Exact Boundary Controllability for the Boussinesq Equation with Variable Coefficients

In this paper we study the exact boundary controllability for the following Boussinesq equation with variable physical parameters: \begin{array}{lll} \rho(x)y_{tt}=-(\sigma(x)y_{xx})_{xx}+(q(x)y_x)_x-(y^2)_{xx},&&t>0,~x\in(0,l),\\ y(t,0)=\sigma(l)y_{xx}(t,0)=y(t,l)=0,~~\sigma(l)y_{xx}(t,l)=u(t)&&t>0, \end{array} where $l>0$, the coefficients $\rho(x)>0,\sigma(x)>0 $, $q(x)\geq0$ in $[0,l]$ and $u$ is the control acting at the end $x=l$. We prove that the linearized problem is exactly controllable in any time $T>0$. Our approach is essentially based on a detailed spectral analysis together with the moment method. Furthermore, we establish the local exact controllability for the nonlinear problem by fixed point argument.


Introduction
Let T > 0 and l > 0. The classical Boussinesq equation on the bounded domain (0, l) is of the form y tt + σy xxxx − y xx + y 2 xx = 0, (t, x) ∈ (0, T ) × (0, l), (1.1) known as the "good" or the "bad" Boussinesq equation. In fact, the "bad" Boussinesq equation is linearly unstable and admits the inverse scattering approach [10,25]. For this reason, we only consider the version of the "good" Boussinesq equation with variable coefficients. These equations arise as a model of nonlinear vibrations along a beam [25], and also for describing electromagnetic waves in nonlinear dielectric materials [23]. From a mathematical point of view, well-posedness and dynamic properties of "good" Boussinesq equations have a huge literature, see the paper by Bona and Sachs [6], see also [1,19] and references therein.
We are concerned with the exact boundary controllability for the "good" Boussinesq equation with variable coefficients. In this direction, the case of the linear "good" Boussinesq equation with constant coefficients has been investigated by Lions in [17]. In that reference, by Hilbert Uniqueness Method "Lions'HUM" (cf. Lions [16,17]), it was proved that the linearized "good" Boussinesq system          y tt = −y xxxx + y xx , (t, x) ∈ (0, T ) × (0, l), y(t, 0) = y xx (t, 0) = 0, t ∈ (0, T ), y(t, l) =ũ(t), y xx (t, l) = u(t), t ∈ (0, T ), y(0, x) = y 0 , y t (0, x) = y 1 , x ∈ (0, l), is exactly controllable in any time T > 2(l + 1 √ λ 0 ), where the two controls (ũ, u) ∈ L 2 (0, T ) × H 1 (0, T ) and λ 0 denotes the first eigenvalue of the operator −∆ with the Dirichlet boundary conditions. Later on, Zhang [24] studied the question of distributed control for the generalized "good" Boussinesq equation with constant coefficients on a periodic domain. A few years after, Crépeau extended in [8] the results obtained in [17]. More precisely, by a detailed spectral analysis and the use of nonharmonic Fourier series, it was shown that System (1.2)-(1.5) (forũ ≡ 0) is exactly controllable at any time T > 0, where the control u ∈ L 2 (0, T ). Furthermore, with the help of the fixed point theorem, it was also proved the local controllability for the nonlinear control problem (1.1), (1.3) and (1.4) (for σ ≡ 1 andũ ≡ 0). Concerning the control of the approached "bad" Boussinesq equation, the controllability properties for the so called improved Boussinesq equation have been obtained recently by Cerpa and Crépeau [9].
In the previous studies of controllability, the coefficients of the "good" Boussinesq equation are supposed to be constant. In the present paper, we address the problem of exact boundary controllability for the "good" Boussinesq equation with variable coefficients. More precisely, we consider the following control problem x ∈ (0, l), where u is a control placed at the extremity x = l, and the functions y 0 , y 1 are the initial conditions. Here and in what follows, we assume that the coefficients and there exist constants ρ 0 , σ 0 > 0, such that In this paper we prove that the linearized problem x ∈ (0, l), is exactly controllable in any time T > 0, where the control u ∈ L 2 (0, T ) and the initial conditions (y 0 , y 1 ) taken in H 1 0 (0, l) × H −1 (0, l). Our approach is essentially based on a detailed spectral analysis and the qualitative theory of fourth-order linear differential equations due to Leighton and Nehari [20]. More precisely, we prove that all the eigenfrequencies √ λ n n≥1 associated System (1.9) (without control) are simple, and by a precise computation of its asymptotics we show that the spectral gap " λ n+1 − √ λ n " is of order O(n). Moreover, we prove that the first derivative of each eigenfunction φ n , n ≥ 1, associated with uncontrolled system does not vanish at the end x = l. As a consequence of the theory of non-harmonic Fourier series and an extension of Ingham's Theorem due to Haraux [12], we establish the equivalence between the H 1 0 × H −1 -norm of the initial data (ỹ 0 ,ỹ 1 ) and the quantity T 0 |ỹ x (t, l)| 2 dt, whereỹ is the solution of System (1.9) without control. Finally, we apply the Lions'HUM to deduce the exact controllability result for the system (1.9). At the end of this paper, we will discuss the local exact controllability for the nonlinear control system (1.6). To this end, we use some results obtained by Crépeau [8] together with standard fixed-point method (e.g., [7,Chapter 4] and [22]).
The rest of the paper is divided in the following way : In section 2, we establish the well-posedness of System (1.9) without control. In the next section, we prove the simplicity of all the eigenvalues (λ n ) n≥1 and we determinate the asymptotics of the associated spectral gap. In section 4, we prove the exact controllability result for the linear control problem (1.9). The last section is devoted to the local controllability for the nonlinear control problem (1.6).
First of all, let us define by L 2 ρ (0, l) the space of functions f such that l 0 |f | 2 ρ(x)dx < ∞, and we denote by H k (0, l) the L 2 ρ (0, l)−based Sobolev spaces for k > 0. We consider the following Sobolev space endowed with the norm u H 2 (0,l)∩H 1 0 (0,l) = u ′′ L 2 ρ (0,l) . It is easily seen from Rellich's theorem that the space H 2 (0, l) ∩ H 1 0 (0, l) is densely and compactly embedded in the space L 2 ρ (0, l). In the sequel, we introduce the operator A defined in L 2 ρ (0, l) by setting : which is dense in L 2 ρ (0, l).

Lemma 2.1
The linear operator A is positive and self-adjoint such that A −1 is compact. Moreover, the linear operator A 1 2 generates a strongly continuous semi-group on L 2 ρ (0, l).
Proof. Let y ∈ D(A), then by integration by parts, we have since σ > 0 and q ≥ 0 then Ay, y L 2 ρ (0,l) > 0, and hence the quadratic form has a positive real values, so the linear operator A is symmetric. Furthermore, it is easy to . Lemma 2.1 leads to the following corollary.

Corollary 2.2
The spectrum of the operator A is discrete. It consists of a sequence of positive eigenvalues (λ n ) n∈N * tending to +∞ : Moreover, the corresponding eigenfunctions (Φ n ) n≥1 can be chosen to form an orthonormal basis in L 2 ρ (0, l).
We give now a characterization of some fractional powers of the linear operator A which will be useful to give a description of the solutions of Problem (2.1) in terms of Fourier series. According to Lemma 2.1, the operator A is positive and self-adjoint, and hence it generates a scale of interpolation spaces H θ , θ ∈ R. For θ ≥ 0, the space H θ coincides with D(A θ ) and is equipped with the norm u 2 θ = A θ u, A θ u L 2 ρ (0,l) , and for θ < 0 it is defined as the completion of L 2 ρ (0, l) with respect to this norm. Furthermore, we have the following spectral representation of space H θ , where θ ∈ R, and the eigenfunctions (Φ n ) n≥1 are defined in Corollary 2.2. In particu- . Obviously, the linear problem (2.1) can be rewritten in the abstract form . As a consequence of the spectral decomposition of the operator A and by [14, Theorem 1.1], we have the following existence and uniqueness result for Problem (2.1) and is given by the following Fourier series where y 0 = n∈N\{0} a n φ n and y 1 = n∈N\{0} b n φ n .

Spectral Analysis
In this section, we shall establish the spectral proprieties related to System (2.1). To this end we need some results of the qualitative theory of fourth-order linear differential equations due to Leighton and Nehary [20], see also [2,4,5]. We consider the following spectral problem which arises by applying separation of variables to System (2.1), Our first main result in this section is the following : In order to prove this theorem, we need the following result.
In the case q ≥ 0 or if the second-order equation transform (3.1) into the equation where σ(x), h(x), ρ(x) are taken as functions of s, · := d ds and ϕ := φ(x(s)). Furthermore, we have the following relations : We are now ready to prove Theorem 3.1.
It follows from the second statement of Lemma 3.2, that ϕ n (0) = 0, but this contradicts the first boundary condition in (3.17). Hence, Φ ′ n (l)TΦ n (l) < 0 for all n ∈ N\{0}. This finalizes the proof of the theorem.
The proof is complete.

Exact controllability for the Linear Problem
The goal of this section is to prove the exact controllability for the linear control problem (1.9). To this end, we first prove the observability results which are consequence of the spectral properties given in Section 3.  where y is the solution of Problem (2.1).
In order to prove Proposition 4.1, we need the following variant of Ingham's inequality due to Haraux [12].
Lemma 4.2 [12] Let f (t) = n∈Z c n e −iλnt , where (λ n ) n∈Z is a sequence of real numbers.
We assume that there exist N ∈ N, β > 0 and ̺ > 0 such that
Proof of Proposition 4.1. Let us first recall from the spectral representation of the space H θ that Furthermore, by the second statement of Theorem 3.1, we have Φ ′ n (l) ≡ 0 for all n ∈ N\{0}. Thus by (3.31), there exists m, M > 0 such that Consequently, which proves (4.1). This completes the proof.
Let us now state the existence and uniqueness result for the control system (1.9). Following [14, Theorem 2.14] and [8], we define a weak solution to the control system (1.9) using the method of transposition. This result is basically well known for ρ = σ = q = 1 (see [8,Proposition 10]). The proof can be also easily extended to the variable coefficient case.
We are now ready to state the main result of this section. Notice that, in view of the fact that (1.9) is linear and reversible in time, this system is exactly controllable if and only if the system is null controllable. Proof Following [8,Proposition 11], we apply the Lions'HUM [17], then the control problem is reduced to the obtention of the observability inequalities (4.1) for the uncontrolled system (2.1). Therefore, Theorem 4.4 immediately follows from Proposition 4.1.

Controllability for the Nonlinear Problem
In this section we prove the local exact controllability for the nonlinear control problem (1.6). First of all, we introduce the following space : The main result of this section is stated as follows : In order to prove Theorem 5.1, we need the following result whose the proof is similar to that [8, Proposition 12].
To prove the theorem it suffices to show that F has a fixed point. Furthermore, using (5.5), (5.7), (5.9) and (5.10), it can be shown by a straightforward computation that there exists a constantk > 0 such that Since we aim to use Banach fixed point theorem to the restriction of F to the ball B (0, R), then the constants r > 0 and R > 0 can be chosen in (5.11) and (5.12), such thatk(2r + R 2 ) ≤ R and 2Rk < 1. Let R = 1 4k and r = 3R 2 2 , then by (5.11) and (5.12), one gets which implies by Banach fixed point theorem that F has a unique fixed point. The proof of the theorem is completed.