Pointwise control of the linearized Gear-Grimshaw system

In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability when we observe two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability.


Introduction
Wave phenomena occur in many branches of mathematical physics and due to the wide practical applications it has become one of the most important scientific research areas. During the past several decades, many scientists developed mathematical models to explain the wave behavior. The Korteweg-de Vries equation (KdV) u t + u xxx + uu x = 0 was first proposed as a model for propagation of unidirectional, one-dimensional, smallamplitude long waves of water in a channel. A few of the many other applications include internal gravity waves in a stratified fluid, waves in a rotating atmosphere, ionacoustic waves in a plasma, among others. Starting in the latter half of the 1960s, the mathematical theory for such nonlinear dispersive wave equations came to the fore as a major topic within nonlinear analysis. Since then physicists and mathematicians were led to derive sets of equations to describe the dynamics of the waves in some specific physical regimes and much effort has been expended on various aspects of the initial and boundary value problems. For instance, since the first coupled KdV system was proposed by Hirota and Satsuma [18,19], it has been studied amply and some important coupled KdV models have been derived. In particular, general coupled KdV models were applied in different fields, such as in shallow stratified liquid: where u = u(x, t) and v = v(x, t) are real-valued functions of the real variables x and t, and a 1 , a 2 , a 3 , b 1 , b 2 and r are real constants with b 1 > 0 and b 2 > 0. System (1.1) was proposed by Gear and Grimshaw [13] as a model to describe strong interactions of two long internal gravity waves in a stratified fluid, where the two waves are assumed to correspond to different modes of the linearized equations of motion. It has the structure of a pair of KdV equations with linear coupling terms and has been object of intensive research in recent years. We refer to [3] for an extensive discussion on the physical relevance of the system in its full structure.
1.1. Setting of the problem. In this paper we are mainly concerned with the study of the pointwise controllability of the linearized Gear-Grimshaw system posed on the unit circle T: where a, c, d, r are given positive constants, δ x 0 denotes the Dirac delta function centered in a given point x 0 ∈ T and f, g are the control functions. More precisely, the purpose is to see whether one can force the solutions of those systems to have certain desired properties by choosing appropriate control inputs. Consideration will be given to the following fundamental problem that arises in control theory, as proposed by Haraux in [15]: Pointwise control problem: Given x 0 ∈ T, T > 0 and (u 0 , v 0 ), (u T , v T ) in L 2 (T) × L 2 (T), can we find appropriate f (t) and g(t) in a certain space such that the corresponding solution (u, v) of (1.2) satisfies u(T ) = u T and v(T ) = v T ?
If we can always find control inputs to drive the system from any given initial state (u 0 , v 0 ) to any given terminal state (u T , v T ), then the system is said to be pointwise controllable.
1.2. State of the art. As far as we know, the internal controllability problem for the system (1.1) remains open. By contrast, the study of the boundary controllability properties is considerably more developed. Indeed, the first result was obtained in [28], when the model is posed on a periodic domain and r = 0. In this case, a diagonalization of the main terms allows us to decouple the corresponding linear system into two scalar KdV equations and to use the previous results available in the literature. Later on, Micu et al. [29] proved the local exact boundary controllability property for the nonlinear system, posed on a bounded interval. Their result was improved by Cerpa and Pazoto [9] and by Capistrano-Filho et al. [7]. By considering a different set of boundary conditions, the same boundary control problem was also addressed by the authors in [8]. We note that, the results mentioned above were first obtained for the corresponding linearized systems by applying the Hilbert Uniqueness Method (HUM) due to J.-L. Lions [25], combined with some ideas introduced by Rosier in [31]. In this case the problem is reduced to prove the so-called "observability inequality" for the corresponding adjoint system. The controllability result was extended to the full system by means of a fixed point argument.
The internal stabilization problem has also been addressed; see, for instance, [5,10,30] and the references therein. Although controllability and stabilization are closely related, one may expect that some of the available results will have some counterparts in the context of the control problem, but this issue is open. Particularly, when the model is posed on a periodic domain, Capistrano-Filho et al. [5], designed a time-varying feedback law and established the exponential stability of the solutions in Sobolev spaces of any positive integral order by using a Lyapunov approach. This extends an earlier theorem of Dávila [10] also obtained in H s (T), for s ≤ 2. The proof follows the ideas introduced in [24] for the scalar KdV equation by using the infinite family of conservation laws for this equation. These conservation laws lead to the construction of suitable Lyapunov functions that give the exponential decay of the solutions. In [5] the Lyapunov approach was possible thanks to the results established by Dávila and Chavez [11]. They proved that, under suitable conditions on the coefficients of the system, the system also has an infinite family of conservation laws.
1.3. Main results. As we mentioned before, no results are available in the literature on the internal controllability of the Gear-Grimshaw system. In this work we use spectral analysis and Fourier series to prove some results of pointwise controllability for the system (1.2).
Fourier series are considered to be very useful in linear control theory (see, e.g. [32] and its references). In particular, a classical generalization of Parseval's equality, given by Ingham [20], and its many recent variants are very efficient in solving many control problems where other methods do not seem to apply. An outline of this theory is presented in [1,23,26].
We also prove some new results concerning the use of harmonic analysis in the framework of dispersive systems. In this spirit, we derive the controllability of the linearized Gear-Grimshaw system posed on the unit circle T. As it was pointed out earlier by Haraux and Jaffard [15,16,17] by studying some other systems, for non-periodic boundary conditions the controllability properties may heavily depend on the location of the observation or control point.
One of the main result provides a sharp positive answer for the controllability issue mentioned in the beginning of this introduction. Theorem 1.1. For almost all quadruples (a, c, d, r) ∈ (0, ∞) 4 the following property holds. For any fixed there exist control functions f, g ∈ L 2 loc (R) such that the solution of (1.2) satisfies the final conditions u(T ) = u T and v(T ) = v T .
We prove this theorem by applying the Hilbert Uniqueness Method (HUM) due to J.-L. Lions [25] (see also Dolecki and Russell [12]) that reduces the controllability property to the observability of the homogeneous dual problem More precisely, Theorem 1.1 will be obtained as a consequence of Theorem 1.2. For almost all quadruples (a, c, d, r) ∈ (0, ∞) 4 the following properties hold.
(i) Given any (u 0 , v 0 ) ∈ H, the system (1.3) has a unique solution (u, v) ∈ C b (R, H), and the linear map The energy of the solution, defined by the formula does not depend on t ∈ R. (iii) For every solution and x 0 ∈ T the functions u(·, x 0 ) and v(·, x 0 ) are well defined in L 2 loc (R). (iv) For every non-degenerate bounded interval I there exist two positive constants α, β such that for all solutions of (1.3) and for all x 0 ∈ T.
By applying a general method [22], analogous to HUM, Theorem 1.2 will also imply the pointwise exponential stabilizability of (1.2): (i) Given any H), and the linear map for all solutions and for all t ≥ 0.
Another relevant result of this work is a uniqueness theorem when only one function, u(·, x 0 ) or v(·, x 0 ), is observed. It is known that Fourier series and, in particular, Ingham type inequalities, are very efficient in solving many control problems when the frequencies satisfy a uniform gap condition. However, in some control problems only some weakened gap conditions are satisfied; see, for instance, [1,4,21]. Under such assumptions we may still get some weakened observability theorems. Theorem 1.4 is obtained by applying a general result of this type. Similarly to Theorem 1.2, Theorem 1.4 implies some exact controllability and exponential stabilizability results by acting in only one of the equations. (ii) The above results remain valid for the scalar KdV equation. We will not present the proofs because they are similar (and simpler) than the proofs given in this paper. (iii) If we require more regularity on the initial data, say (u 0 , v 0 ) ∈ H 2 (T) × H 2 (T), then the results obtained for the linear system allow us to prove the local controllability of the nonlinear system by means of a fixed point argument. The proof is similar to that of [6, Theorem 2.2], and hence it will be omitted.
The plan of the present article is the following. In Section 2 we present some known and new vectorial Ingham type theorems which will form the basis of the proofs of our observability and uniqueness theorems. Then Theorems 1.2, 1.1, 1.3 and 1.4 will be proved (in strengthened forms) in Sections 3, 4, 5 and 6, respectively.

A review of Ingham type theorems
Given a family Ω = (ω k ) k∈K := {ω k : k ∈ K} of real numbers, we consider functions of the form where I is some given bounded interval.
We recall the definition of the upper density D + = D + (Ω) of Ω. For each ℓ > 0 we denote by n + (ℓ) the largest number of exponents ω k that we may find in an interval of length ℓ, and then we set It can be shown (see, e.g., [1, p. 57] or [23, p. 174]) that It follows from the definition that D + is subadditive: for any families Ω 1 and Ω 2 . If Ω is uniformly separated, i.e., if then D + ≤ 1/γ, and hence D + < ∞. Henceforth, unless stated otherwise, by a complex family we mean a square summable family of complex numbers.
First we recall a classical theorem of Ingham and Beurling [20,2].
In fact, the constants α(I) and β(I) depend only on the length of I. The inequalities in (ii) and (iii) are called direct and inverse inequalities, respectively.
Remark 2.2. If A : X → R and B : X → R are two nonnegative functions, defined on some set X, then we will write, following Vinogradov, For example, in the estimates (ii) and (iii) above the integral and the sum may be considered as real-valued functions defined on the vector space of the complex families (c k ) k∈K , and we may write the inequalities in the form respectively, if we do not want to indicate explicitly the constants α(I) and β(I).
We will need an extension of Theorem 2.1 to the more general where Ω has a finite upper density, but it is not necessarily uniformly separated. If D + (Ω) < ∞, then we may enumerate the elements of Ω into an increasing sequence (ω k ), where k runs over a (finite or infinite) subset of Z, formed by consecutive integers. Assume for simplicity that k runs over all integers. Given an arbitrary bounded interval I of length |I|, by [1, Proposition 1.4] or [23, Proposition 9.3] there exists a positive integer M such that Ω satisfies the weakened gap condition Fix 0 < ε ≤ γ M arbitrarily. For each maximal chain ω k , . . . , ω n satisfying ω j+1 − ω j < ε for j = k + 1, . . . , n (note that it has at most M elements), we introduce the divided differences e k (t), . . . , e n (t) by Newton's formulas (see [1] or [23] for more details), and we rewrite the usual exponential sums in the form There exists always a positive constant θ such that for all complex families (c k ) k∈K . On the other hand, there exists a positive constant η such that for all complex families (c k ) k∈K if and only if Ω is uniformly separated.
then choose an arbitrary ε ∈ (0, γ M ], and introduce the functions e k (t) according to (2.5). Then there exist two positive constants α(I) and β(I) such that for all complex families (c k ) k∈K .
As before, the constants α(I) and β(I) depend only on the length of I.
(i) Mehrenberger [27] proved that 2πD + is the critical length for the validity of (iii).
(ii) If the sequence (ω k ) has a uniform gap γ > 0, then choosing ε ∈ (0, γ] every maximal chain is a singleton, so that b k = c k for all k. In this special case Theorem 2.3 reduces to Theorem 2.1. We will also need a vectorial extension of Theorem 2.1 (i), (ii).
Theorem 2.5. Let Ω = (ω k ) k∈K be a uniformly separated family of real numbers, and (Z k ) k∈K a uniformly bounded family of vectors in some Hilbert space H, so that Proof. This is an easy adaptation of the usual proof of Theorem 2.1 (ii); see, e.g., the proof of [23, Theorem 6.1, p. 90].
Finally, we will also need a vectorial generalization of a powerful estimation method of Haraux [14]. Let us consider a finite number of families of real numbers: and corresponding nonzero vectors Z j,k in some Hilbert space H. The following proposition is a special case of [23, Theorem 6.2]: Theorem 2.6. Assume that the families (2.7) are uniformly separated. Furthermore, assume that there exist finite subsets F j ⊂ K j , a bounded interval I 0 , and two positive constants α ′ , β ′ such that Finally, assume that for each exceptional index (j, k) such that k ∈ F j , the exponent ω j,k has a positive distance from the sets {ω j,n : n ∈ K j , n = k} and {ω ℓ,n : n ∈ K ℓ } for all ℓ = j.
Then for each bounded interval I of length > |I 0 | there exist two positive constants α(I) and β(I) such that holds for all complex complex families Remark 2.7. The assumption (2.8) is stronger than the assumption (2.6) of Theorem 2.5. Indeed, applying (2.8) to one-element sums we obtain that for all j and k ∈ K j \ F j . Since F 1 ∪· · ·∪F J is a finite set and the vectors Z j,k are different from zero, we may choose two other positive constants β ′′ ≤ β ′ and α ′′ ≥ α ′ such that for all j and k ∈ K j . Hence (2.6) is satisfied with ∆ = α ′′ / |I 0 |. 8

Pointwise observability
Given four positive constants a, c, d, r, we consider the following system of linear partial differential equations with 2π-periodic boundary conditions: It will be more convenient to write u(t)(x) := u(t, x), and to work on the unit circle T without boundary conditions, i.e., to rewrite our system in the form Let us write (3.1) in the abstract form

and the linear operator
where Z ′ is the time derivative and D is the spatial derivative. The well posedness of (3.1) in most cases follows from the following result.
Proposition 3.1. If ad = 1, then A is an anti-adjoint operator in H for the Euclidean norm given by and hence it generates a group of isometries in H.

Proof. It is clear that
Indeed, for any (ϕ, ψ), (u, v) ∈ D(A), we obtain by integrating by parts the equality It remains to show that D(A * ) = D(−A), i.e., that each (ϕ, ψ) ∈ D(A * ) belongs to D(A).
Pick any (ϕ, ψ) ∈ D(A * ). Then there exists a constant C > 0 satisfying for all (u, v) ∈ D(A) the inequality Choosing v = 0 and u ∈ C ∞ (T) hence we infer from that the distributional derivative ϕ xxx + aψ xxx belongs to L 2 (T). Similarly, choosing u = 0 and v ∈ C ∞ (T) we obtain that the distributional derivative dϕ xxx + rψ x + ψ xxx belongs to L 2 (T). Combining the two relations we obtain that (1 − ad)ψ xxx + rψ x ∈ L 2 (T), and in case ad = 1 this implies that ψ ∈ H 3 (T). Combining this property with the relation ϕ xxx + aψ xxx ∈ L 2 (T) we conclude that ϕ xxx ∈ L 2 (T), and therefore ϕ ∈ H 3 (T).
In order to establish the well posedness of (3.1) in the general case we determine the eigenvalues and eigenfunctions of the operator A. There exists an orthogonal basis There exist non-trivial solutions if and only if 2cω ± k = (c + 1)k 3 − rk ± k 4acdk 4 + [(c − 1)k 2 + r] 2 . If k = 0, then ω − k = ω + k , and two corresponding non-zero eigenvectors are given by the formula If k = 0, then both eigenvalues are equal to zero, and two linearly independent eigenvectors are given for example by the formula Furthermore, we have According to Remark 2.2, the notation Z ± k ≍ 1 means that there exist two positive constants θ and η such that Proof. The orthogonality follows from the orthogonality of the functions e −ikx in L 2 (T) and from the orthogonality relations The latter equalities may be checked directly: we have The limit relations readily follow from (3.4) because k −2 → 0. Since Z ± = 0, they imply the property Z ± k ≍ 1.
Next we establish the well posedness of (3.1). Let us denote by C b (R, H) the Banach space of bounded continuous functions R → H for the uniform norm. (i) Given any (u 0 , v 0 ) ∈ H, the system (3.1) has a unique solution (u, v) ∈ C b (R, H), and the linear map The energy of the solution, defined by the formula does not depend on t ∈ R. (iii) The solution is given by the explicit formula with suitable square summable complex coefficients c ± k satisfying the equality and hence the relation Proof. If ad = 1, then the theorem follows from Proposition 3.1 and Lemma 3.2.
The following alternative proof works even if ad = 1. The functions (2π) −1 e −ikx , k ∈ Z form an orthonormal basis in L 2 (T). Furthermore, by Lemma 3.2 the non-zero vectors Z ± k are orthogonal in C 2 , and Z ± k ≍ 1. Hence the functions e ± k (x) := e −ikx Z ± k , k ∈ Z form an orthogonal basis in H, and e ± k ≍ 1.
By a standard method, the theorem will be proved if we show that for any given square summable sequences (c ± k ) the series H) is a Banach space, it suffices to check the uniform Cauchy criterium. Setting the following equality holds for all n > m > 0 and t ∈ R: Using the above mentioned orthogonality properties and the relation Z ± k ≍ 1 hence we infer that The Cauchy property follows by observing that the last expression is independent of t, and that it converges to zero as n > m → ∞ by the convergence of the numerical series The above proof also yields (by taking m = −1) the equalities (3.6)-(3.8).
Now we turn to the question of observability. We need some additional information on the eigenvalues: Proof. Since ω + −k = −ω + k , it suffices to consider the case k → ∞. Since Hence and therefore This implies the first two cases of the lemma because B and 1 − ad have the same sign. If ad = 1, then (3.9) implies the relation Corollary 3.6. We have Proof. By using the definition of the upper density, this follows from Lemmas 3.4 and 3.5.
The rest of this section is devoted to the proof of the following Fix x 0 ∈ T arbitrarily, and consider the solutions of (3.1).

13
(i) The direct inequality holds for all non-degenerate bounded intervals I. (ii) If ad = 1, then the inverse inequality also holds for all non-degenerate bounded intervals I. (iii) If ad = 1, then (3.11) holds for all bounded intervals I of length |I| > 2π(c + 1)/r, and it fails if |I| < 2π(c + 1)/r.
Here we use the notation ≪ introduced in Remark 2.2: two sides of the direct and inverse inequalities may be considered as real-valued functions of the initial data (u 0 , v 0 ) of the problem (3.1).) Remark 3.8. The assumption (3.10) is will not be needed in the proof of the direct inequality. We will show in Lemma 3.9 below that (3.10) is satisfied for almost all (a, c, d, r) ∈ (0, ∞) 4 .
We proceed in several steps.
Proof of the direct inequality in Theorem 3.7. Fix an arbitrary bounded interval I and for brevity, using the Young inequality, applying Theorem 2.1 (ii), and finally using the relation Z ± k ≍ 1 from Lemma 3.2, we have Using (3.8) we conclude that Proof of a weakened version of the inverse inequality in Theorem 3.7. We fix a positive number K whose value will be precised later, and we consider only solutions of the form (3.6) satisfying c ± k = 0 whenever |k| ≤ K, i.e., functions of the form with square summable complex coefficients c ± k . Using the Young inequality and applying Theorem 2.5 we have Since Z + and Z − are orthogonal, and Z ± ≥ 2ac by Lemma 3.2, it follows that Assume that ad = 1. If K is sufficiently large, say K ≥ K 0 , then by Corollary 3.6 we may apply Theorem 2.1 (iii) to conclude with some constant β(I, K) > 0 the estimate Let us observe that if K ≥ K 0 , then β(I, K) ≥ β(I, K 0 ) α(I, K) ≤ α(I, K 0 ) because we have less complex families to consider for K than for K 0 . Therefore we have for every K ≥ K 0 and for all complex families (c ± k ) |k|>K . Since δ K → 0 as K → ∞, we may choose a sufficiently large K ≥ K 0 such that β(I, K 0 ) − α(I, K 0 )δ 2 K > 0. Then using (3.8) we conclude that under the assumption ad = 1 the inverse inequality holds for all functions of the form (3.13).
If ad = 1, then by Corollary 3.6 we may repeat the last reasoning for every bounded interval of length |I| > 2π(c + 1) r .
Proof of the inverse inequality in Theorem 3.7. Thanks to our assumption (3.10) and to Corollary 3.6 we may apply Theorem 2.6 to infer the inverse inequality from the weakened inverse inequality, established in the preceding step.
Proof of the lack of the inverse inequality if ad = 1 and |I| < 2π(c+1) r . For any fixed positive integer K, by Corollary 3.6 and Remark 2.4 (i) we may find square summable complex numbers c − k satisfying the inequality Moreover, since the removal of finitely many frequencies does not alter the upper density, we may also assume that Then, applying the Young inequality and then Theorem 2.5 (ii) with J = 1 and 16 we have the following estimate: Since the expression in front of the sum tends to zero as K → ∞, this estimate shows that the inverse inequality (3.11) does not hold. Theorem 1.2 (iv) now follows from Theorem 3.7 and from the following lemma, showing that the hypothesis (3.10) holds for almost all choices of the parameters a, c, d, r: Lemma 3.9. For almost every quadruple (a, c, d, r) ∈ (0, ∞) 4 , we have ω + k = ω + n and ω − k = ω − n whenever k = n.
For any fixed (a, c, d) ∈ (0, ∞) 3 and for any (k, n) ∈ Z 2 with k = n the polynomial equation (3.16) vanishes for at most three values of r. Therefore for any fixed (a, c, d) ∈ (0, ∞) 3 , ω + k = ω + n and ω − k = ω − n for all k = n for all but countable many values of r. The lemma follows by applying Fubini's theorem.

Pointwise controllability
In this section we study the pointwise controllability of the system (4.1) where a, c, d, r are given positive constants, δ x 0 denotes the Dirac delta function centered in a given point x 0 ∈ T, and f, g are the control functions. We will prove the following Theorem 4.1. Fix x 0 ∈ T arbitrarily, and choose a, c, d, r such that (3.10) be satisfied. Set Given T > T 0 arbitrarily, for every (u 0 , v 0 ), (u T , v T ) ∈ H there exist control functions f, g ∈ L 2 loc (R) such that the solution of (4.1) satisfies the final conditions u(T ) = u T and v(T ) = v T .
Remark 4.2. In case ad = 1 the system is not controllable for any T < T 0 . This follows from Theorem 3.7 and from a general theorem of duality between observability and controllability; see, e.g., [12].
By a general argument of control theory, it suffices to consider the special case of null controllability, i.e., the case where u T = v T = 0. We prove this by applying the Hilbert Uniqueness Method (HUM) of J.-L. Lions [25] as follows. Fix (u 0 , v 0 ) ∈ H arbitrarily. Choose (ϕ 0 , ψ 0 ) ∈ D(A) arbitrarily, and solve the homogeneous adjoint system Then solve the non-homogeneous problem with zero final states (instead of the non-zero initial states in (4.1)) by applying the controls (4.3) f := −ϕ(·, x 0 ) and g := −ψ(·, x 0 ) : and then from Theorem 3.7 that Applying the Lax-Milgram Theorem we conclude that Λ is an isomorphism of H onto itself; in particular, it is onto. (iii) there exist a non-degenerate bounded interval I and a positive constant c I such that the solutions of (5.1) satisfy the inequalities

Pointwise stabilizability
The operator B is usually called an observability operator: We may think that we can observe only BU and not the whole solution U.
Under the assumptions (i), (ii), (iii) we may define the solutions of the dual problem by transposition, were H ′ , G ′ denote the dual spaces of H, G, and A * , B * denote the adjoints of A and B.
The operator B * is usually called a controllability operator: we may think that we can act on the system by choosing a control W .
Motivated by a formal computation, by a solution of (5.2) we mean a function V : R → H ′ satisfying the identity for all U 0 ∈ H and for all s ∈ R. Then for every V 0 ∈ H ′ and W ∈ L 2 loc (R; G ′ ), (5.2) has a unique solution. Moreover, the function V : R → H ′ is weakly continuous. 20 The Hilbert Uniqueness Theorem states that if the inverse inequality of (iii) also holds: (iv) there exist a bounded interval I ′ and a constant c ′ such that the solutions of (5.1) satisfy the inequality then the system (5.2) is exactly controllable in the following sense: Theorem 5.1. Assume (i), (ii), (iii), (iv), and let T > |I ′ | (the length of I ′ ). Then for all V 0 , V 1 ∈ H ′ there exists a function W ∈ L 2 loc (R; G ′ ) such that the solution of (5.2) satisfies V (T ) = V 1 .
We may of course assume that W vanishes outside the interval (0, T ).
Under the assumptions of the theorem we may also construct feedback controls yielding arbitrarily fast decay rates: Theorem 5.2. Assume (i), (ii), (iii), (iv), and fix ω > 0 arbitrarily. There exists a bounded linear map F : H ′ → G ′ and a constant M > 0 such that the problem has a unique weakly continuous solution V : R → H ′ , and the solutions satisfy the estimates Let us observe that Theorems 5.1 and 5.2 have the same assumptions. These assumptions have been verified during the proof of Theorem 1.1. Therefore we may also apply Theorem 5.2 for the Gear-Grimshaw system, and Theorem 1.3 follows.

Use of one control
In this section we establish a variant of Theorem 3.7 when we observe only one of the functions u(·, x 0 ) and v(·, x 0 ). Such observations do not allow us to determine completely the initial data in (3.1). Indeed, if (u, v) solves (3.1), then for any constant c the couple (u, v + c) also solves (3.1) with some other initial data, so that the observation of the component u may allow to determine v up to an arbitrary additive constant. An analogous situation occurs by observing v. Theorem 6.2 and Corollary 6.3 below will show that up to this indeterminacy the determination of the solutions is possible by observing only one component. Since the solutions of (3.1) are given by the formulas where we use the notation (3.12), we need an Ingham type theorem for the family ω ± k : k ∈ Z . It does not have a uniform gap, because ω + 0 = ω − 0 = 0 and because ω + k may be close to ω − n for many couples (k, n), but it satisfies the weakened gap condition of Theorem 2.3 with M = 2.
Given a positive number ε we consider in the set Ω := ω ± k : k ∈ Z the following equivalence relation: x ∼ y if x = y, or if there exists a finite sequence x 1 , . . . , x n in Ω such that x = x 1 , x n = y, and |x j+1 − x j | < ε for j = 1, . . . , n − 1. Lemma 6.1.
(i) We have z ± k,j ≍ 1, j = 1, 2. (ii) For almost every quadruple (a, c, d, r) ∈ (0, ∞) 4 , we have (iii) Assume (6.2) and (6.3). If ε sufficiently small, then each equivalence class of Ω has one or two elements. Moreover, if it has two elements, then one of them belongs to ω + k : k ∈ Z , and the other one belongs to ω − k : k ∈ Z . Proof. (i) This follows from the explicit expression (3.4) of these vectors.
(ii) In view of Lemma 3.9 we only need to consider the property (6.3). For this we adapt the proof of Lemma 3.9 as follows.
(iii) For any fixed ε > 0, by Lemmas 3.4 and 3.5 there exists a sufficiently large positive integer K such that ω + k − ω + n ≥ 2ε and ω − k − ω − n ≥ 2ε whenever |k| , |n| > K and k = n. Then each equivalence class in the restricted set ω ± k : |k| > K has at most two elements. Indeed, if two elements are equivalent, then they have to belong to the different families ω + k and ω − k , say ω + k − ω − n < ε. Then we infer from our choice of K that ω + k − ω − m > ε for all m = n and ω + m − ω − n > ε for all m = k, so that no other exponent is equivalent to ω + k and ω − n . This property remains valid if we change ε to a smaller positive value. Indeed, each one-point equivalence class remains the same, while the others either remain the same or they split into two one-point equivalence classes.
Using this and Lemma 6.1, the theorem follows by applying Theorem 2.3 with M = 2.
(i) If u(t, x 0 ) = 0 for all t ∈ I, then u = 0 and v is an arbitrary constant function.
(ii) If v(t, x 0 ) = 0 for all t ∈ I, then v = 0 and u is an arbitrary constant function.
Similarly, if u(t, x 0 ) = 0 for all t ∈ I, then we obtain that c + 0 − c − 0 = 0, and c ± k = 0 for all k ∈ Z * . This implies that v = 0 and u is an arbitrary constant function.
We end this paper by proving two variants of Theorem 4.1 where we apply only one control. Let us observe that if f = 0 in (4.1), then T u(t, x) dx does not depend on t ∈ R because d dt T u dx = − T u xxx + av xxx dx = 0 24