INTERNAL NULL CONTROLLABILITY OF THE GENERALIZED HIROTA-SATSUMA SYSTEM

. The generalized Hirota-Satsuma system consists of three coupled nonlinear Korteweg-de Vries (KdV) equations. By using two distributed controls it is proven in this paper that the local null controllability property holds when the system is posed on a bounded interval. First, the system is linearized around the origin obtaining two decoupled subsystems of third order dispersive equations. This linear system is controlled with two inputs, which is optimal. This is done with a duality approach and some appropriate Carleman estimates. Then, by means of an inverse function theorem, the local null controllability of the nonlinear system is proven.


Introduction
In the eighties, Hirota and Satsuma introduced in [15] the set of two coupled Korteweg-de Vries (KdV) equations, describing the interaction of two long waves with different dispersion relations.They studied the existence of soliton solutions and conserved quantities.Later, in [22] the same authors introduced a new system, coupling now three KdV equations, (1.2) This set of equations was called in the literature the generalized Hirota-Satsuma (HS) system and has attracted the attention of many researchers mainly interested in soliton or explicit solutions.See for instance [13,23] and the references therein.
As far as we know, there is no studies of the control properties of this kind of coupled systems.Thus, in this article the goal is to fill this gap focusing on the null controllability with distributed controls.An important point is that we obtain our results on the control of this three-equation system using only two control inputs.
Regarding dispersive systems, we find papers dealing with the boundary controls of either KdV systems on a bounded domain [6][7][8]17] or KdV equations posed on a network [2,9].Concerning the internal control of dispersive systems, the closest works are [18] where Ingham theorems are used to prove some observability inequalities for Boussinesq systems and [3] where a Carleman estimates approach is used to get the null controllability of a linear system coupling a KdV equation with a Schrödinger equation.
Summarizing the links with the existent literature, in this paper we follow the same methods than in [3,5] to study the null controllability property of a dispersive system with less controls than equations.
Let us go back to the control of system (1.5).The first step in our strategy is to linearize the system (1.5) around the origin, getting the linear system where f 1 , f 2 and f 3 will play later the role of the nonlinearities.In order to study the null controllability of (1.6) we apply a duality approach that leads us to prove that the solutions of the adjoint system satisfy an appropriate observability inequality.This is realized proving a Carleman estimate for system (1.7)where functions g 1 , g 2 and g 3 are useful to get information on the solutions of (1.6) when using duality arguments.Finally, the last step in our strategy is to go back to the original nonlinear system by using an inverse function theorem.In this way we will get our main result, stating the local null controllability of (1.5).
Theorem 1.1.Let γ ⊂ (0, L) and ω = (a, L), with a ∈ (0, L).Assume that (u 0 , v 0 , w 0 ) ∈ [L 2 (0, L)] 3 .Then, for every T > 0 there exists δ > 0 such that if (u 0 , v 0 , w 0 ) [L 2 (0,L)] 3 < δ, there are controls p ∈ L 2 (0, T ; L 2 (γ)) and The organization of this paper is the following.We start giving in Section 2 the well-posedness framework in which we work along this paper.Then, Section 3 is devoted to the proof of a Carleman estimate that is used to prove an appropriate observability inequality.Section 4 contains the control results for both the linear and nonlinear systems.Finally, we end this paper with some comments and related open problems.

Well-posedness results
In this section, we give the functional framework and some well-possedness results for the KdV equation.Additionaly, we present some regularity results for the system (1.6).

Functional spaces
We introduce the following functional spaces: and These spaces are equipped with their usual norms.Moreover, we define for each θ ∈ [0, 1] the interpolation spaces (see [4]): A sample of spaces that will be often used in the following is

Regularity results for a single equation
We first consider a single KdV equation with a source term: in (0, L). (2.3) For this equation we have the following known results.
3) admits a unique solution χ ∈ Y 1 .Moreover, there exists a constant C > 0 such that Notice that the same results are valid for the (backward-in-time) adjoint equation and the reverse-in-space equation, for any dispersive coefficient d > 0.

Regularity results for the linear system
We first consider the linear system (1.6).Taking advantage of its cascade structure, notice that we can apply the results for a single equation in order to get the solutions v and w (Prop.2.1, for instance).Then, we can see the term 3w x as a source term in the equation satisfied by u.Therefore, we can easily obtain the following result.
This result can be applied to the adjoint system (1.7) with appropriate functions g 1 , g 2 , and g 3 .To do that we only need to perform a change of variable in space x ≈ L − x and time t ≈ T − t.

Regularity results for the nonlinear system
In this section we apply a fixed point argument in order to establish the well-posedness of the nonlinear system (1.5).First of all, we prove the following lemma inspired from [19].
We can now prove the following well-posedness result.
Proposition 2.6.Let L > 0 and T > 0. There exist ε > 0 and C > 0 such that for every where ε will be chosen small enough later.Let (u, v, w) ∈ (Y 1/4 ) 3 and consider the map Φ : (Y 1/4 ) 3 → (Y 1/4 ) 3 defined by Φ(u, v, w) = (ũ, ṽ, w) where (ũ, ṽ, w) is the solution of the linear problem, By Proposition 2.4 we have (2.10) By Lemma 2.5, we obtain, We also have, for any (u R} where R > 0 will be chosen later, we have the estimate, Then if we take R and ε such that R < 1 2C and ε < R 2C , we can apply the Banach fixed point theorem and Φ admits a unique fixed point, which ends the proof of Proposition 2.6.

Carleman inequalities
This section is dedicated to Carleman estimates.First, we present a general estimate for a KdV equation with observation in an interior domain.Then, we will prove a new Carleman estimate for the whole adjoint system (1.7).

Carleman weights
Let ω 0 = (a 0 , b 0 ) ⊂ (0, L), and set c 0 = (a 0 + b 0 )/2.Consider the weight functions defined in [3], namely for and Notice that, for any K 1 , K 2 > 0, we have ) Furthermore, K 1 and K 2 can be chosen such that Indeed, property (3.6) holds for since the extremum of the interval where the maximum is achieved depends on the location of c 0 .Thus, if we call then, it suffices to take K 1 = (110 C(K 2 , c 0 )) −1 for (3.7) to hold.

Carleman estimate for a single KdV equation
In this section, we establish a Carleman estimate for the general backward in time KdV equation of the following type, for ν ∈ R * : To begin, we recall a Carleman estimate for the linear KdV equation (3.8) obtained in ([3], Thm.3.1) and ( [5], Prop.3.1).Their results are obtained in the case ν > 0, but they can easily be converted in the case ν < 0 by using the change of variables x → L − x.We can rewrite that estimate as follows.
Using the properties of the weight functions, we have Now, for φ we apply the second inequality of Proposition 3.2 with ω 0 = ω 1 , ϕ = ϕ 1 , ν = −1/4, and g = −g 1 .In this way, from (3.11) we get, after using the properties of the weight functions, the estimate s 5 ξ 13 e −2s(4 φ1−3 φ1) |φ| 2 dxdt. (3.20) Lastly, we apply Proposition 3.2 to the equation in (1.7) satisfied by η, with ω 0 = ω 1 , ϕ = ϕ 1 , ν = 1/2, and g = −g 3 + 3φ x .From (3.10), we obtain s 5 ξ 25 e −2s(7 φ1−6 φ1) |η| 2 dxdt, from where we deduce To finish the proof of estimate (3.18), it remains to absorb the last term of this inequality.The idea is to use the coupling of the equation satisfied by η in system (1.7) to express φ in terms of η.However, since the coupling is of first order, this cannot be done directly.Here, we will need the fact that ω "touches" the boundary of (0, L).Let us call J := T 0 ω1 s 5 ξ 13 e −2s(4 φ1−3 φ1) |φ| 2 dxdt, and consider ω 2 := (δ, L), with δ ∈ (0, L) such that ω 1 ⊂ ω 2 ⊂ ω, where all the inclusions are strict.Since φ(t, L) = 0, we have with Poincaré's inequality that We concentrate on this term.Let θ ∈ C ∞ ([0, L]) a non-negative function such that θ(x) = 0 for x ∈ [0, L] \ ω, and θ(x) = 1 for x ∈ ω 2 .Then, using the equation satisfied by η in system (1.7), we have (3.23) Let ε > 0. We estimate each one of these terms.Using Young's inequality, we have For J 2 , taking into account that φ x (t, L) = 0, we integrate by parts in space: where we use Young's inequality to obtain The third and last term is the most difficult one.We integrate by parts once in time and space in the term J 3 .We get For the first term, since ξ 13 e −2s(4 φ1−3 φ1) t ≤ Csξ 15 e −2s(4 φ1−3 φ1) , we have that For the second one, we use the fact that φ t = 1 4 φ xxx − g 1 and integrate by parts in space.This is: We observe the following: Going back to the expression of J 3 , we obtain We estimate now the local term of η.Regarding H 2 (ω) as the interpolation of the spaces H 7/3 (ω) and L 2 (ω), and Young's inequality, we obtain Then, finally, we get Going back to (3.22), we deduce (3.18) by choosing the biggest weight functions and ε sufficiently small.

Control results
In this section, we establish an observability inequality for the solutions of system (1.7) and deduce a null controllability result for the linear system (1.6).Moreover, we prove our main result getting the local null controllability of system (1.5).

Observability inequality
The observability inequality will be deduced from Carleman estimate (3.18), but first, to be able to deduce null controllability, we need to change the weight functions in such a way that they do not vanish at t = 0. Before that, let us deduce a somewhat simpler version of the Carleman estimate (3.18) which will be useful in what follows. Let and Notice that if we call φ(t) := ξ(t)ϕ M and φ(t) := ξ(t)ϕ m , under the assumptions of Theorem 3.3 we can deduce from (3.18) the following inequality: Now, let β ∈ C 1 (0, T ) be defined by and let us call α(t) := β(t)ϕ M and α(t) := β(t)ϕ m .
Furthermore, we will assume also that g 1 , g 2 , and g 3 in system (1.7) belong to L 2 (0, T ; H 1 0 (0, L)).This will make the analysis of the controllability of system (1.6) simpler later on.Proposition 4.1.Let s be fixed such that Carleman estimate (3.18) holds.Assume that g 1 , g 2 , g 3 ∈ L 2 (0, T ; H 1 0 (0, L)).Then, every solution (φ, ψ, η) of system (1.7) Then, from the system satisfied by (λφ, λψ, λη) and the estimate in Proposition 2.1, we deduce that Since e −2s φ ≥ C > 0 in (T /2, 3T /4), the last term of this estimate can be bounded from above by the left-hand side of (4.1).Thus, we get where we have also used the fact that ξ ≡ β in (T /2, T ).Actually, using this last property again, we see that is bounded from above by the left-hand side of (4.1).
The following result establishes the null controllability of the linearized system (1.6).
Recall that the space E is the Banach space defined at the beginning of Section 4.2.
Let us check the two points above.
It is fairly clear to see that it suffices to prove that the bilinear terms in F are bounded.Indeed, let y and z two functions in E. We have -F (0) : E → [L 2 (β −3/2 e s α(0, T ); H −1 (0, L) × L 2 (0, L))] 3 is surjective.

Final comments
We finish our paper with some comments and open problems.
-We have proven in Theorem 1.1 the local null controllability of the generalized HS system (1.5).Given the strategy followed in this paper, we have done the best possible: to control the three-equation system with two internal controls.This optimality is clear from the fact that when we linearize we obtain two decoupled subsystems and consequently we need two controls to achieve our results.-A very nice open problem is to get the control of the generalized HS system (1.5) using only one control input.To do that, the strategy used here is not good enough as explained in the previous point.A possible strategy is the use of nonlinear arguments as the return method as done for instance in [10,11] for parabolic systems and in [24] for hyperbolic systems.This strategy should be also useful to control the HS system ) and 56 φ(t) > 55 φ(t) in (0, T ), (3.7)where φ(t) := min x∈[0,L] ϕ(t, x) and φ(t) := max x∈[0,L] ϕ(t, x).