On the controllability of a Boussinesq system for two-way propagation of dispersive waves

Abstract

In this paper, we are concerned with a Boussinesq system of Benjamin–Bona–Mahony type equation, posed on a bounded interval, modeling the two-way propagation of surface waves in a uniform horizontal channel filled with an irrotational, incompressible and inviscid liquid under the influence of gravitation. The main focus is on the boundary controllability property, which corresponds to the question of whether the solutions can be driven to a given state at a given final time by means of controls acting at one endpoint of the interval. We first show that the system is not spectrally controllable. This means that no finite linear combination of eigenfunctions associated with the state equations, other than zero, can be steered to zero. Although the system is not spectrally controllable, it can be shown that it is approximately controllable, i.e., any state can be steered arbitrarily close to another state.

Appendices

Appendix

The results presented in this section were obtained in [16]. For the sake of completeness, they were included in the paper.

Appendix A: Study of some initial value problems

This section is devoted to present some explicit formulas and properties of a family of initial values problems depending on several parameters. These results will allow us to obtain the asymptotic behavior of the eigenvalues and eigenfunctions of the differential operator associated with (4) in the previous sections. Firstly, we study the properties of the following simple initial value problem, where $\sigma _1\in {\mathbb {C}}^*$ is a complex nonzero parameter:

$$\begin{aligned} \left\{ \begin{array}{ll} -b\varphi _{xx}+\sigma _1 v_{x}=f, \quad x\in (0,2\pi )\\ -dv_{xx}+\sigma _1 \varphi _{x}=g, \quad x\in (0,2\pi ) \\ \varphi (0)=\varphi ^0,\, \, \varphi _x(0)=\varphi ^1\\ v(0)=v^0,\,\, v_x(0)=v^1. \end{array} \right. \end{aligned}$$

(40)

In (40) and in the remaining part of the article, b and d denote two positive real numbers. We have the following result.

Lemma A.1

Given $\left( \begin{array}{cc} \varphi ^0 \\ \varphi ^1 \\ v^0\\ v^1 \end{array}\right) \in {\mathbb {C}}^4$ and $\left( \begin{array}{cc} f\\ g\end{array}\right) \in (L^2(0,2\pi ))^2$, there exists a unique solution $\left( \begin{array}{c} \varphi \\ v \end{array}\right)$ of problem (40) given by the following formula

$$\begin{aligned} \begin{array}{c} \left( \begin{array}{cc} \varphi (x)\\ \\ v(x)\end{array}\right) = \left( \begin{array}{cc} \varphi ^0+ \displaystyle \frac{\sqrt{bd}}{\sigma }\sinh \left( \frac{\sigma x}{\sqrt{bd}}\right) \varphi ^1+ \frac{d}{\sigma }\left( \cosh \left( \frac{\sigma x}{\sqrt{bd}}\right) -1\right) v^1\\ \\ v^0+ \displaystyle \frac{b}{\sigma }\left( \cosh \left( \frac{\sigma x}{\sqrt{bd}}\right) -1\right) \varphi ^1 +\frac{\sqrt{bd}}{\sigma }\sinh \left( \frac{\sigma x}{\sqrt{bd}}\right) v^1 \end{array}\right) \\ \\ \\ -\left( \begin{array}{cc} \displaystyle \frac{1}{\sigma } \int _0^x \left[ \sqrt{\frac{d}{b}}\sinh \left( \frac{\sigma (x-s)}{\sqrt{bd}}\right) f(s)+\left( \cosh \left( \frac{\sigma (x-s)}{\sqrt{bd}}\right) -1\right) g(s)\right] \,\mathrm{d}s\\ \\ \displaystyle \frac{1}{\sigma }\int _0^x \left[ \left( \cosh \left( \frac{\sigma (x-s)}{\sqrt{bd}}\right) -1\right) f(s)+ \sqrt{\frac{b}{d}}\sinh \left( \frac{\sigma (x-s)}{\sqrt{bd}}\right) g(s)\right] \,\mathrm{d}s\end{array}\right) . \end{array} \end{aligned}$$

(41)

In the remaining part of the paper, C denotes a positive constant that may change from one line to another, but it is independent of the parameter $\sigma$ and the initial data.

We define the set

$$\begin{aligned} Z=\left\{ z\in {\mathbb {C}}\,:\, |z|\ge \frac{1}{2},\,\, |\mathfrak {R}(z)|\le 1\right\} . \end{aligned}$$

(42)

Next, we consider system (40) with $f\equiv g\equiv 0$

$$\begin{aligned} \left\{ \begin{array}{ll} -b\varphi _{xx}+\sigma v_{x}=0, \quad x\in (0,2\pi )\\ -dv_{xx}+\sigma \varphi _{x}=0, \quad x\in (0,2\pi ) \\ \varphi (0)=\varphi ^0,\,\,\, v(0)=v^0\\ \varphi _x(0)=\varphi ^1,\,\,\, v_x(0)=v^1, \end{array} \right. \end{aligned}$$

(43)

and the following system

$$\begin{aligned} \left\{ \begin{array}{ll}\xi -b\xi _{xx}+\sigma \zeta _{x}=0, \quad x\in (0,2\pi )\\ \zeta -d\zeta _{xx}+\sigma \xi _{x}=0, \quad x\in (0,2\pi ) \\ \xi (0)=\xi ^0,\, \, \xi _x(0)=\xi ^1,\\ \zeta (0)=\zeta ^0,\,\, \zeta _x(0)=\zeta ^1. \end{array} \right. \end{aligned}$$

(44)

Then, the difference between the solutions of (44) and (40) is given by the following result.

Proposition A.1

Let $\left( \begin{array}{c} \varphi \\ v \end{array}\right)$ and $\left( \begin{array}{cc} \xi \\ \zeta \end{array}\right)$ solutions of (40) and (44), respectively, with $f\equiv g\equiv 0$. Then, there exists a positive constant $C > 0$, such that

$$\begin{aligned} \displaystyle | \xi _x(x) -\varphi _x (x)| + \displaystyle | \zeta _x(x) - v_x (x)|&\le \frac{C}{|\sigma _1|} ( |\xi ^1| +|\zeta ^1|) + C | \sigma _1 -\sigma | \left( |\varphi ^1|+|v^1| \right) \nonumber \\ \nonumber \\&\qquad + C\left[ \displaystyle | \xi ^1 -\varphi ^1| + \displaystyle | \zeta ^1 - v^1| + \vert \xi ^0 \vert + \vert \zeta ^0 \vert \right] , \end{aligned}$$

(45)

where $\sigma , \sigma _1\in Z$.

Appendix B: Spectral analysis of the operator ${\mathcal {A}}$

Given $b,d>0$, we define the operators $\widetilde{{\mathcal {A}}},\,\, {\mathcal {B}} :(H_0^1(0,2\pi ))^2\rightarrow (H_0^1(0,2\pi ))^2$ given by

$$\begin{aligned}&\widetilde{{\mathcal {A}}}=-\left( \begin{array}{cc} 0 &{}\quad \left( I-b\partial ^2_{x}\right) ^{-1}\partial _x \\ \\ \left( I-d\partial ^2_{x}\right) ^{-1}\partial _x &{}\quad 0 \end{array}\right) ,\nonumber \\&\quad \, {\mathcal {B}}=-\left( \begin{array}{cc} 0 &{} \left( -b\partial ^2_{x}\right) ^{-1}\partial _x \\ \\ \left( -d\partial ^2_{x}\right) ^{-1}\partial _x &{} 0 \end{array}\right) . \end{aligned}$$

(46)

Recall that, for $\alpha >0$, the operator $(-\alpha \partial _x^2)^{-1}: L^2(0,2\pi ) \rightarrow L^2(0,2\pi )$ defined by

$$\begin{aligned} (- \alpha \partial _x^2)^{-1}\varphi = v \Leftrightarrow \left\{ \begin{array}{l} - \alpha v_{xx}= \varphi \\ v(0)=v(2\pi )=0, \end{array} \right. \end{aligned}$$

is a well-defined, compact operator in $L^2(0,2\pi )$.

In this section and the rest of the paper, $\mu \in {\mathbb {C}}$ is called eigenvalue of the operator $\widetilde{{\mathcal {A}}}$ (${\mathcal {B}}$) if there exists a nontrivial vector $\Phi =\left( \begin{array}{c} \varphi \\ v\end{array}\right) \in (H_0^1(0,2\pi ))^2$, called eigenfunction corresponding to$\mu$, such that $\widetilde{{\mathcal {A}}}\Phi =\mu \Phi$ (${\mathcal {B}} \Phi =\mu \Phi$, respectively). The following two theorems are devoted to the spectral analysis of these operators.

Theorem B.1

The eigenvalues of the operator ${\mathcal {B}}$ are ${\widetilde{\mu }}_n=\sqrt{bd}n\,i$ with $n\in {\mathbb {Z}}^*$. Each eigenvalue ${\widetilde{\mu }}_n$ is double and has two independent eigenfunctions given by

$$\begin{aligned} {\widetilde{\Phi }}_n^1=\frac{b}{{\widetilde{\mu }}_n} \left( \begin{array}{cc} \displaystyle \sqrt{\frac{d}{b}}\sinh \left( \frac{{\widetilde{\mu }}_n x}{\sqrt{bd}}\right) \\ \displaystyle \cosh \left( \frac{{\widetilde{\mu }}_nx}{\sqrt{bd}}\right) -1 \end{array}\right) ,\,\, {\widetilde{\Phi }}_n^2=\frac{d}{{\widetilde{\mu }}_n} \left( \begin{array}{cc} \displaystyle \cosh \left( \frac{{\widetilde{\mu }}_nx}{\sqrt{bd}}\right) -1\\ \displaystyle \sqrt{\frac{b}{d}}\sinh \left( \frac{{\widetilde{\mu }}_nx}{\sqrt{bd}}\right) \end{array}\right) \quad (n\in {\mathbb {Z}}^*).\nonumber \\ \end{aligned}$$

(47)

Moreover, the sequence $({{\widetilde{\Phi }}}_n^j)_{n\in {\mathbb {Z}}^*,\, j\in \{1,2\}}$ forms an orthonormal basis of $(H^1_0(0,2\pi ))^2$.

We pass to analyze the spectral properties of the operator $\widetilde{{\mathcal {A}}}$. The main difference with respect to ${\mathcal {B}}$ is that we do not have an explicit representation formula as (47) for the eigenfunctions of $\widetilde{{\mathcal {A}}}$. In order to complete the task, it was used a strategy which combines two-dimensional versions of the shooting method and Rouché’s Theorem.

Theorem B.2

The eigenvalues of the operator $\widetilde{{\mathcal {A}}}$ are purely imaginary numbers $(\mu ^{j}_n)_{n\in {\mathbb {Z}}^*,\, j\in \{1,2\}}$ with the property that

$$\begin{aligned} \mu _n^{j}={\widetilde{\mu }}_n+ {\mathcal {O}}\left( \frac{1}{|n|}\right) \qquad \left( n\in {\mathbb {Z}}^*,\, j\in \{1,2\}\right) . \end{aligned}$$

(48)

Moreover, to each eigenvalue, $\mu ^{j}_n$ corresponds an eigenfunction $\Phi _n^{j}$ given by

$$\begin{aligned} \Phi _n^{j}={\widetilde{\Phi }}_n^j+{\mathcal {O}}\left( \frac{1}{|n|^2}\right) \qquad \left( n\in {\mathbb {Z}}^*,\,\, j\in \{1,2\}\right) , \end{aligned}$$

(49)

with the property that the sequence $(\Phi _n^{j})_{n\in {\mathbb {Z}}^*,\, j\in \{1,2\}}$ forms an orthogonal basis of $(H^1_0(0,2\pi ))^2$.