Stabilization of the wave equation with moving boundary

TL;DR

This paper presents a feedback control method to exponentially stabilize the wave equation with a periodically moving boundary, ensuring energy decay, based on a reduction theorem of Yoccoz.

Contribution

It introduces a novel feedback control strategy for wave equations with moving boundaries, leveraging Yoccoz's reduction theorem for stability analysis.

Findings

Exponential decay of energy achieved under the proposed feedback.

Stability holds for boundaries moving slower than the speed of light.

Provides insights into moving-pointwise stabilization problems.

Abstract

We deal with the wave equation with assigned moving boundary ($0<x<a(t)$) upon which Dirichlet-Neuman boundary conditions are satisfied, here $a(t)$ is assumed to move slower than the light and periodically. We give a feedback which guarantees the exponential decay of the energy. The proof relies on a reduction theorem of Yoccoz. At the end we give a remark on the moving-pointwise stabilization problem.

Equations

$$\left\{\begin{array}[]{lll}u_{tt}-u_{xx}=0\;\;\;\;\mbox{for}\;\;\;\;0<x<a(t),\,t>0,\\ u(0,t)=0\;\;\;\;\mbox{and}\;\;\;\;u_{t}(a(t),t)+f(t)u_{x}(a(t),t)=0,\,t>0,\\ u(x,0)=\phi(x),u_{t}(x,0)=\psi(x),\;\;0<x<a(0),\\ \end{array}\right.$$

$$H_l^1((0,a(0)))=\left\{v\in H^1((0,a(0))),v(0)=0\right\}.$$

$$E_u(t)=\frac{1}{2}{\int }_0^{a(t)}\left[{\left|u_t(x,t)\right|}^2+{\left|u_x(x,t)\right|}^2\right]dx$$

$$L(F):=\underset{x,y\in \text{mbox}\text{sr}R,x\ne y}{\sup }\left|\frac{F(x)-F(y)}{x-y}\right|.$$

$$u(x,t)=h(t+x)-h(t-x)\quad\mbox{a.e. in $Q$},$$

$$F:=(I+a)\circ (I-a{)}^{-1}$$

$$\rho (F)=\underset{n\to \infty }{\lim }\frac{F^n(x)-x}{n}$$

$$H^{-1}\circ F\circ H(\xi )=\xi +\rho (F).$$

$$f(t)=\frac{(\mu -1)H^{\prime }(a(t)+t)+(\mu +1)H^{\prime }(-a(t)+t)}{(1-\mu )H^{\prime }(a(t)+t)+(\mu +1)H^{\prime }(-a(t)+t)}$$

$${\lambda }_1\le H^{\prime }(t)\le {\lambda }_2,t\in \mathbb{R}.$$

$$E_u(t)\le C{e}^{-\omega t}{E}_u(0),$$

$$E_{u}(t)=0,\mbox{ for all }t\geq T_{0}=:\big{(}I+a\big{)}^{-1}\circ H^{-1}\left(\frac{3\rho(F)}{2}\right).$$

$$a(t):=\left\{\begin{array}[]{ll}\alpha t+\frac{\alpha(1-\alpha)(1+\beta)}{2(\alpha-\beta)}&\mbox{if $\frac{\alpha(1+\beta)}{2(\alpha-\beta)}\leq t\leq\frac{\alpha(1+\beta)-2\beta}{2(\alpha-\beta)},$}\\ \beta t-\beta+\frac{\alpha(1-\beta^{2})}{2(\alpha-\beta)}&\mbox{if $\frac{\alpha(1+\beta)-2\beta}{2(\alpha-\beta)}\leq t\leq\frac{\alpha(3+\beta)-2\beta}{2(\alpha-\beta)}$,}\end{array}\right.$$

$$F(x):=(I+a)\circ(I-a)^{-1}(x)=\left\{\begin{array}[]{ll}l_{1}x+F_{0}&\mbox{if $0\leq x\leq x_{0},$}\\ l_{2}x+F_{0}+1-l_{2}&\mbox{if $x_{0}<x<1,$}\end{array}\right.$$

$$\rho (F)=\frac{\ln l_1}{\ln \left(\frac{l_1}{l_2}\right)},$$

$$H(x)=h_0\ln \left(|x+h_1|\right)+h_2,$$

$$\frac{1}{\ln (\frac{l_1}{l_2})}\frac{l_1-l_2}{l_1}\le H^{\prime }(x)\le \frac{1}{\ln (\frac{l_1}{l_2})}\frac{l_1-l_2}{l_2}.$$

$$\Phi :{\mathbb{R}}^2\to {\mathbb{R}}^2,$$

$$\left\{\begin{array}[]{ll}\xi=(H(x+t)-H(-x+t))/2,\\ \tau=(H(x+t)+H(-x+t))/2,\end{array}\right.$$

$$({\partial }_t^2-{\partial }_x^2)u(x,t)=K(\xi ,\tau )({\partial }_{\tau }^2-{\partial }_{\xi }^2)V(\xi ,\tau )$$

$$4H^{\prime }\circ H^{-1}(\xi +\tau )H^{\prime }\circ H^{-1}(-\xi +\tau )\circ H^{-1}(\xi +\tau ).$$

$$\left\{\begin{array}[]{lll}u_{tt}-u_{xx}=0\;\;\;\;\mbox{for}\;\;\;\;0<x<a(t),\,t>0,\\ u(0,t)=0\;\;\;\;\mbox{and}\;\;\;\;u_{t}(a(t),t)+f(t)\,u_{x}(a(t),t)=0,\,t>0,\\ u(x,0)=\phi_{1}(x),u_{t}(x,0)=\psi_{1}(x),\;\;0<x<a(0)\\ \end{array}\right.$$

$$\left\{\begin{array}[]{lll}v_{\tau\tau}-v_{\xi\xi}=0\;\;\;\;\mbox{for}\;\;\;\;0<\xi<\rho(F)/2,\,\tau>0,\\ v(0,\tau)=0\;\;\;\;\mbox{and}\;\;\;\;v_{\tau}(\rho(F)/2,\tau)+\mu v_{\xi}(\rho(F)/2,\tau)=0,\,\tau>0,\\ v(\xi,0)=\phi(\xi),v_{\tau}(\xi,0)=\psi(\xi),\;\;0<\xi<\rho(F)/2).\end{array}\right.$$

$$\left\{\begin{array}[]{ll}u_{x}(a(t),t)=V_{\xi}(\rho(F)/2,\tau)\xi_{x}(a(t),t)+V_{\tau}(\rho(F)/2,\tau)\tau_{x}(a(t),t),\\ u_{t}(a(t),t)=V_{\xi}(\rho(F)/2,\tau)\xi_{t}(a(t),t)+V_{\tau}(\rho(F)/2,\tau)\tau_{t}(a(t),t).\end{array}\right.$$

$${\xi }_x=({\partial }_x\xi )=({\partial }_t\tau )={\tau }_t=[H^{\prime }(x+t)+H^{\prime }(-x+t)]/2,$$

$${\xi }_t=({\partial }_t\xi )=({\partial }_x\tau )={\tau }_x=[H^{\prime }(x+t)-H^{\prime }(-x+t)]/2,$$

$$u_t(a(t),t)+\frac{{\xi }_t-\mu {\tau }_t}{\mu {\tau }_x-{\xi }_x}{u}_x(a(t),t)=0.$$

$${\xi }_t{u}_x(a(t),t)-{\xi }_x{u}_t(a(t),t)+\mu ({\tau }_x{u}_t(a(t),t)-{\tau }_t{u}_x(a(t),t))=0.$$

$$C_1{E}_V\left(H(a(t)+t)-\frac{\rho (F)}{2}\right)\le E_u(t)\le C_2{E}_V\left(H(a(t)+t)-\frac{\rho (F)}{2}\right),$$

$$E_V(\tau )={\int }_0^{\rho (F)/2}\left({\left|V_{\xi }(\xi ,\tau )\right|}^2+{\left|V_{\tau }(\xi ,\tau )\right|}^2\right)d\xi .$$

Full text

1. 1 Introduction and main result
2. 2 Proof of the main result
3. 3 Moving pointwise stabilization

1. Introduction and main result

In this note, we analyze the stabilization property of solutions for the wave equation with a moving boundary. More precisely, we consider the following system:

[TABLE]

$(\phi,\psi)\in H^{1}_{l}((0,a(0)))\times L^{2}((0,a(0))),$ where

[TABLE]

Here $a$ is a strictly positive real function which is continuous, 1-periodic and f\in L^{\infty}\big{(}\mathbb{R}_{+}^{*}\big{)}.

Denote by

[TABLE]

the energy of the field $u$. Our major concern will be to detect the feedback $f(t)$ necessary to obtain the exponential decay of $E_{u}(t).$

We start with some notations and known results. Let $\mbox{Lip}(\mbox{\sr R})$ be the space of Lipschitz continuous functions on ℝ. We shall denote the Lipschitz constant of a function $F$ by

[TABLE]

On the existence of solutions to the system (1.1), we refer the reader to [9]. We have the following proposition:

Proposition 1.1**.**

*If $a\in\mbox{Lip}(\mbox{\sr R})$, $L(a)\in[0,1)$, $a>0$, f\in L^{\infty}\big{(}\mathbb{R}_{+}^{*}\big{)} and

$(\phi,\psi)\in H_{l}^{1}((0,a(0)))\times L^{2}((0,a(0))),$ denote by $Q:=(0,a(t))\times\mbox{\sr R}_{+}$ and

$Q_{\tau}:=(0,a(t))\times(0,\tau),\tau\in\mbox{\sr R}_{+}$. There exists a unique weak solution111$u\in H^{1}(Q_{\tau})$ is called a weak solution of (1.1) if $u_{tt}-u_{xx}=0$ in ${\mathcal{D}}^{\prime}(Q)$ and the boundary conditions are satisfied. $u$ of the system (1.1). Moreover there exists $h\in H_{\mbox{loc}}^{1}(\mbox{\sr R})\cap L^{\infty}(\mbox{\sr R})$ such that*

[TABLE]

and $u\in L^{\infty}(Q)\cap H^{1}(Q_{\tau})$.

We denote by $D_{p}$ the set of continuous functions and strictly increasing of the form $x+g(x)$, where $g(x)$ is a 1-periodic continuous function.

We recall the following results.

Proposition 1.2**.**

\bigr{(}[10, Herman]** and [12, Yamaguchi]\bigr{)} Let $a$ be a 1-periodic function. Then

[TABLE]

belongs to $D_{p}$. Moreover, the rotation number $\rho(F)$ defined by

[TABLE]

exists, and the limit is equal for all $x\in\mbox{\sr R}$.

After, we construct a transformation of the time-dependent domain $[0,a(t)]\times\mbox{\sr R}$ onto $[0,\rho(F)/2]\times\mathbb{R}$ that preserves the D’Alembertian form of the wave equation. For this purpose, we use the following proposition:

Proposition 1.3**.**

\bigr{(}[10, Herman, section II]\bigr{)}* Assume that $a(t)$ is a 1-periodic function, $a(t)>0$, $a\in\mbox{Lip}(\mbox{\sr R})$ such that $L(a)\in[0,1)$. Assume also that $|a^{\prime}(t)|<1$ for all $t\in\mbox{\sr R}$ and $\rho(F)\in\mbox{\sr R}\setminus\mbox{\sr Q}$ such that there exists a function $H\in D_{p}$ and*

[TABLE]

Our main result is stated now as follows:

Theorem 1.4** (Exponential stability).**

Let

[TABLE]

where $\mu$ is a nonnegative constant and assume that there exist $\lambda_{1}>0$ and $\lambda_{2}>0$ such that

[TABLE]

Then, in the case where $\mu\neq 1$, there exists a positive constant $C$ such that

[TABLE]

for every solution $u$ of (1.1) with initial data $(\phi,\psi)\in H_{l}^{1}((0,a(0)))\times L^{2}((0,a(0)))$ and where $\omega=\ln\left(\displaystyle\left|\frac{1+\mu}{1-\mu}\right|\right)$.

In the case $\mu=1$ which corresponds to $f(t)=1$, we obtain

[TABLE]

We give an example where assumption (1.6) is guaranteed (see [1, 8] for more details).

Example 1.5**.**

Let $a$ be continuous and 1-periodic on ℝ, $a>0$, be such that

[TABLE]

with $-1<\beta<0<\alpha<1$. The function $F$ is defined by:

[TABLE]

with $l_{1}:=\frac{1+\alpha}{1-\alpha}$, $l_{2}:=\frac{1+\beta}{1-\beta}$, $F_{0}:=\frac{l_{2}(l_{1}-1)}{l_{1}-l_{2}}$ and $x_{0}:=\frac{1-l_{2}}{l_{1}-l_{2}}$.

*We extend $F$ through the formula: $F(x+1)=F(x)+1$ for any $x\in\mathbb{R}.$

Also the rotation number is given by the expression:*

[TABLE]

and the function $H$ given by (1.4) is done by

[TABLE]

where $h_{0}=\frac{1}{\ln\left(\frac{l_{1}}{l_{2}}\right)}$, $h_{1}=\frac{l_{2}}{l_{1}-l_{2}}$ and $h_{2}=-\ln\left(h_{1}\right)$. $H$ satisfies the inequalities:

[TABLE]

Here $\displaystyle{f(t)=\frac{2a(t)+2\mu t+2\mu h_{1}}{2\mu a(t)+2t+2h_{1}}}$ and T_{0}=\big{(}I+a\big{)}^{-1}\circ H^{-1}\left(\frac{3\rho(F)}{2}\right).

Remark 1.6**.**

For the moving pointwise stabilization of the wave equation, see section 3.

2. Proof of the main result

Before starting the proof, we begin by defining a domain transformation

[TABLE]

using $H$ given by (1.4), as follows:

[TABLE]

for $(x,t)\in\mathbb{R}^{2}$.

Proposition 2.1**.**

\bigr{(}[13, Yamaguchi]\bigr{)}* The transformation $\Phi$ is a bijection of $[0,a(t)]\times\mathbb{R}$ to $[0,\rho(F)/2]\times\mbox{\sr R}$ and $\Phi$ maps the boundaries $x=0$ and $x=a(t)$ onto the boundaries $\xi=0$ and $\xi=\rho(F)/2$.*

Proposition 2.2**.**

\bigr{(}[13, Yamaguchi]\bigr{)}* Let $u(x,t)$ satisfying $(\partial_{t}^{2}-\partial_{x}^{2})u(x,t)=0$ and $V(\xi,\tau)$ defined by $u(\Phi^{-1}(\xi,\tau))$. Then the following identity holds*

[TABLE]

where $K(\xi,\tau)$ is defined by

[TABLE]

Now we consider the system:

[TABLE]

where $\displaystyle{f(t)=\frac{(\mu-1)H^{\prime}(a(t)+t)+(\mu+1)H^{\prime}(-a(t)+t)}{(1-\mu)H^{\prime}(a(t)+t)+(\mu+1)H^{\prime}(-a(t)+t)}}$ and $\mu$ is a nonnegative constant.

Proposition 2.3**.**

The transformation of the system (2.13) is

[TABLE]

Proof.

We have:

[TABLE]

Starting from (2.13) and make use of:

[TABLE]

[TABLE]

we conclude:

[TABLE]

So we have

[TABLE]

Finally we get: $v_{\tau}(\rho(F)/2,\tau)+\mu v_{\xi}(\rho(F)/2,\tau)=0.$

Note that $f(t)=1$ in the special case $\mu=1$. ∎

The next lemma shows that the energy of the solution (1.1) and the energy of the corresponding static system are equivalent.

Lemma 2.4**.**

\bigr{(}[1, Ammari et al.]\bigr{)}* Under the assumption (1.6), there are two positive constants $C_{1}$ and $C_{2}$ such that*

[TABLE]

where $E_{V}(\tau)$ is the energy of the field $V$ defined by:

[TABLE]

Note that we can write H(a(t)+t)=a(t)+t+g\big{(}a(t)+t\big{)}, where $g(x)$ is a 1-periodic continuous function.

Finally, the stabilization of the system (1.1) is a direct combination of Proposition 2.3, Lemma 2.4 and the following Lemma 2.5.

Lemma 2.5**.**

\bigr{(}[7, Cox and Zuazua]\big{)}* For $\mu\neq 1$, there exists a positive constant $C$ such that*

[TABLE]

where $V$ is the solution of the following system:

[TABLE]

It is well known that for $\mu=1$, $E_{V}(t)=0$ for all $t\geq\rho(F),$ see [7] for more details.

3. Moving pointwise stabilization

We consider the following problem:

[TABLE]

The aim of this section is to determine the functions $f_{1}$, $f_{2}$ and $b$ to get after transformation the vibrations of a string with the static pointwise damping and conclude the asymptotic behavior of the energy.

Proposition 3.1**.**

The transformation of the system:

[TABLE]

is

[TABLE]

where $\Lambda_{t}$ is defined by $\Lambda_{t}(y)=\frac{H(y+t)-H(-y+t)}{2}.$

Proof.

We recall that if $u(x,t)$ satisfying $(\partial_{t}^{2}-\partial_{x}^{2})u(x,t)=0$ and $v(\xi,\tau)$ defined by $u(\Phi^{-1}(\xi,\tau))$. Then the following identity holds

[TABLE]

where $K(\xi,\tau)$ is defined by

[TABLE]

On the other hand, make use of (2.15) we obtain:

[TABLE]

We combine (3.22) and (3.23) to obtain the result of the Proposition 3.1. ∎

Remark 3.2**.**

If we return to the Example 1.5 and after some computation we get that the transformation of the system:

[TABLE]

is the system (3.21).

As above and according to [4, 5] we have the following:

• •

$\displaystyle\lim_{t\rightarrow+\infty}E_{u}(t)=0,\,\forall\,(\phi_{1},\psi_{1})\in H^{1}_{0}(0,b(0))\times L^{2}(0,b(0))\Leftrightarrow\rho(F)\notin\mathbb{Q}.$

• •

For any $\rho(F)/2\in(0,1)$ the system (3.20) is not exponentially stable in $H^{1}_{0}(0,b(0))\times L^{2}(0,b(0))$.

• •

For all $\rho(F)/2\in\mathcal{S}$ 222Denote by $\mathcal{S}$ the set of all numbers $\xi\in(0,1)$ such that $\xi\notin\mathbb{Q}$ and if $[0,a_{1},...,a_{n},...]$ is the expansion of $\xi$ as a continued fraction, then $(a_{n})$ is bounded. Let us notice that $\mathcal{S}$ is is obviously uncountable and, by classical results on diophantine approximation (cf. [6], p. 120), its Lebesgue measure is equal to zero. In particular, by Euler–Lagrange theorem (see Lang [11], p. 57) S contains all $\xi\in(0,1)$ such that $\xi$ is an irrational quadratic number (i.e. satisfying a second degree equation with rational coefficients). and for all $(\phi_{1},\psi_{1})\in\mathcal{D}$ we have according to [2, 3] that there exists $C>0$ such that:

[TABLE]

• •

If $\varepsilon>0$ then, for almost all $\rho(F)/2\in(0,1)$ and for all $(\phi_{1},\psi_{1})\in\mathcal{D}$ we have according to [2, 3] that there exists $C>0$ such that:

[TABLE]

where

[TABLE]

[TABLE]