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periodic solutions,pendulum equation,variational methods
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\title{PERIODIC SOLUTIONS OF THE FORCED}
\endtitle
\bigskip
\centerline{\bf PENDULUM
EQUATION WITH FRICTION}
\bigskip
\centerline{\bf Pablo Amster and Mar\'\i a Cristina Mariani}
\medskip
\centerline{Universidad de Buenos Aires}
\medskip
\medskip
\lema {Abstract: }
This paper is devoted
to the study of the general forced pendulum equation in the
presence of friction,
$$u'' + a(t)u' + b(t) \sin u = f(t)$$
with $a,b\in C([0,T])$ and
$f\in L^2(0,T)$.
We'll show that $T$-periodic solutions may be obtained as zeroes of a
$2\pi$-periodic
continuous real function. Furthermore, the existence of infinitely many solutions is
proved under appropiate conditions on $a,b$ and $f$.
\rm
\tit {Introduction}
The periodic problem for the forced pendulum equation has been studied by
different authors.
In 1922 Hamel [H] found that the equation
$$u'' + b \sin u = \beta \sin (t)$$
with $b,\beta$ constants
admits a $2\pi$-periodic solution that can be obtained as a minimum in
$C_{per}[0,2\pi]$
of the functional
$$J(u)=\int_0^{2\pi} {u'^2\over 2} + b \cos u + u\beta \sin t$$
and the same argument can be extended for
$T$-periodic solutions of
the equation
$u'' + b \sin u = f (t)$
where $f$ is $T$-periodic and orthogonal to constants (see [M1-2]).
If we allow the presence of friction, namely the equation
$$u'' + au' + b \sin u = f (t)\tag1$$
(where $a$ is a positive constant) then variational methods are not appliable
to the
periodic problem.
In fact, examples of $T$-periodic functions $f$ such that $f\perp 1$
and (1) is
not solvable are given in [A] and [O], under some conditions on the
parameters $a,b$ and $T$. In a recent work
a rather general class of
counterexamples is constructed
for arbitrary $a,b,T$ [OST].
Periodic type conditions for a general system are
studied in [AM] by fixed point methods.
The
purpose of this work is to
give conditions for the
existence of $T$-periodic solutions
of the general equation
$$u'' + a(t)u' + b(t) \sin u = f(t)\tag2$$
In the first section we'll show that
the
Dirichlet problem for (2) is
solvable
for any boundary data,
and uniquely solvable under an
appropiate
condition on $b$.
In the second section
we prove that the
$T$-periodic
solutions of (2) are the zeroes of a $2\pi$-periodic continuous
real function.
Finally, in the third section
we give explicit conditions on $a,b$ and $f$ in order to obtain
$T$-periodic
solutions of (2).
\tit{1. Unique solvability of the Dirichlet problem}
The existence
of solutions
of (2) for
arbitrary
Dirichlet boundary data $\varphi$ is easily
obtained
from
classical results available in the literature (see e.g.
[M3]). We sketch a simple
proof here which makes use of Schauder
fixed point theorem.
For any
$\overline u\in L^2(0,T)$ let $K \overline u=u $ be the unique
solution
of the linear problem
$$\cases
u''+a(t)u' = f - b(t)\sin(\overline u) &\\
u|_{\partial I} = \varphi &
\endcases$$
As $\sin \overline u \in L^2(0,T)$ then
$K: L^2\to L^2$ is well defined and compact.
Moreover,
using a standard apriori bound for the
operator
$Lu=u''+a(t)u'$, we obtain:
$$\| K\overline u - K0\|_2 \le c \| L(K \overline u - K0)\|_2 =
c\| b\sin \overline u\|_2\le
c\| b\|_2$$
Hence, taking $M = c\| b\|_2$ we
obtain that $R(K) \subset B_M(K0)$
and the result follows.
A sufficient condition for the
uniqueness is:
$\| b\|_\infty < \frac 1c$.
However, for further
applications
we'll state a slightly more precise uniqueness result providing
also an apriori bound for the nonlinear operator
$Qu:= u'' + au' + b\sin u$.
\lema{Lemma 1}
Let $p$ be a positive solution of the equation
$p' = a(t) p - k(t)$ for
$k\in H^1(0,T)$ with $k'\ge 0$ a.e. and
$\lambda_p$ the first eigenvalue of the
problem
$-(pu')' = \lambda u$.
Then, if
$\| bp\|_\infty <\lambda_p$,
$$\| u- v\|_2 \le \frac 1{\lambda_p -\| pb\|_\infty}\| p(Qu-Qv)\|_2$$
and
$$\left(\int_0^T p[(u- v)']^2\right)^{1/2} \le
\frac {\sqrt{\lambda_p}}{\lambda_p -\| pb\|_\infty} \| p(Qu-Qv)\|_2$$
for any $u,v\in H^2(0,T)$ such that $u=v$ on $\partial I$.
In particular, Dirichlet problem is uniquely solvable for any
boundary data $\varphi$.
\demost{Proof}
Let $w=u-v$, then
$$\|p(Qu-Qv)\|_2\| w\|_2\ge -\int_0^T p(Qu-Qv).w =
\int p(w')^2 - kw'w - pb(\sin u - \sin v)w$$
As $-\int kw'w = \int k'\frac {w^2}2 \ge 0$, and
$|\int pb(\sin u - \sin v)w| \le \|pb\|_\infty \|w\|_2^2$, it follows that
$$\|p(Qu-Qv)\|_2\| w\|_2\ge
\int p(w')^2 -\|pb\|_\infty \|w\|_2^2$$
and the result holds since $\|w\|_2^2 \le \frac 1{\lambda_p}
\int p(w')^2$.
\lema{Remark:}
If $a\in H^1(0,T)$ with $a'\ge 0$ a.e., we
may take $p \equiv 1$,
and the result holds
for $\| b\|_\infty < (\frac \pi T)^2$.
\rm
\tit{2. An embedded
curve in $H^1(0,T)$}
In this section
we'll assume that $b$ satisfies the assumptions of lemma 1 for some
$\widetilde p>0$, and
prove that all the possible
$T$-periodic solutions of (2) belong
to a continuous
curve $\Cal C\subset H^1(0,T)$.
More precisely, if the functional
$I: H^1(0,T)\to \R$ is given by
$I(u)= \int_0^T a(t)u'+ b(t)\sin u$, then the set of
$T$-periodic solutions of (2) is
$\Cal C \cap I^{-1} (\int_0^T f)$.
Indeed, we may prove directly the following
\newpage
\lema{Theorem 2}
Let $\psi:\R\to \R$
given by
$\psi (s) = I(u_s) - \int_0^T f$, where $u_s$ is
the unique solution of the problem
$$\cases
u''+a(t)u' + b(t)\sin(u)= f &\\
u|_{\partial I} = s &
\endcases$$
Then $\psi$ is continuous
and $2\pi$-periodic.
Moreover, the set of $T$-periodic solutions of (2)
can be characterized as
$$\{ u_s : \psi (s) =0\}$$
\demost{Proof}
Let
$s\to s_0$, and define $w_s=u_s-u_{s_0}$.
In order
to prove that
$I(u_s)\to I(u_{s_0})$, by dominated
convergence
it will suffice to show that $w_s, w_s'\to 0$ a.e.
As
$$0 = Q u_s - Qu_{s_0} = w_s'' + a w_s'+ b(\sin u_s - \sin u_{s_0})=
w_s'' + a w_s'+ b\cos \xi w_s$$
it follows
that
$\| w_s\|_{1,2}\le c|s-s_0|$ for some constant $c$.
Hence, $\psi$ is continuous.
Moreover, by uniqueness $u_{s+2\pi}=u_s + 2\pi$,
and by definition of $I$ we conclude that $\psi$ is $2\pi$-periodic.
Finally, $u$ is a $T$-periodic solution of (2) if and only if
$u\in \Cal C=
\{ u_s : s\in \R\}$, and
$0 = u'(T)-u'(0) = \int_0^T f - (au'+b\sin u)$. Hence, the proof is complete.
\lema{Remarks:}
i) With the notation of the preceeding theorem, it's immediate
that
$\Cal C$ is an embedded curve of $H^1(0,T)$.
ii) A more general version of Theorem 2 may be obtained if we define
$p>0$ verifying:
$$p' = a(t)p - k(t), \qquad p(0) = p(T)\tag{P}$$
and
$$\psi_p (s) = \int_0^T p(t) (b(t)\sin u_s - f) + k(t)u_s' := I_p(u_s) - \int_0^T pf $$
iii) By periodicity, extremals $s_{min}$ and
$s_{max}$ of $\psi_p$
are achieved in
$[0,2\pi]$.
Using
dominated convergence, it can be shown that if
$w = \frac {\partial}{\partial s} u_s$ then
satisfies the linear equation
$$\cases
(pw')'+kw' + b(t)\cos(u_s)w= 0 &\\
w|_{\partial I} = 1 &
\endcases$$
For a solution of
the equation $\psi_p '(s)=0$,
if $k$ is constant then
$w$ is periodic.
\rm
\medskip
As an immediate consequence of theorem 2
we have:
\lema{Corollary 3}
Let
$p>0$ verify (P)
and
assume that
$I_p(u_{s_-})\le \int_0^T pf \le I_p(u_{s_+})$
for some
$s_-\le s_+\in \R$.
Then there exist
$s_1\in [s_-,s_+]$,
$s_2\in [s_+,s_- + 2\pi]$
such that
$u_{s_i} + 2n\pi$ is a $T$-periodic solution of
(2) for any integer $n$ and $i=1,2$.
\rm
\tit{3. Sufficient conditions
for the existence of $T$-periodic solutions of (2) }
In this section
we'll
obtain
$T$-periodic
solutions of (2) under
rather
explicit conditions on $a,b$ and $f$.
For a straightforward application of corollary 3 we first note that
a positive $p$ satisfying (P) for some
constant $k$
may be constructed in a unique way up to a constant factor.
Indeed, from the equation
$p'= ap - k$ we obtain that
$$p(t) = \left(c_0 - k\int_0^t e^{-\int_0^sa}ds\right) e^{\int_0^ta}$$
Without loss of generality
we may assume that $c_0 = 1$, and
as $p(0)=p(T)$ we deduce that
$$k=\frac { e^{\int_0^Ta}-1}{\int_0^T e^{\int_s^Ta}ds}$$
Now, if $k\le 0$ it's immediate that $p>0$. On the other hand, if
$k > 0$ and $p$ vanishes in $(0,T)$,
since $p(0)=p(T)$
there exists $t_0\in (0,T)$ such that
$p(t_0)=0$ and $p'(t_0)\ge 0$. Then
$ k = -p'(t_0) \le 0$, a contradiction.
\lema {Remark:} in particular,
if $a\perp 1$ then
$p(t)= e^{\int_0^ta}$, and if $a$ is a constant then $p\equiv 1$.
\rm
\medskip
In the following $p$ will be considered
as in the previous construction,
and we'll assume
that
$\| \widetilde p b\|_\infty < \lambda_{\widetilde p}$ for some
$\widetilde p$ such that lemma 1 holds.
As $Qu_s - Qs = f - b\sin s$,
we may denote by $\delta$ the best
possible constant such that
$$\| u_s -s\|_\infty \le \delta \| p(f-b\sin s)\|_2$$
A simple computation shows that
if $\| pb\|_\infty < \lambda_p$, then
$$\delta \le
\left( \frac {T\lambda_p}{\inf_t p}\right) ^{1/2} \frac 1{\lambda_p -
\| pb\|_\infty}$$
and if $\| pb\|_2 \le \| pb\|_\infty$ (for example, when $T\le 1$)
then $\| pb\|_\infty$ may be replaced in the formula by $\| pb\|_2$.
We obtain the following
\lema {Theorem 4}
Let us assume that $b$ does not vanish in $[0,T]$, and
\noi i) $\| \widetilde p b\|_\infty < \lambda_{\widetilde p}$
for some $ \widetilde p$
satisfying the hypothesis of lemma 1
\noi ii)
$\| p(f \pm b) \|_2 \le \frac c\delta $ for some constant $c<\frac {\pi}2$.
\noi iii) $|\int_0^T pf| \le \| pb\|_1 \cos (c)$
Then
(2)
has infinitely many
$T$-periodic
solutions.
More precisely,
there exist
$s_1\in [\frac {-\pi}2, \frac {\pi}2]$,
$s_2\in [\frac {\pi}2, \frac {3\pi}2]$
such that
$\{ u_{s_i} + 2n\pi : n\in Z\}$ is a family of
$T$-periodic
solutions of (2).
\demost{Proof}
With the previous notations,
for $s_n = \frac {(2n-1)\pi}2$ we obtain that
$\| u_{s_n} - s_n\|_\infty\le c $.
Furthermore, as
$$I_p(u_{s_{n}})=
\int_0^T k u_{s_n}' + pb\sin u_{s_n} =
(-1)^n \int_0^T pb \cos (u_{s_n}-s_n),$$
taking $n$ such that $(-1)^nb > 0$ we obtain:
$$I_p(u_{s_{n}})\ge \| pb\|_1 \cos (c)$$
and in the same way
$$I(u_{s_{n \pm 1}})\le -\| pb\|_1 \cos (c)$$
Hence, by iii)
$$I_p(u_{s_{n \pm 1}}) \le \int_0^T pf \le I_p(u_{s_n})$$
and from corollary 3 the result holds.
\medskip
For constant $a$, conditions $i)-iii)$ can be written in a more explicit way:
\lema{Corollary 5}
Let
us assume that $a$ is a constant,
and $b$ does not vanish in $[0,T]$. Then, if
\noi i) $\| b\|_\infty< (\frac {\pi}{T})^2$
\noi ii) $\| f \pm b \|_2
\le c\frac {\pi^2- \|b\|_\infty
T^2}{T^{3/2}}$ for some constant $c<\frac {1}2$.
\noi iii) $|\int_0^\a f|\le \| b\|_1 \cos (c\pi)$
\noi there exist
$s_1\in [\frac {-\pi}2, \frac {\pi}2]$,
$s_2\in [\frac {\pi}2, \frac {3\pi}2]$
such that
$\{ u_{s_i} + 2n\pi : n\in Z\}$ is a family of
$T$-periodic
solutions of (2).
\lema{Remarks:}
1) Condition ii) in Theorem 4
may be easily generalized in the following way:
\noi
ii') There exist $s_0, s_1$ such that
$\delta \| p(f - b\sin s_i) \|_2 + |s_i - (-1)^i \frac {\pi}2| \le c$
for some constant $c<\frac {\pi}2$.
Replacing ii) by ii')
a family $\{ u_{s} + 2n\pi\}$
of
$T$-periodic
solutions of (2) is obtained
for some $s \in [s_1,s_0]$
and also
for some $s \in [s_0,s_1+2\pi]$.
Furthermore, the existence
of one of the numbers $s_i$ may be avoided
if the following extra condition on $f$
holds:
$$\| pb\|_2\| pf\|_2\le \frac
1\delta_2 |\int pf|\tag{F} $$
where $\delta_2$ is the best constant such that
$\| u_s - s\|_2 \le \delta_2 \| pf\|_2$.
Indeed, for $s = 0,\pi$
$$|\int_0^T pb \sin u_s| = |\int_0^T pb \sin (u_s-s)| \le
\delta_2 \| pb \|_2\| pf \|_2\le |\int_0^T pf|,$$
and one of the inequalities required in corollary 3 is
proved.
For example, (F) holds for
$sg(f)$ constant, if $0\le |f| \le \frac 1p$ and
$|\int pf| \ge \delta_2 \| pb\|_2$.
2) As a particular case, condition iii) in Theorem 4
(Corollary 5) holds for
$f \perp p$ (respectively: $\int_0^T f = 0$).
3) The results of this paper may be
simplified and constants can be computed in a
sharper way by the use of a weighted Hilbert space $L_p^2(0,T)$.
For example, if $\|\cdot \|$ denotes the $p$-weighted norm, and $\mu_p$ is
the first eigenvalue of the
problem $(-pu')' = \mu pu$, then the inequalities of
lemma 1 can be transformed in:
$$\| u- v\| \le \frac 1{\mu_p -\|b\|_\infty}\|(Qu-Qv)\|$$
and
$$\|(u- v)'\| \le \frac {\sqrt{\mu_p}}{\mu_p -\| b\|_\infty} \| (Qu-Qv)\| $$
\newpage
\tit {References}
[A] Alonso, J.: Nonexistence of periodic solutions for a damped
pendulum equation. Diff. and Integral Equations, 10 (1997), 1141-8.
[AM] Amster, P., Mariani, M.C.:
Resolution of Semilinear Equations by Fixed Point Methods.
To appear in the Bulletin of the Belgian
Math. Society, Simon Stevin.
[H] Hamel, G.: \"Uber erzwungene Schwingungen bei endlichen Amplituden.
Math. Ann., 86 (1922), 1-13.
[M1] Mawhin, J.: Periodic oscillations of forced pendulum-like equations. Lecture
Notes in Math., Springer, 964 (1982), 458-76.
[M2] Mawhin, J.: The forced pendulum: A paradigm for nonlinear analysis and
dynamical systems. Expo. Math., 6 (1988), 271-87.
[M3] Mawhin, J.: Boudary value problems for nonlinear
ordinary differential equations: from successive approximations to topology.
[O] Ortega, R.: A counterexample for the damped pendulum equation.
Bull. Classe des Sciences, Ac.Roy. Belgique, LXXIII (1987), 405-9.
[OST] Ortega, R., Serra, E., Tarallo, M.: Non-continuation
of the periodic oscillations of a forced pendulum in the presence of
friction. To appear.
\bigskip
{\bf P.Amster and M. C. Mariani}
Dpto. de Matem\'atica
Fac. de Cs. Exactas y Naturales, UBA
Pab. I, Ciudad Universitaria (1428), Buenos Aires, Argentina
CONICET
\bigskip
{\bf Address for correspondence:}
\noi P.Amster and M. C. Mariani,
\noi Dpto. de Matem\'atica,
Fac. de Cs. Exactas y Naturales, UBA
\noi Pab. I, Ciudad Universitaria
\noi (1428) Buenos Aires, Argentina
\medskip
{\bf E-mail:} \quad pamster\@dm.uba.ar - mcmarian\@dm.uba.ar
\end
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