\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations, {\em Electronic Journal of Differential Equations}, Conf. 17 (2009), pp. 33--38.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{33} \title[\hfilneg EJDE-2009/Conf/17/\hfil Second-order differential equations] {Second-order differential equations with asymptotically small dissipation and piecewise flat potentials} \author[A. Cabot, H. Engler, S. Gadat \hfil EJDE/Conf/17 \hfilneg] {Alexandre Cabot, Hans Engler, S\'ebastien Gadat} % in alphabetical order \address{Alexandre Cabot \newline D\'epartement de Math\'ematiques, Universit\'e Montpellier II, CC 051\\ Place Eug\`ene Bataillon, 34095 Montpellier Cedex 5, France} \email{acabot@math.univ-montp2.fr} \address{Hans Engler \newline Department of Mathematics, Georgetown University\\ Box 571233\\ Washington, DC 20057, USA} \email{engler@georgetown.edu} \address{S\'ebastien Gadat \newline Institut de Math\'ematiques de Toulouse, Universit\'e Paul Sabatier\\ 118, Route de Narbonne 31062 Toulouse Cedex 9, France} \email{Sebastien.Gadat@math.ups-tlse.fr} \dedicatory{Pour Alban, n\'e le 27 mars 2008} \thanks{Published April 15, 2009.} \subjclass[2000]{34G20, 34A12, 34D05} \keywords{Differential equation; dissipative dynamical system; \hfill\break\indent vanishing damping; asymptotic behavior} \begin{abstract} We investigate the asymptotic properties as $t\to \infty$ of the differential equation $$ \ddot{x}(t)+a(t)\dot{x}(t)+ \nabla G(x(t))=0, \quad t\geq 0 $$ where $x(\cdot)$ is $\mathbb{R}$-valued, the map $a:\mathbb{R}_+\to \mathbb{R}_+$ is non increasing, and $G:\mathbb{R} \to \mathbb{R}$ is a potential with locally Lipschitz continuous derivative. We identify conditions on the function $a(\cdot)$ that guarantee or exclude the convergence of solutions of this problem to points in $\mathop{\rm argmin} G$, in the case where $G$ is convex and $\mathop{\rm argmin} G$ is an interval. The condition $$ \int_0^{\infty} e^{-\int_0^t a(s)\, ds}dt<\infty $$ is known to be necessary for convergence of trajectories. We give a slightly stronger condition that is sufficient. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{claim}[theorem]{Claim} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this note, we study the differential equation \begin{equation} \ddot{x}(t)+a(t)\dot{x}(t)+ \nabla G(x(t))=0, \quad t\geq 0 \label{eqcS} \end{equation} where $x(\cdot)$ is $\mathbb{R}$-valued, the map $G:\mathbb{R} \to \mathbb{R}$ is at least of class $\mathcal{C}^1$, and $a:\mathbb{R}_+\to \mathbb{R}_+$ is a non increasing function. In a previous paper \cite{CabEngGad}, we studied this differential equation in a finite- or infinite-dimensional Hilbert space $\mathcal{H}$. We are interested in the case where $a(t) \to0$ as $t \to\infty$. Broadly speaking, convergence of solutions can be expected if $a(t)\to 0$ sufficiently slowly. One of the questions left open in that paper was whether solutions converge to a limit if the property \begin{equation} \label{eq.exp.int.infinite} \int_0^\infty e^{-\int_0^t a(s) ds} dt = \infty \end{equation} does \emph{not} hold and if $\mathop{\rm argmin} G$ consists of more than just one point. In this note, we give a positive answer to this question, in the one dimensional case. \section{Preliminary Facts}\label{se.preli} Throughout this paper, we will denote by $G: \mathbb{R} \to \mathbb{R}$ a $\mathcal{C}^1$ function for which the derivative $G'$ is Lipschitz continuous, uniformly on bounded sets. The function $a:\mathbb{R}_+ \to \mathbb{R}_+$ will always be assumed to be continuous and non-increasing. We also define the energy \begin{equation*} \mathcal{E}(t) = G(x(t)) + \frac12 |\dot{x}(t)|^2 \,. \end{equation*} Here are some basic results for solutions of \eqref{eqcS} from \cite{CabEngGad}. For any $(x_0, x_1) \in \mathbb{R}^2$, the problem \eqref{eqcS} has a unique solution $x(\cdot) \in \mathcal{C}^2([0,T),\mathbb{R})$ satisfying $x(0) = x_0, \dot{x}(0) = x_1$ on some maximal time interval $[0,T) \subset [0,\infty)$. For every $t\in [0,T)$, the energy identity holds $$\frac{d}{dt} \mathcal{E}(t) = -a(t)|\dot{x}(t)|^2.$$ If in addition $G$ is bounded from below, then \begin{equation}\label{energy3} \int_0^T a(t)|\dot{x}(t)|^2 dt < \infty \, , \end{equation} and the solution exists for all $T > 0$. If also $G(\xi) \to \infty$ as $|\xi| \to \infty$ (i.e. if $G$ is \emph{coercive}), then all solutions to \eqref{eqcS} remain bounded together with their first and second derivatives for all $t > 0$. The bound depends only on the initial data. If a solution $x$ to \eqref{eqcS} converges toward some $\overline{x}\in \mathbb{R}$, then $\lim_{t\to \infty}\dot x(t)=\lim_{t\to \infty}\ddot x(t)=0$ and $G'(\overline{x})=0$. If $\int_0^{\infty} a(s)\, ds<\infty$ and if $\inf G > - \infty$, then solutions $x(\cdot)$ of \eqref{eqcS} for which $(x(0),\dot{x}(0))\not \in \mathop{\rm argmin} G\times \{0\}$ cannot converge to a point in $\mathop{\rm argmin} G$. For the remainder of this note we shall assume that $\mathop{\rm argmin} G \ne \emptyset$. Without loss of generality, we may assume that $\min_\mathbb{R} G = 0$ and $G(0) = 0$. If for some $\rho \in \mathbb{R}_+$ and $z\in \mathop{\rm argmin} G$ $$\forall x\in \mathbb{R}, \quad G(x)-G(z)\leq \rho\, G'(x)(x-z)$$ then it is possible to show that any solution $x$ to the differential equation \eqref{eqcS} satisfies $$\int_0^{\infty}a(t) \, \mathcal{E}(t)\, dt <\infty.$$ Since $t \mapsto \mathcal{E}(t)$ is decreasing, this estimate implies that $\mathcal{E}(t) \to \min G = 0$ as $t\to \infty$, provided that $\int_0^{\infty}a(t) \, dt=\infty$. If now $\mathop{\rm argmin} G = \{\overline{x}\}$ is a singleton, then trajectories must converge to $\overline{x}$ under fairly weak additional conditions. The reader is referred to \cite{CabEngGad} for details. \section{Convex potentials with non-unique minima}\label{ConvTraj} In this section, we investigate the convergence of the trajectories of \eqref{eqcS} when $\mathop{\rm argmin} G$ is \emph{not} a singleton. While the previous discussion shows that $\int_0^\infty a(s) ds = \infty$ is a necessary condition for trajectories to converge to a point in $\mathop{\rm argmin} G$, this condition is clearly not sufficient, as the particular case $G\equiv 0$ shows. In this case, the solution is given by $$ x(t)= x(0)+\dot x(0)\,\int_{0}^{t}e^{-\int_{0}^s a(u)\, du}ds $$ and the solution $x$ converges if and only if \eqref{eq.exp.int.infinite} does not hold. Therefore it is natural to ask whether for a general potential $G$, the trajectory $x$ is convergent if this condition does not hold. The potential $G$ is assumed to have all the properties listed in the previous section. A general result of non-convergence of the trajectories under the condition \eqref{eq.exp.int.infinite} is shown in \cite{CabEngGad}. There, we assume that $G$ is coercive, $\inf_{\mathbb{R}} G = 0$, $\mathop{\rm argmin} G = [\alpha,\beta]$ for some $\alpha < \beta$, and that $G$ is non-increasing on $(-\infty,\alpha]$ and non-decreasing on $[\beta,\infty)$. It is also assumed that $a$ satisfies condition \eqref{eq.exp.int.infinite}. Then either a solution satisfies $(x(0),\dot{x}(0))\in [\alpha,\beta]\times \{0\}$, or else the $\omega$- limit set $\omega(x_0,\dot x_0)$ contains $[\alpha,\beta]$ and hence the trajectory $x$ does not converge. We now ask if the converse assertion is true: do the trajectories $x$ of \eqref{eqcS} converge if \eqref{eq.exp.int.infinite} does not hold? We give a positive answer when the map $a$ satisfies the following stronger condition \begin{equation}\label{cond-theta} \int_0^\infty e^{-\theta\, \int_0^s a(u)\, du} ds<\infty, \end{equation} for some $\theta\in (0,1)$. \begin{theorem}\label{pr.conv_dim1} Let $G:\mathbb{R}\to \mathbb{R}$ be a convex function of class $\mathcal{C}^1$ such that $G'$ is Lipschitz continuous on the bounded sets of $\mathbb{R}$. Assume that $\mathop{\rm argmin} G=[\alpha,\beta]$ with $\alpha < \beta$ and that there exists $\delta>0$ such that $$ \forall \xi\in (-\infty,\alpha], \quad G'(\xi)\leq 2\, \delta\, (\xi-\alpha) \quad \mbox{and} \quad \forall \xi\in [\beta, \infty), \quad G'(\xi)\geq 2\, \delta\, (\xi-\beta). $$ Let $a:\mathbb{R}_+\to \mathbb{R}_+$ be a differentiable non increasing map such that $\lim_{t\to\infty}a(t)= 0$ and such that condition \eqref{cond-theta} holds for some positive $\theta< 1$. Then, for any solution $x$ to the differential equation \eqref{eqcS}, $\lim_{t\to\infty} x(t)$ exists. \end{theorem} \begin{proof} We may assume without loss of generality that $\alpha = 0, \beta = 1$. The conditions on $G$ imply that it is coercive, hence $\lim_{t\to\infty}\mathcal{E}(t) = 0$ and $|x(t)|\le M$ for some $M>0$, for all $t\in \mathbb{R}_+$. Define the set $\mathcal{T}=\{t\geq 0\,|\,\dot x(t)=0\}$. We shall show that either $\mathcal{T} = [0,\infty)$ or $\mathcal{T}$ is a finite set. Assume first that $\mathcal{T}$ has an accumulation point $t^*$. Then $\dot x(t^*)=0$ and $\ddot x(t^*)=0$ by Rolle's Theorem. Since then $\dot x(t^*)=\ddot x(t^*)=G'(x(t^*))=0$, $x(\cdot)$ must be constant by forward and backward uniqueness, $\mathcal{T} = [0,\infty)$, and clearly the limit exists. Therefore we may now assume that $\mathcal{T}$ is discrete. If $\mathcal{T}$ is a finite set, then $\dot x$ does not change sign for sufficiently large $t$, and the trajectory $x$ has a limit. It remains to consider the case $\mathcal{T}=\{t_n\,| \, n\in \mathbb{N}\}$, where the $t_n$ are increasing and tend to $\infty$. We want to show that this is impossible. Observe that at each $t_n$, $\dot x$ must change its sign and $G'(x(t_n)) \ne 0$, since otherwise also $\ddot x(t_n) = 0$ and we would again have a stationary solution. Without loss of generality, we can assume that $\dot x(0) < 0, \, x(0) < 0$ and therefore $x(t_0)< 0$. Since $G'(x(t_0))<0$, equation \eqref{eqcS} shows that $\ddot x(t_0)>0$, hence the map $\dot x$ is positive on $(t_0, t_1)$, $x(t_1) > 1$, $\dot x$ is negative on $(t_1,t_2)$, and so on. The argument so far shows that $G'(x(t))$ vanishes on a union of infinitely many disjoint closed intervals, $$ \{t \, | \, 0 \le x(t) \le 1\} = \cup_{k\ge 0} [u_{2k},u_{2k+1}] $$ where $0 < t_0 < u_0$ and $u_{2k-1} < t_k < u_{2k}$ for $k = 1, \, 2, \dots$. Let us observe that, for every $k\in \mathbb{N}$, $$ 1=|x(u_{2k+1})-x(u_{2k})|=\int_{u_{2k}}^{u_{2k+1}}|\dot x (t)|\, dt\leq |u_{2k+1}-u_{2k}|\, \max_{t\ge u_{2k}}|\dot x (t)|. $$ Since $\lim_{t\to \infty}\dot x (t)=0$, we deduce that $\lim_{k\to \infty} |u_{2k+1}-u_{2k}|=\infty$. We next observe that for $u_{2k} \le t \le u_{2k+1}$ the function $v = \dot x$ satisfies $\dot v(t) + a(t)v(t) = 0$ and hence \begin{equation}\label{expres.dotx} \forall t\in [u_{2k},u_{2k+1}], \quad \dot x(t) = \dot x(u_{2k}) e^{-\int_{u_{2k}}^t a(\tau) d \tau} \,. \end{equation} \begin{claim}\label{clm_1} There is a constant $\gamma$ such that $u_{2k+2} - u_{2k+1} \le \gamma$ for all $k\in \mathbb{N}$. \end{claim} To show this claim, fix $k\in \mathbb{N}$ and assume that $t\in [u_{2k+1},u_{2k+2}]$. Assume for now that $k$ is odd and thus $x(t)\le 0$. Define the quantity $A(t)=\exp\left(\frac{1}{2}\int_0^t a(s)\, ds\right)$ and set $y(t)= A(t)\, x(t)$. Then $y$ is the solution of the differential equation \begin{equation}\label{diff_eq_y} \ddot y(t)+ A(t) \, G'\left( \frac{y(t)}{A(t)}\right) -\left(\frac{a^2(t)}{4}+ \frac{\dot a (t)}{2}\right) y(t)=0, \end{equation} and satisfies $y(u_{2k+1}) = y(u_{2k+2})=0$ and $\dot y(u_{2k+1})=A(u_{2k+1})\, \dot x(u_{2k+1}) < 0$. Since the map $a$ converges to $0$, we can choose $k$ large enough so that $a(t)<2\,\sqrt{\delta}$ for every $t\in [u_{2k+1},u_{2k+2}]$. On the other hand, the assumption on $G'$ shows that, for every $t\in [u_{2k+1},u_{2k+2}]$, $$ A(t) \, G'\left( \frac{y(t)}{A(t)}\right)\leq 2\, \delta\, y(t). $$ Recalling finally that $\dot a (t)\leq 0$ for every $t\geq 0$, we deduce from (\ref{diff_eq_y}) that $$ \forall t\in [u_{2k+1},u_{2k+2}], \quad \ddot y(t)+\delta\, y(t)\geq 0. $$ The unique solution $z$ of the differential equation $\ddot z(t)+\delta\, z(t)=0$ with the same initial conditions as $y$ has the first zero larger than $u_{2k+1}$ at $u_{2k+1}+\frac{\pi}{\sqrt{\delta}}$. By a standard comparison argument, we deduce that $y$ vanishes before $z$ does, hence $$ u_{2k+2}\leq u_{2k+1}+ \gamma, \quad \gamma = \frac{\pi}{\sqrt{\delta}} \,. $$ The same argument applies if $k$ is even. This proves the claim. \begin{claim}\label{clm_2} There is a $k_0\in \mathbb{N}$ such that for $k \ge k_0$ $$ |\dot x(u_{2k+2})|\leq |\dot x(u_{2k})|\, e^{-\theta \int_{u_{2k}}^{u_{2k+2}} a(s)\, ds}. $$ where $\theta$ is as in \eqref{cond-theta}. \end{claim} To prove this, pick $k_0$ so large that for all $k \ge k_0$, $$ (1-\theta)(u_{2k+2} - u_{2k}) \ge \gamma\theta \,. $$ This is possible since $u_{2k+2} - u_{2k} \to \infty$ as $k \to \infty$. Since $a$ is non-increasing, this implies that \begin{align*} \theta \int_{u_{2k+1}}^{u_{2k+2}} a(\tau) d \tau &\leq \gamma \theta a(u_{2k+1}) \\ &\le (1 - \theta)(u_{2k+1} - u_{2k})a(u_{2k+1}) \\ &\leq (1-\theta) \int_{u_{2k}}^{u_{2k+1}} a(\tau) d \tau \end{align*} and hence $$ \theta \int_{u_{2k}}^{u_{2k+2}}a(\tau) d \tau \le \int_{u_{2k}}^{u_{2k+1}} a(\tau) d\tau \,. $$ Then for $k \ge k_0$, \begin{align*} |\dot x(u_{2k+2})| &\leq |\dot x(u_{2k+1})| = |\dot x(u_{2k})| e^{-\int_{u_{2k}}^{u_{2k+1}} a(s)\, ds} \\ &\leq |\dot x(u_{2k})| e^{-\theta \int_{u_{2k}}^{u_{2k+2}} a(s)\, ds} \end{align*} proving the claim. \begin{claim}\label{clm_3} If the set $\mathcal{T}$ is unbounded, there must exist a constant $C$, depending on $\mathcal{T}$ and on $x(0), \dot x(0)$ such that for all $t \ge 0$ \begin{equation} \label{majo_dotx_all} |\dot x(t)|\leq C\, e^{-\theta \int_0^{t} a(s)\, ds}. \end{equation} \end{claim} By making sure that $C$ is sufficiently large, we only have to prove the estimate for $t \ge u_{2k_0}$. First assume that $u_{2k} \le t \le u_{2k+1}$ for some $k$. Then from \eqref{expres.dotx} $$ |\dot x(t)|\leq |\dot x(u_{2k})| \, e^{-\int_{u_{2k}}^t a(s)\, ds} \le |\dot x(u_{2k})| \, e^{-\theta \int_{u_{2k}}^t a(s)\, ds} \,. $$ Using induction, we deduce from Claim \ref{clm_2} that $$ |\dot x(t)|\leq |\dot x(u_{2k_0})| \, e^{-\theta \int_{u_{2k_0}}^t a(s)\, ds} = C_1\, e^{-\theta \int_{0}^t a(s)\, ds} $$ with $C_1 = |\dot x(u_{2k_0})| \, e^{\theta \int_0^{u_{2k_0}} a(s)\, ds}$. Next consider the case where $u_{2k+1} < t \le u_{2k+2}$ for some $k$. Then $$ |\dot x(t)|\leq |\dot x(u_{2k+1})| \le C_1\, e^{-\theta \int_{0}^{u_{2k+1}} a(s) \, ds} \le C_1 e^{\theta \int_{u_{2k+1}}^{u_{2k+2}}a(\tau) d \tau} \, e^{-\theta \int_0^t a(s)\, ds} \,. $$ Due to Claim \ref{clm_1}, $e^{\theta\,\int_{u_{2k+1}}^{u_{2k+2}}a(\tau) d \tau} \le C_2$ for all $k$, for some constant $C_2$. Estimate (\ref{majo_dotx_all}) now follows for $t \ge u_{2k_0}$ with $C = C_1 C_2$. By enlarging $C$ further, the estimate follows for all $t\geq 0$. Let us now conclude the proof of the theorem. From assumption \eqref{cond-theta} and estimate (\ref{majo_dotx_all}), we derive that $\dot x \in L^1(0,\infty)$. Hence $\lim_{t \to \infty} x(t)$ exists, contradicting the initial assumption. Therefore $\lim_{t \to \infty} x(t)$ exists after all, and the theorem has been proved. \end{proof} \begin{remark}\label{re.power_neg} Note that the map $t\mapsto \frac{c}{t+1}$ with $c>1$ satisfies condition \eqref{cond-theta} for every $\theta\in (\frac{1}{c},1)$. In fact, if merely $a(t)\geq \frac{c}{t+1}$ for $t$ large enough for some $c>1$, then condition \eqref{cond-theta} is satisfied. Consider next the family of maps $a:\mathbb{R}_+\to \mathbb{R}_+$ defined by $$ a(t)= \frac{1}{t+1}+ \frac{d}{(t+1)\,\ln (t+2)}, $$ for some $d>0$. It is immediate to check that condition \eqref{eq.exp.int.infinite} holds if and only if $d\in (0, 1]$. Thus non-stationary trajectories of \eqref{eqcS} do not converge when $d\in (0, 1]$. But condition \eqref{cond-theta} is never satisfied, for any $\theta\in (0,1)$ and $d>0$, and the convergence of trajectories remains an open question. Thus there remains a ``logarithmic'' gap between the criteria for existence and non-existence of limits. \end{remark} We conclude with some remarks on convergence results in dimension $n>1$. It is possible to extend the non-convergence result given at the beginning of this section to the case where the differential equation is given in a Hilbert space $\mathcal{H}$, see \cite{CabEngGad}. However, it is not clear how to prove that $\lim_{t \to \infty} x(t)$ exists, in a general Hilbert space $\mathcal{H}$ and for the case where $G$ is convex and $\mathop{\rm argmin} G$ is not a singleton. Since in this case $|\dot x(t)| \le \sqrt{2 \mathcal{E}(t)}$, it appears natural to derive convergence results from suitable estimates for $\mathcal{E}(t)$. In \cite{CabEngGad}, we give conditions that imply $\mathcal{E}(t) \le D a(t)$ for all $t$, for some constant $D>0$. However, since we must also assume that $\int_0^\infty a(s) ds = \infty$, these estimates are not strong enough to guarantee the convergence of trajectories. One could try to extend the proof of Theorem \ref{pr.conv_dim1}. Set $a_1(t) = a(t) \cdot \chi_{S}(x(t))$, where $\chi_S$ is the characteristic function of $S= \mathop{\rm argmin} G$, then $\frac{d}{dt}\mathcal{E}(t) \le - 2a_1(t)\mathcal{E}(t)$, and hence $\mathcal{E}(t) \le \mathcal{E}(0) e^{-2\int_0^t a_1(s) ds}$. If the function $t \mapsto e^{-\int_0^t a_1(s) ds}$ can be shown to be in $L^1(0,\infty)$, it would follow that $|\dot{x}|$ is integrable, implying the convergence of trajectories. This works in the one-dimensional case since the behavior of trajectories is quite simple. However, if $\dim \mathcal{H} > 1$, it is difficult to satisfy this property, since trajectories corresponding to \eqref{eqcS} can be expected to behave like trajectories of a billiard problem in $S = \mathop{\rm argmin} G$ for large times. When the map $a$ is constant and positive, it is established in \cite{Alv,AttGouRed} that the trajectories of \eqref{eqcS} are weakly convergent if the potential $G:\mathcal{H} \to \mathbb{R}$ is convex and $\mathop{\rm argmin} G \neq \emptyset$, in an arbitrary Hilbert space $\mathcal{H}$. The key ingredient of the proof is the Opial lemma \cite{Opi}, which allows the authors of these papers to prove convergence even if $|\dot{x}(\cdot)|$ is only in $L^2(0,\infty)$ and not in $L^1(0,\infty)$. However, if e.g. $a(t) = \frac{c}{t+1}$, then Opial's lemma requires that we show $\int_0^\infty (t+1) |\dot x(t)|^2 \, d t < \infty $, while (\ref{energy3}) implies only $\int_0^\infty \frac{1}{t+1} |\dot x(t)|^2 \, d t < \infty $. Hence there remains a gap if arguments similar to those in \cite{Alv} or \cite{AttGouRed} are to be used. It is unclear how this gap can be closed. \begin{thebibliography}{99} \bibitem{Alv} F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert spaces, {\em SIAM J. on Control and Optimization}, 38 (2000), n$^\circ$ 4, 1102-1119. \bibitem{AttGouRed} H. Attouch, X. Goudou, P. Redont, The heavy ball with friction method: I the continuous dynamical system, {\em Communications in Contemporary Mathematics}, 2 (2000), n$^\circ$ 1, 1-34. \bibitem{CabEngGad} A. Cabot, H. Engler, S. Gadat, On the long time behavior of second order differential equations with asymptotically small dissipation, {\em Trans. of the Amer. Math. Soc.}, in press. {\tt http://arxiv.org/abs/0710.1107} \bibitem{Opi} Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, {\em Bull. of the American Math. 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