\def\papertitle{Control of an Euler-Bernoulli beam with a nonlinear tension and an end-mass}
\def\paperauthorA{Billal Lekdim$^{1,2}$ \qquad Ammar Khemmoudj$^{2}$}
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\title{\papertitle}
\affiliation{\paperauthorA}	% This command replaces \name{The DAFx Crew} 
{$^{1}$Department of Mathematics, University Ziane Achour of Djelfa,\\
	 Djelfa 17000, Algeria\\
$^{2}$Laboratory of SDG, Faculty of Mathematics, University of Science\\
      and Technology Houari Boumediene, P.O. Box 32, El-Alia 16111,\\ Bab Ezzouar, Algiers, Algeria}

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	\maketitle
	
\begin{abstract}
In this article, we intend to study the vibrations of a nonlinear Euler-Bernoulli beam without internal damping, fixed at one end and a mass attached at the other end. By applying a suitable control at the free end, we can quickly dampen these vibrations. In fact, we give the result of the exponential stability of the solutions, and we base our method on the multiplier technique.
\end{abstract}
\textbf{Keywords}: exponential stability, Euler-Bernoulli beam, boundary control, nonlinear tension, multiplier method.

\textbf{Mathematics Subject Classification (2010)}: 35L05, 35B40, 93D20

\section{Introduction}
In this paper, we consider a Euler-Bernoulli beam with a nonlinear tension and an mass attached to its free end, The beam is modeled through the following equation:
\begin{equation}
	\begin{cases}
		\rho w_{tt} +EIw_{xxxx}-Tw_{xx} -\frac{3EA}{2}w_{x}^{2}w_{xx}=0, \ 
		&x \in \left( 0,L\right) ,t\geq 0 ,\\[5pt]
		w(0,t) =w_{x}(0,t) =	w_{xx}(L,t) =0,\\
		Mw_{tt}( L,t) -EIw_{xxx}( L,t)+Tw_{x}( L,t)+\frac{EA}{2}w_{x}^{3}(L,t)=U(t)  & t\geq 0 ,\\[5pt]
		w(x,0)=w^{0}(x), \qquad w_{t}(x,0)=w^{1}(x), & x \in \left( 0,L\right),
	\end{cases}  \label{sys}
\end{equation}
where $w(x,t)$  represents the transverse displacements of the beam at a position $x$ for time $t$, $\rho >0$ is the uniform mass
per unit length of the beam, and $L$ is the length of the beam. Also, $T$ is the
axial tension of the beam, $EI$ is the bending stiffness, $EA$ is the
axial stiffness, and the control force $U(t)$ will be specified later.

Recent years have witnessed an increasing cognizance of the problem of controlling thinner structures in various industries, especially slender ones. Domains like transportation and construction have particularly benefited from it. The purpose behind attempts to control these structures is to suppress or at least reduce transverse vibrations which often occur due to the irregular material property or environmental disturbances.

The issue of stability and stabilization has been discussed by many researchers who relied in their studies on several damping terms that induce dissipation. For works that relied on internal damping in order to achieve stability of the system, \cite{Dad2004,Lek,Lek2,Lek3,Li98,Mil2014,Wan2019,Zha1998} can offer decent insights. 

As for the boundary  stabilization case, we have \cite{Guo2001,Guo2002,Lek1,Mor1992,Sha2012}.
Furthermore, for studies that tackle the stabilization and control of Euler-Bernoulli beams with damping, we may mention the works of Gao et al. \cite{Gao}, Karagiannis et al. \cite{Kar} and Miletić et al. \cite{Mil}.

We note also that in \cite{He11} He et al. adopted the linear system of \eqref{sys} with internal viscous damping term $cw_t$. Thus, they established exponential stability under a robust control.

 Seghour et al. \cite{Se15} investigated the linear system of \eqref{sys} with the viscoelastic damping term $\int_{0}^{t}h(t-s)w_{xxxx}(x,s)ds$. The result was establishing an exponential stability result under a nonlinear control. 

The main contribution of this work can be summarized in achieving an exponential stability result result strictly with linear boundary control and without internal damping for the nonlinear problem.


\vspace{0.5em}
\section{Preliminary}
In this paper, $\|.\|$  represents the norm and $(.;.)$ is
the inner product of $L^{2}(0, L)$.
We introduce the energy associated to \eqref{sys} by
\begin{equation}\label{en}
	E(t)=\dfrac{\rho}{2} \|w_{t} \|^{2}+\dfrac{M}{2}w_t(L,t)+\dfrac{EI}{2} \|w_{xx} \|^{2}+\dfrac{T}{2} \|w_{x} \|^{2}+\dfrac{EA}{8} \|w^{2}_{x} \|^{2}.
\end{equation}
Observe that this is the usual classical energy. The first two terms represent kinetic energy  while the rest of the terms represent potential energy.

\subsubsection*{Control}
The control objective is to reduce the free transverse vibrations of the beam. 
Lyapunov’s direct method is used to construct a suitable boundary control  at the free boundary of the flexible beam.

To stabilize system \eqref{sys}), we propose the following control:
\begin{equation}\label{col}
	U(t)=-k_1w_t(L,t)-k_2w_{xt}(L,t),
\end{equation}
where $k_1$ and $k_2$ are positive constants.
\begin{lemma}
	The energy functional \eqref{en} satisfies
	\begin{equation}\label{den}
		E'(t)=w_t(L,t)U(t), \qquad \forall t \geq 0.
	\end{equation}
\end{lemma}	
\begin{proof}
	Taking the inner product of the first equation of \eqref{sys} with $w_{t}$ in 
	$L^{2}(0,L)$, integrating by parts and taking into account the boundary, we
	get \eqref{den}.
\end{proof}	
%
\vspace{0.5em}
\section{Stability Result}
In order to prove the energy decay result.
let us define the Lyapunov functional by
\begin{equation}\label{L}
	\mathcal{L}(t)=E(t)+\beta V(t),
\end{equation}
where $\beta$ is a positive constant and 
\begin{equation}\label{V1}
	V(t)=(xw_x,w_t)+LMw_tw_x(L,t)+Lk_2w_x^{2}(L,t).
\end{equation}
%
%
\begin{proposition}
	There exist two positive constants $a$ and $b$, such that
	\begin{equation}\label{eqv}
		aE(t) \leq \mathcal{L}(t) \leq bE(t), \qquad \forall t\geq 0,
	\end{equation} 
	i.e. $E(t) \sim \mathcal{L}(t).$
\end{proposition}

\begin{proof} This Proposition can be proved by applying Young, Cauchy-Schwarz and Poincaré’s inequalities, as follows
	\begin{eqnarray}
		|V(t)| &\leq &\frac{L}{2}\Vert w_{t}\Vert ^{2}+\frac{L}{2}\Vert w_{x}\Vert
		^{2}+\frac{LM}{2}w_{t}^{2}(L,t)+\left( \frac{LM}{2}+Lk_{2}\right)
		w_{x}^{2}(L,t)  \nonumber \\
		&\leq &\frac{L}{2}\Vert w_{t}\Vert ^{2}+\frac{L}{2}\Vert w_{x}\Vert ^{2}+%
		\frac{LM}{2}w_{t}^{2}(L,t)+\left( \frac{LM}{2}+Lk_{2}\right) L\Vert
		w_{xx}\Vert ^{2}  \nonumber \\
		&&\theta E(t),
	\end{eqnarray}
	where $\theta=\max \{\frac{L}{\rho},\frac{L}{T},L,\frac{LM+2Lk_{2}}{EI}\}.$
	
	Considering $\beta<1/\theta$ and using a proper selections, 
	 we can get \eqref{eqv} with $a=1-\beta \theta$ and $b=1+\beta \theta$.
\end{proof}

\begin{lemma}
	The derivative of $V_1(t)$ satisfies along solutions of system \eqref{sys}
	\begin{eqnarray}\label{dv1}
		V^{\prime }(t) &\leq&-\frac{1}{2}\Vert w_{t}\Vert ^{2}-\frac{3EI}{2}\Vert
		w_{xx}\Vert ^{2}-\frac{T}{2}\Vert w_{x}\Vert ^{2}-\frac{3EA}{8}\Vert
		w_{x}^{2}\Vert ^{2}+MLw_{t}w_{tx}(L,t)  \nonumber \\
		&&+\left[ \frac{L}{2}+\frac{Lk_{1}}{\delta }\right] w_{t}^{2}(L,t)-\left[ 
		\frac{TL}{2}-\frac{Lk_{1}}{4}\delta \right] w_{x}^{2}(L,t)-\frac{EAL}{8}%
		w_{x}^{4}(L,t)
	\end{eqnarray}
\end{lemma}
\begin{proof}
	By differentiating $V_1$, taking into account system \eqref{sys} and integrating by parts, we get 
	\begin{eqnarray}
		V^{\prime }(t) &=&\frac{L}{2}w_{t}^{2}(L,t)-\frac{1}{2}\Vert w_{t}\Vert ^{2}-%
		\frac{3EI}{2}\Vert w_{xx}\Vert ^{2}-\frac{T}{2}\Vert w_{x}\Vert ^{2}-\frac{%
			3EA}{8}\Vert w_{x}^{2}\Vert ^{2}  \nonumber \\
		&&-\frac{TL}{2}w_{x}^{2}(L,t)-\frac{			EAL}{8}w_{x}^{4}(L,t)+MLw_{t}w_{tx}(L,t)-Lk_{1}w_{t}w_{x}(L,t).
	\end{eqnarray}
	After applying the inequality of young to the last term of  the above equation, for $\delta>0$, we obtain \eqref{dv1}.
\end{proof}

\begin{theorem} The energy $E(t)$ satisfies  along the solution of system \eqref{sys}
	\begin{equation}\label{th}
		E(t)\leq A e^{-\lambda t}, \qquad t\geq 0,
	\end{equation}  
	where $A$ and $\lambda$ are two positive constants.
\end{theorem}
\begin{proof}
	Taking the derivative of $\mathcal{L}(t)$, \eqref{col}, \eqref{den} and \eqref{dv1}, and picking $k_2=ML$, we obtain
	\begin{equation}\label{key}
		\mathcal{L}'(t)\leq-c E(t).
	\end{equation}
	Using equivalence relation \eqref{eqv}, the above inequality becomes
	\begin{equation}
		\mathcal{L}'(t)\leq-\dfrac{c}{b} \mathcal{L}(t).
	\end{equation}  
	which gives
	\begin{equation}
		\mathcal{L}(t)\leq \mathcal{L}(0) e^{-\frac{c}{b}t}.
	\end{equation}
	By reusing relation \eqref{eqv}, we find \eqref{th}, 
	in which $A=\frac{\mathcal{L}(0)}{b} $ and $\lambda=\frac{c}{b} $.
\end{proof}

\vspace{0.5em}
\section{conclusions}
This research succeeded in obtaining a linear boundary control in order to achieve the exponential stability of the nonlinear Euler-Bernoulli beam  under free vibrations.

\vspace{0.5em}
\section*{Acknowledgment}
The author thanks DGRSDT for its support.

%\begin{thebibliography}
	%include bibliography file or list
	%\end{thebibliography}
\vspace{0.5em}
\begin{thebibliography}{9}
	
	
	\bibitem{Dad2004} M. Dadfarnia, N. Jalili, B. Xian and D. M. Dawson. 
	\newblock Lyapunov-based vibration control of translational Euler-Bernoulli beams using the stabilizing effect of beam damping mechanisms. 
	\newblock{\em Journal of Vibration and Control}, 10(7), 933-961, 2004.
	
	\bibitem{Gao} K. Gao, W. Gao, D. Wu and C. Song. 
	\newblock Nonlinear dynamic stability analysis of Euler–Bernoulli beam–columns with damping effects under thermal environment. 
	\newblock{\em Nonlinear Dynamics}, 90(4): 2423--2444, 2017.
	
	\bibitem{Guo2001} B. Z. Guo and K. Y. Chan. 
	\newblock Riesz basis generation, eigenvalues distribution, and exponential stability for a Euler–Bernoulli beam with joint feedback control. 
	\newblock{\em Rev. Mat. Complut}, 14(1): 205--229, 2001.
	
	\bibitem{Guo2002} B. Z. Guo. 
	\newblock Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients. 
	\newblock{\em  SIAM Journal on Control and Optimization}, 40(6): 1905--1923, 2002.
	
	\bibitem{He11}
	W. He, S. S. Ge, B. V. E. How, Y. S. Choo and K. S. Hong.  ,.
	\newblock Robust adaptive boundary control of a flexible marine riser with vessel dynamics.
	\newblock {\em Automatica},  47(4): 722--732, 2011.
	
	\bibitem{Kar} D. Karagiannis and V. Radisavljevic-Gajic. \newblock Exponential stability for a class of boundary conditions on a Euler-Bernoulli beam subject to disturbances via boundary control. 
	\newblock{\em Journal of Sound and Vibration}, 446: 387--411, 2019.
	
	\bibitem{Lek} B. Lekdim and A. Khemmoudj. 
	\newblock Uniform decay of a viscoelastic nonlinear beam in two dimensional space. 
	\newblock{\em Asian Journal of Mathematics and Computer Research}, 25(1): 50--73, 2018.
	
	\bibitem{Lek1} B. Lekdim and A. Khemmoudj. 
	\newblock General decay of energy to a nonlinear viscoelastic two-dimensional beam. 
	\newblock{\em Applied Mathematics and Mechanics}, 39(11): 1661--1678, 2018.
	
	\bibitem{Lek2} B. Lekdim and A. Khemmoudj.   
	\newblock Existence and energy decay of solution to a nonlinear viscoelastic two-dimensional beam with a delay. 
	\newblock{\em Multidimensional Systems and Signal Processing}, 32(3): 915--931, 2021.
	
	\bibitem{Lek3} B. Lekdim and A. Khemmoudj. 
	\newblock Existence and general decay of solution for nonlinear viscoelastic two-dimensional beam with a nonlinear delay. 
	\newblock{\em Ricerche di Matematica}, 1--22, 2021.
	
	
	\bibitem{Li98} K. Liu and Z. Liu. 
	\newblock Exponential decay of energy of the Euler--Bernoulli beam with locally distributed Kelvin--Voigt damping. 
	\newblock{\em SIAM journal on control and optimization}, 36(3): 1086--1098, 1998.
	
	\bibitem{Mil2014} M. Miletić, D. Stürzer and A. Arnold. 
	\newblock An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. 
	\newblock{\em  arXiv preprint arXiv:1411.7946.}, 2014.
	
	
	\bibitem{Mil} M. Miletic, D. Sturzer, A. Arnold and A. Kugi. \newblock Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system. 
	\newblock{\em IEEE Transactions on Automatic Control}, 61(10): 2782--2795, 2015.
	
	\bibitem{Mor1992} O. Morgul. 
	\newblock Dynamic boundary control of a Euler-Bernoulli beam. 
	\newblock{\em IEEE Transactions on automatic control}, 37(5): 639--642, 1992.
	
	\bibitem{Se15}
	L. Seghour, A. Khemmoudj and N. E. Tatar.
	\newblock Control of a riser through the dynamic of the vessel.
	\newblock {\em Applicable Analysis}, 95(9): 1957--1973, 2016.	
	
	
	
	
	
	
	\bibitem{Sha2012} Y. F. Shang, G. Q. Xu and Y. L. Chen. 
	\newblock Stability analysis of Euler‐Bernoulli beam with input delay in the boundary control. 
	\newblock{\em Asian Journal of Control}, 14(1): 186--196, 2012.
	
	
	
	\bibitem{Wan2019} P. Wang and J. Hao. 
	\newblock Asymptotic Stability of Memory-Type Euler-Bernoulli Plate with Variable Coefficients and Time Delay. 
	\newblock{\em Journal of Systems Science and Complexity}, 32(5): 1375-1392, 2019.
	
	
	\bibitem{Zha1998}H.H. Zhang, H. Matsuo, H. Morita and H. Yamakawa. 
	\newblock Stability and stabilization of a deployable flexible structure. 
	\newblock{\em 37th IEEE Conference on Decision
		and Control}, 4: 4541--4542, 1998.
	
	
		
\end{thebibliography}

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