 2,l | 2 + |γ 1,l | 2 + |γ 1,l | 2 .
From this and the estimate (34), we conclude that Moreover, we also have the following modification of previous result for s ≥ 0. such that for any (U 0 , U T ) ∈ Y s , the solution U (t) of the initial value problem associated with the linear system for initial data U 0 and F = (0, LΥ T (U 0 , U T )) t , satisfies that Moreover, we also have that The proof follows by the same type of arguments as in previous theorem. In this case, we have that Υ T (U 0 , U T ) has the form Global case. We first discuss the exact controllability issue for the system (10) in the case that the support is the whole S. Theorem 5.1. (Exact Controllability) Let T > 0 and s ≥ 2 be given and suppose that g is a smooth function such that Then for any (q 0 , r 0 ) t , (q T , r T ) t ∈ H s such that there exists a control function H ∈ L 2 ((0, T ), H s−2 (S)) such that the initial value problem associated with the nonlinear system has a solution U ∈ C((0, T ), H s ) C 1 ((0, T ), H s−1 ) that satisfies for t ≥ 0, Proof. From Theorem 4.1, for s ≥ 2 and a given T > 0, there exist a bounded linear operator such that for any (U 0 , U T ) ∈ Y s , there exists U ∈ C((0, T ), H s ) C 1 ((0, T ), H s−1 ) satisfying So, we know that U = (q, r) t ∈ H s (S) × H s (S) and G(q, r) ∈ H s (S) × H s (S). Now, as done by D. Russell and B. Zhang in [17] for the case the KdV equation, we define the linear operatorL : H s g (S) → H s g (S) as where the weighted space H s g (S) is the space H s (S) endowed with the norm ||v|| H s g (S) = gv, v H s (S) , under the assumption that g > 0 on S (otherwise, we change g for −g). For this operator, we have the following properties:L is a Fredholm operator,L * =L, N (L) = {1}, andL has a bounded inverse from H s g (S)/{1} → H s g (S). Now, we see that From these facts, we conclude that there are functions h 2 (·, t), h 3 (·, t) ∈ H s−2 (S) such that for each fixed t, So, adding and subtracting G(U ) in the right side of equation (45), we conclude that (46) So, we set H ∈ L 2 ([0, T ], H s−2 ) as follows Local case. Now, we consider the local exact controllability problem. From the semigroup theory, we know that the solution of the initial value problem associated with the system with F = (0, B −1 Lh) t has the general form Note that, if we define and choose h using Theorem 4.1 with G(U ) = 0 as then, we see that due to the definition of Ψ T , we have that Theorem 5.2. (Exact Controllability) For T > 0 and s ≥ 2 be given. Then there exists δ > 0 such that for (U 0 , U T ) ∈ Y s and (U 0 , U T ) Ys < δ, there exists a control function h ∈ L 2 ((0, T ), H s−2 (S)) such that the initial value problem associated with nonlinear system with F = (0, B −1 Lh) t has a solution U ∈ C((0, T ), H s ) C 1 ((0, T ), H s−1 ) that satisfies Proof. According to previous discussion, in order to obtain the exact controllability in H s , we need to prove that the operator from H s to H s defined by the right side of formula (49) has a fixed point. Recall that for T > 0 and s ≥ 0, we define the space Z s T = C([0, T ], H s ). Then, if R > 0, we consider the bounded set of the space Z s T defined by IfF is given byF

JOSÉ R. QUINTERO AND ALEX M. MONTES
using the estimate in Theorem 4.1, we have that Moreover, from the nonlinear estimates (18) in Lemma 2.4, we have that sup t∈(0,T ) Then we see that (see proof Theorem 2.5). If we choose δ > 0 and R > 0 such that we conclude for any U ∈ X R T that sup provided that U 0 H s (S) < δ, U T H s (S) < δ. Now, using the same type of computations in Theorem 2.5 and using the nonlinear estimates (18) in Lemma 2.4, we have that Φ 1 is a contraction on X R T . From the Contraction Mapping Theorem, there is a unique local solution (nonlinear control) for the integral equation (49) in the space Z s T . On the exact controllability for the generalized Benney-Luke model. As we discussed in the well-posedness section, it is possible to build solutions for the forced generalized Benney-Luke model (2) from solutions for the Boussinesq type system (10). So, we derive the global and the local exact controllability for the Benney-Luke model from the analysis for the Boussinesq type system (10).
Proof of Theorem 1.1. From the hypothesis, we have that Then, we have that [q 0 ] = [q T ] = 0. From the global exact controllability result (Theorem 5.1), we have that has a solution U = (q, r) t ∈ H s (S) that satisfies U (0) = U 0 and U (T ) = U T . If we set u(x, t) = ∂ −1 x q(x, t), which makes sense since q(·, t) has average zero for any t, then from the first component we see that We point out that B −1 Lh 2 and B −1 Lh 3 have mean zero on S, so ∂ −1 x (B −1 Lh 2 ) and ∂ −1 x (B −1 Lh 3 ) make complete sense. Using this in the second equation, we get that where the control H is given by with h 4 ∈ H s (S) being taking in such a way that (Lh 2 (x, t)) , using the same argument in the proof of Theorem 5.1, due to the fact that the term (Lh 2 (x, t)) ∈ H s (S) has mean zero on S for any t ∈ R. Proof of Theorem 1.2. From the hypothesis, we have that Then, we have that [u 0 ] = [u T ] = 0. From the local exact controllability result (Theorem 5.2), there exists a function h ∈ L 2 ((0, T ), H s−2 (S)) such that the initial value problem associated with the system has a solution U = (q, r) t ∈ H s that satisfies U (0) = U 0 and U (T ) = U T . As we discussed in Remark (1) in Section 2, we know that u(x, t) = ∂ −1 x q(x, t) satisfies the forced generalized Benney-Luke equation Moreover, we also have that u x (x, 0) = u 0 and u x (x, T ) = u T . On the other hand, we have that r(·, 0) = v 0 + B −1 1 p+1 u p+1 0 = r 0 , which implies that 6. Stabilization problem. In this section, we study the stabilization problem associated with the nonlinear system with the conditions having the property [q 0 ] = [r 0 ] = 0 and K > 0.
6.1. Linear problem. We start to show the exponential decay result for the linear equation with K being a positive constant and the conditions having the property [q 0 ] = 0.
Theorem 6.1. Given s ≥ 1, for any (q 0 , r 0 ) t ∈ H s with [q 0 ] = 0, the problem (53)-(54) has a unique solution (q, r) t ∈ C(R, H s ). Moreover, there exist constants C, γ > 0 such that Proof. Without loss of generality, we perform the proof assuming that [q 0 ] = [r 0 ] = 0, since the couple (q, r − [r 0 ]) t is also a solution of the system (53)-(54). We must note that the local existence result is a direct consequence of the semigroup properties in Theorem 2.1. In other words, given (q 0 , r 0 ) ∈ H s , there is T * > 0 (depending on (q 0 , r 0 ) Hs ) and a unique solution We start to show that the estimate (55) holds for s = 1. A direct computation shows that the energy of the solution (q, r) t is given by We note for t ∈ [0, T * ) that there is α ∈ (0, 1) such that To see this, we note that for given t ∈ [0, T * ), using integration by parts yields, On the other hand, using this estimate where the last equality holds since that On the other hand, for t ∈ [0, T * ) we have that (q(·, t), r(·, t)) ∈ H 1 (S) and [q(·, t)] = [r(·, t)] = 0. So, from Corollary 1 there is h ∈ L 2 ((0, t), H 1 (S)) such that the system with the conditionsq(x, 0) =r(x, 0) = 0, has a solution satisfying q(x, t) = q(x, t),r(x, t) = r(x, t), Now, if we set then, we see directly that Now, using that R(0) = 0, and the final conditions forq andr, we see that On the other hand, we also have that Moreover, where we denote g * > 0 as the least upper bound of g in S. Thus, Then we see that Then we obtain that so, we have for some 0 < α < 1 that meaning that the solution can be extended to R in the case s = 1. Now, we claim that In fact, for a given T > 0, we have that Repeating this estimate on successive intervals [(k − 1)T, kT ], for k = 2, 3, . . . , with then it follows that, Then, we complete the proof for s = 1. Now, we consider the case s = 2. Set Then (q 1 , r 1 ) solves the system with the conditions Note that we have [q 1 (·, 0)] = [r 1 (·, 0)] = 0. By taking q 0 , r 0 , ∈ H 2 (S), then we see that r 0 , B −1 Aq 0 − KL(r 0 ) ∈ H 1 (S).
So, we have that the estimate (55) holds for s = 2. Moreover, for 1 < s < 2, the estimate (55) can be gotten by interpolation, and for s > 2, it can be obtained by an inductive argument.
We note that the system (53) can be written as the first order system with the condition U (x, 0) = (q 0 (x), r 0 (x)) = U 0 (x), (62) where U = (q, r) t , M and F are given by .
We see that the solution U can be written as where S K (t) is the semigroup on H s associated to (61)-(62). Then we have the following corollary.
Corollary 2. Given s ≥ 1, for any U 0 = (q 0 , r 0 ) t ∈ H s (S) with [q 0 ] = 0, the problem (61)-(62) admits a unique solution U ∈ C(R, H s (S)). Moreover, there exists C, γ > 0 such that 6.2. Stability of the nonlinear system. In this section we consider the nonlinear system or the equivalent first order system in the variable U = (q, r) t with the condition U (x, 0) = U 0 (x) = (q 0 (x), r 0 (x)) t , where M and F are defined as in previous section and the components G 1 and G 2 of G are given by (8) and (9), respectively.
A direct observation shows that there is an explicit relation between the semigroups S(t) and S K (t), as we state in the following result without including the proof.
Lemma 6.2. For s ≥ 1, the semigroup S K (t) defined in (63) satisfies the following relation for any U 0 ∈ H s , where S(t) is defined by (12). In addition, for V ∈ L 1 (R, H s (S)), Using the definition of S K (t), the system (65) can be written as the integral equation Proof. From Corollary 2, for U 0 ∈ H s , we have that by choosing T > 0 satisfying that We want to have a solution U to the integral equation In order to get this result, we need U 0 Hs(S) ≤ δ, where δ is determined later. Moreover, to reach the exponential stability, we need to have δ and R be chosen such that  From similar estimates done for S(t), we are able to establish that where η p (y) = y 2p + +y p (see Lemma 2.4 and proof of local existence Theorem 2.5).
In addition, from the boundedness of L we obtain that We define δ = 4Cη p (R)R where R > 0 is chosen so that (4C 2 + C)η p (R) ≤ 1, CR ≤ 1 2 .
Thus, we have that Φ is a contraction mapping in X R . Moreover, using (71), the unique fixed point U ∈ X R satisfies U (T ) Hs = Φ(U )(T ) Hs ≤ δ 2 .
As in the estimate (59), we are able to establish that U (·, t) Hs ≤ M e −γt U 0 Hs , for some constant M > 0 and γ > 0, as desired.
Finally, we are able to establish the corresponding stabilization result related with the Benney-Luke model. In fact, if we assume that [r 0 ] = c = 0, then we need to observe that (q(·, t), r(·, t)) t is conserved for any t, due to the structure of the system (65)-(66). We note that L(c) = 0. So, if we setr(x, t) = r(x, t)−c, then we have (q(·, t),r(·, t)) t solves the new system As we discussed in section 2, we know that u(x, t) = ∂ −1 x q(x, t) is such that r = h 1 (u t , u x ) and that satisfies the forced Benney-Luke equation u tt − u xx + au xxxx − bu xxtt + pu t u p−1 x u xx + 2u p x u xt + KBL(h 1 (u t , u x )) = 0.

7.
Conclusions. In this work we investigated the problems of controllability and stabilization for the Benney-Luke equation on the unit circle on the plane S, with internal control f supported on an arbitrary sub-domain of S. We showed that the model is exactly controllable when S is the support of f , without any assumption on the size of the initial and terminal states, and that is locally exactly controllable when the support of f is a proper sub-domain of S, in the case of small initial and terminal states. Moreover, assuming that the initial data is small and f is a special internal linear feedback, the solution of the model must have uniform exponential decay to a constant state. The results are obtained by re-writing the Benney-Luke equation as a first order system in special variables. This allowed us to perform the spectral analysis and the existence of the linear and nonlinear control in a simpler form. The results obtained in the paper are in concordance with the results of controllability and stabilization for model like the KdV, the Benjamin-Ono, the best Boussinesq equation, among others.