\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 44, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2011/44\hfil Optimizing SODE systems]
{Optimizing second-order differential equation systems}

\author[T. Hajba\hfil EJDE-2011/44\hfilneg]
{Tam\'as Hajba}

\address{Tam\'as Hajba \newline
Department of Mathematics and Computer Science
Faculty of Engineering Sciences, Sz\'echenyi Istv\'an
University, Egyetem t\'er 1., H-9026 Gy\H or, Hungary}
\email{hajbat@sze.hu, Tel +36 (96) 503-400, Fax +36 (96) 613-657}

\thanks{Submitted October 25, 2010. Published March 31, 2011.}
\subjclass[2000]{90C25, 65K05, 34D05}
\keywords{Fletcher-Reeves iteration; second-order differential
equation; \hfill\break\indent minimizing trajectory;
stationary point in limit; Lyapunov-type methods}

\begin{abstract}
 In this article we study some continuous versions of the
 Fletcher-Reeves iteration for minimization
 described by a system of second-order differential equations.
 This problem has been studied in earlier papers
 \cite{hajba-1,hajba-2} under the assumption that the minimizing
 function is strongly convex. Now instead of the strong convexity,
 only the convexity of the minimizing function will be required.
 We will use  the Tikhonov regularization \cite{t-dokl,banach}
 to obtain the minimal norm solution as the asymptotically stable
 limit point of the trajectories.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a  convex, continuously
differentiable function. Let us consider the minimization
problem
\begin{equation}
\min_{\mathbf{x}\in\mathbb{R}^n} f(\mathbf{x}),
\label{problem}
\end{equation}
where the function $f(\mathbf{x})$ satisfies the
following conditions:
\begin{equation}
f_{*}=\inf f(\mathbf{x})>-\infty,\quad
X_{*}=\{\mathbf{x}\in\mathbb{R}^n:
f(\mathbf{x})=f_{*}\}\neq\emptyset.\label{condi-f}
\end{equation}

Several methods have been developed for the solution of this
problem. The methods generated with an iterative process
can be modelled with differential equations. These differential
equations are usually called the continuous version of the method.

 Modelling the iterative numerical methods of
optimization with differential equations has been investigated in
several papers. Some of them deal with either the gradient or the
Newton's method and model the given method by a system of first
order differential equations
(e.g. \cite{antipin,archetti,attouch-cominetti,botsaris,
brown,cominetti-peypouquet-sorin,evt,flam,hauser-nedic,hoo,km1,km2,
venec,zhang} etc.). In this article we investigate two 
models of the continuous version of the Fletcher-Reeves iteraiton. 
Both of  them lead to the analysis of second-order differential equation 
systems (shortly SODE system). One of these models has not been studied 
earlier.

There is another approach  to the study of second order differential
equations  with the optimization that arise in physical problems
such as the heavy ball with friction. Results concerning such
type of second-order differential equation models can be found in
\cite{alvarez,alvarez-attouch,
attouch-goudou-redont, cabot-engler-gadat-a, cabot-engler-gadat-b,
goudou-munier,vasilev-amochkina-nedic, vasilev-nedic-2}.
There are also some papers discussing higher
order methods; e.g. \cite{vasilev-nedic-3,
vasilev-nedic-2,nedic}. However, the mentioned
papers deal with SODE systems that are linear in
$\dot{\mathbf{x}}$.
Since the Fletcher-Reeves
iteration uses the new state point in the construction of the new
direction, our system of second-order differential equations will
not necessary be linear in the first derivative vector
$\dot {\mathbf{x}}$. In connection with the optimization such type of
second-order differential equation has been investigated in
\cite{hajba-1} assuming the minimizing function being strongly
convex. The minimizing property of such type of 
second-order differential equation has not been investigated yet when the 
function is convex but necessary strongly convex. 
Since in this case the uniqueness of the
minimum point can not be guaranteed the
Tikhonov regularization will be used to obtain the so called
 minimal norm solution.

In this paper we  consider the SODE system describing the so
called heavy ball with friction  as a simplification of the
continuous version of the Fletcher-Reeves iteration using the old
state point in the construction of the new direction. Since the
regularized version of this type of differential equation is known
only if the coefficient of $\dot{\mathbf{x}}$ is constant (see
\cite{cabot-engler-gadat-a,cabot-engler-gadat-b}) we will show that
the convergence of the trajectories to the minimal norm solution is
valid with function-coefficient, too.

\section{Second-order differential equation models of minimization}

 As it was pointed out in \cite{km2} the
minimization models modelled by first order differential systems can
be divided into two classes. Those models described by a system of
first order differential equations for which the point
$\mathbf{x}_*$ is a stationary point of the system belong to the
first class. In this case the convergence of the trajectories to
$\mathbf{x}_*$ is equivalent with the asymptotic stability of
$\mathbf{x}_*$, therefore the Lyapunov function methods (see e.g. in
\cite{rouche}) are useful to prove the convergence with an
appropriately chosen Lyapunov function (see e.g.
\cite{evt,flam,venec}). To the second class of the
models belong those continuous first order models, for which the
minimum point is not stationary, but along the trajectories the
right hand side vector of the differential equation system tends to
the null-vector if $t\to \infty$. Following \cite{km2} we say in
this case, that $\mathbf{x}_*$ is \emph{stationary in limit}.

We extend this definition for the SODE systems, too. We will say,
that a point is \emph{stationary point}  or \emph{stationary in limit
point}  of a SODE system if it is stationary  or stationary in limit
point respectively for the equivalent first order system.

We will say, that a SODE system is a \emph{minimizing model} for the
  minimization problem \eqref{problem}-\eqref{condi-f}
if along its trajectories
  $\lim_{t\to \infty}
  f(\mathbf{x}(t))=f_*$. It is \emph{convergent} if   any trajectory  converges in norm to some
  $\mathbf{x}_*\in X_*$; i.e.,
  $\|\mathbf{x}(t)-\mathbf{x}_*\|\to 0$. The trajectories of a convergent minimizing
  model are called \emph{minimizing trajectories}.

It will be seen that the continuous version of the regularized
Fletcher-Reeves iteration belongs to the class of methods stationary
in limit both in the general and in the simplified cases.

As it was shown in \cite{km1,km2}  the Lyapunov-type methods are
also applicable to prove the convergence of the trajectories to a
point stationary in limit. Namely, it has been proved,  if the
chosen Lyapunov function along the trajectory of the differential
equality systems satisfies certain differential inequality on
$[t_0,\infty)$, then it tends to zero if $t\to \infty$. This
technique will be used in our proofs, too.


Here we describe one of the appropriate lemmas from \cite{km1}
which will be fundamental in our investigation to prove the
convergence of the trajectories to a stationary in limit minimum
point.

\begin{lemma}\label{s-lemma}
Suppose that there exists $T_0\ge 0$ such that
\begin{enumerate}
\item for every fixed $\tau\ge T_0$ the
scalar function $g(t,\tau)$ is defined and non-negative for all
$T_0\le t< \tau$ and $g(T_0,\tau)\le K$ uniformly in $\tau$,
furthermore, it is continuously differentiable in $t$;
\item
$g(t,\tau)$ satisfies the following differential inequality:
\begin{equation}\frac{{\rm d}}{{\rm
d}t}g(t,\tau)\le -a(t)g(t,\tau)+b(t)(\tau-t)^s
\label{km-lemma}\end{equation}  for $T_0\le t<\tau$ where $s$ is
nonnegative integer and the functions $a(t)>0$ and $b(t)$ are
defined for all $t\ge T_0$ and integrable on any finite interval of
$[T_0,\infty)$ and they are endowed with  the following properties:
\begin{itemize}
\item[(a)] $\int_{T_0}^{\infty}a(t)dt=\infty$,
\item[(b)] $\lim_{t\to\infty}\frac{b(t)}{a^{s+1}(t)}=0,$
\item[(c)]
In the case $s\geq 1$ the function $a(t)$ is differentiable and
$$
\lim_{t\to\infty}\frac{\dot a(t)}{a^2(t)}=0.
$$
\end{itemize}
\end{enumerate}
 Then
$\lim_{\tau\to\infty}g(\tau,\tau)=0$.
\end{lemma}

\begin{proof}
From \eqref{km-lemma} we have that
\[
0\le g(\tau,\tau)\le
g(T_0,\tau)\mathrm{e}^{-\int_{T_0}^{\tau}{a(\nu)}{\rm d}\nu}
+\int_{T_0}^{\tau}{b(\theta)(\tau-\theta)^s
\mathrm{e}^{\int_{\tau}^{\theta}{a(\nu)}{\rm
d}\nu}}{{\rm d}\theta}.
\]
The convergence of the first term  to zero follows from the
condition 2(a).


By induction on $s$ it can be proved that for all nonnegative
integer $s$ the limit
$\lim_{\tau\to\infty}\exp\big(\int_{T_0}^{\tau}{a(\nu)}{\rm
d}\nu\big)a^{s}(\tau)=\infty$ holds true and hence we can estimate the
second term by applying  $(s+1)$ times the
L'Hospital rule and the conditions 2(b) and 2(c):
\begin{align*}
&\lim_{\tau\to\infty}\frac{\int_{T_0}^{\tau}{b(\theta)(\tau-\theta)^s
\exp\big(\int_{T_0}^{\theta}{a(\nu)}{\rm d}\nu\big)}{{\rm d}\theta}}
{\exp\big(\int_{T_0}^{\tau}{a(\nu)}{{\rm d}\nu}\big){\rm d}\theta}\\
&= \lim_{\tau\to\infty}\frac{b(\tau)s!}{[a(\tau)]^{s+1}}
\lim_{\tau\to\infty}\prod_{j=0}^{s}\frac{1}{1+\frac{j\,\dot
a(\tau)}{[a(\tau)]^2}}=0.
\end{align*}
\end{proof}

The function $g(t,\tau)$ in the lemma constructed for a SODE problem
will be called \emph{Lyapunov-like function} of the model.

The focus of our interest is to formulate such SODE systems which
are convergent and minimizing and for which the minimum point with
the minimal norm is a stationary or stationary in limit point. Our
motivation to construct such mo\-dels of minimization was the
following:

The Fletcher-Reeves iteration to minimize a function of $n$ variables
starting from $\mathbf{x}_0$ and
$\mathbf{p}_0=-f'(\mathbf{x}_0)$ computes the pair of points
\begin{gather*}
\mathbf{x}_{k+1}=\mathbf{x}_k+\alpha_k\mathbf{p}_k\\
\mathbf{p}_{k+1}=-f'(\mathbf{x}_{k+1})+\delta_k\mathbf{p}_k
\quad k=1,2,\dots.
\end{gather*}
To obtain a convergent
process we have to use   well defined (here not detailed)
changing rules for the sequences $\alpha_k$ and $\delta_k$.

Taking into consideration that the Fletcher-Reeves iteration uses
the new state point in the construction of the new direction it is
easy to see that this iteration can be considered as the Euler
discretization with step size 1 of the non-autonomous first-order
differential equation system of $2n$ variables
\begin{gather}
\dot{\mathbf{x}}=\alpha(t)\mathbf{p}\label{complete-1}\\
\dot{\mathbf{p}}=-\nabla
f(\mathbf{x}+\alpha(t)\mathbf{p})+\beta(t)\mathbf{ p},\label{complete-2}\\
\mathbf{x}(t_0)=\mathbf{x}_0,\quad \mathbf{p}(t_0)=\mathbf{
p}_0,\label{kezd-first}
\end{gather}
where the changing rule of  the parameters are described by continuous
functions.

 We will refer to the model \eqref{complete-1}-\eqref{complete-2} as
\emph{general model} (shortly  Model G-FR) of the continuous
version of the Fletcher-Reeves iteration. This model is equivalent
with the SODE system of $n$ variable
\begin{gather}
\ddot{\mathbf{x}}+\gamma(t))\dot
{\mathbf{x}}+\alpha(t)\nabla f(\mathbf{x}+\dot{\mathbf{x}})=0,
\label{g-h}\\
\mathbf{x}(t_0)=\mathbf{x}_0,\quad \dot{
\mathbf{x}}(t_0)=\alpha(t_0){\mathbf{p}}_0,\label{kezd-second}
\end{gather}
where
\begin{equation}
\gamma(t)=-\beta(t)-\frac{\dot\alpha(t)}{\alpha(t)}.\label{gamma}
\end{equation}

If we approximate  $\nabla
f(\mathbf{x}(t)+\alpha(t){\mathbf{p}}(t))$ with
$\nabla f(\mathbf{x}(t))$ then we obtain a much more simple model,
namely
\begin{gather}
\dot{\mathbf{x}}=\alpha(t){\mathbf{p}}\label{simple-1}\\
\dot{{\mathbf{p}}}=-\nabla
f(\mathbf{x})+\beta(t){\mathbf{p}}\label {simple-2}
\end{gather}
with the initial values \eqref{kezd-first}.  This model will be
called
 \emph{simplified model} (shortly  Model S-FR) of the continuous
version of the Fletcher-Reeves iteration.
This model is equivalent with  the SODE system
\begin{equation}\ddot{
\mathbf{x}}+\gamma(t)\dot{ \mathbf{x}}+\alpha(t)\nabla
f(\mathbf{x})=0\label{s-h}\end{equation} with the initial values
\eqref{kezd-second}.


In \cite{hajba-1} the asymptotic behavior of the trajectories of
the  Model G-FR and  Model S-FR have been analyzed. It has
been proved that under the assumption of the strong convexity of the
function $f(\mathbf{x})$ there are such  harmonizing  conditions
between the parameter functions $\alpha(t)$ and $\beta(t)$ which
ensure that the differential equation system
\eqref{simple-1}-\eqref{simple-2} or
\eqref{complete-1}-\eqref{complete-2} is minimizing and the minimum
point $\mathbf{x}_*$ is an asymptotically stable stationary point to
which any trajectory tends if $t\to \infty$. Furthermore,  several
class of pairs of the functions $\alpha(t)$ and $\beta(t)$
satisfying the harmonization conditions has been given in
\cite{hajba-2}.

 The behavior of the trajectories of the second-order differential equation
 \eqref{s-h} has been investigated in
several papers assuming  that $\gamma(t)$ is a positive constant
function and $\alpha(t)\equiv 1$ (e.g.
\cite{alvarez,alvarez-attouch,attouch-cominetti,
attouch-goudou-redont,goudou-munier}). This is the so
called \emph{heavy ball with friction} model.  A detailed discussion
of the minimizing properties of the trajectories of \eqref{s-h} with
positive
 $\gamma(t)$ and $\alpha(t)\equiv 1$ functions have been given in
 the papers \cite{cabot-engler-gadat-a,cabot-engler-gadat-b}.


\section{Convergence theorems of the regularized
SODE models}

The strong convexity is too strict condition for most of the
practical optimization problems.

In this paper we will require only the convexity of the minimizing
function. But under this weaker assumption we can not expect that
the set of minimum points consists of only one point. Therefore, as
it will be shown in a numerical example in Section \ref{example}, it
can happen that either the discrete Fletcher-Reeves method or its
continuous versions  stop in different minimum points starting from
different initial points.

To avoid these problems a regularization technique is generally
used. The regularization means that the minimizing function will be
approximated with a bundle of strong convex functions depending on a
damping parameter.  Choosing the appropriate damping parameter  one
can expect that the sequence of the unique minimum points of the
auxiliary functions tends to one of the well defined minimum point
of the original minimizing function independently from the starting
point. The possibility of this type of regularization is based on
the following lemma due to  Tikhonov \cite{t-dokl,banach}.

\begin{lemma} \label{t-reg}
Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex  function
satisfying  \eqref{condi-f} and $\lambda_k$,
$k=1,2,\dots$ be a positive monotone decreasing sequence for which
$\lim_{k\to\infty}\lambda_k=0$. Let the auxiliary strong
convex function bundle defined by the sequence
\[
F_k(\mathbf{x})=f(\mathbf{x})+\frac{1}{2}\lambda_k\|\mathbf{x}\|^2,\quad
k=1,2,\dots
\]
 and let $\mathbf{x}_k$ denote the unique minimum
point of $F_k(\mathbf{x})$. Then
\[
\lim_{k\to\infty}\|\mathbf{x}_k-\mathbf{x}_*\|=0,
\]
where $\mathbf{x}_*$ is the minimal norm solution of \eqref{problem};
 i.e.,
\[
f(\mathbf{x}_*)=\inf_{\mathbf{x}\in\mathbb{R}^n} f(\mathbf{x})
\quad\text{and}\quad
\inf_{\mathbf{x}\in X_*}\|\mathbf{x}\|=\|\mathbf{x}_*\|,
\]
where $X_*$ is given in \eqref{condi-f}.
\end{lemma}

In lots of minimization methods the damping parameter can be
synchronized with the parameters of the used method modifying it
step by step.  Such regularized method is the Levenberg-Marquard
algorithm \cite{levenberg,marquard} for the Newton's method
which was developed independently from the Tikhonov-regularization.

The regularization of the minimization methods modelled by
differential equation systems  means that instead of the function
$f(\mathbf{x})$ and its first and higher order partial derivatives
the auxiliary function
\begin{equation}
F(\mathbf{x},t)=f(\mathbf{x})+\frac{1}{2}\lambda(t)\|\mathbf{x}\|^2.
\label{c-T}
\end{equation}
and its partial derivatives are used where the damping parameter
$\lambda(t)$  continuously changes in time.

For the continuous gradient method which is modelled by first-order
system of differential equations the regularization technique was applied
in \cite{km1}. Other approaches can be found in
\cite{attouch-cominetti} and \cite{cominetti-peypouquet-sorin}.
Regularized second and higher order models have been examined e.g.
in
\cite{attouch-charnecki,cabot,vasilev-amochkina-nedic,vasilev-nedic-2}.
Since second order dynamics are generally not descent methods hence they allow to overcome some drawbacks of the steepest descent method.

In the following we will discuss the convergence of the regularized
methods mo\-delled by \eqref{simple-1}-\eqref{simple-2}, (resp. by
\eqref{s-h}) and by \eqref{complete-1}-\eqref{complete-2}, (resp. by
\eqref{g-h}).

\subsection{Regularized general model}

The \emph{regularized general model} (shortly  RG-FR model) to
solve the problem \eqref{problem} can be given by   the following
first order system of differential equations of $2n$ variables:
\begin{gather}
\dot{\mathbf{x}}=\alpha(t){\mathbf{p}} \label{rg-1}\\
\dot{{\mathbf{p}}}=-\nabla_{\mathbf{x}}F(\mathbf{x}+\alpha(t){\mathbf{p}},t)+\beta(t){\mathbf{p}}\label{rg-2}
\end{gather}
with the initial values \eqref{kezd-first}, where $F(\mathbf{x},t)$
is given by \eqref{c-T} and the function $\lambda(t)$ is a monotone
decreasing positive function. This system is equivalent with the
SODE system of $n$ variables
\begin{equation}
\ddot {\mathbf{x}}+\gamma(t)\dot{
\mathbf{x}}+\alpha(t)\nabla_{\mathbf{x}}F(\mathbf{x}+\dot{
\mathbf{x}},t)=0.\label{rg-masod}
\end{equation} with the initial
values \eqref{kezd-second}, where $\gamma(t)$ is given by
\eqref{gamma}.

It can be seen that the difference between the  RG-FR model
and the  G-FR model is that instead of the partial derivatives
of the function $f(\mathbf{x})$ the partial derivatives of
the auxiliary function $F(\mathbf{x},t)$ are used.
%%

\begin{proposition} \label{rg}
Let us assume that the following hypotheses are satisfied:
\begin{enumerate}
\item  In the minimization problem
\eqref{problem} $ f$ is defined and continuously differentiable
convex function on $\mathbb{R}^n$ and its gradient $\nabla f$ is
local Lipschitz continuous; i.e., it is Lipschitz continuous on all
bounded subsets of $\mathbb{R}^n$ and the conditions given in
\eqref{condi-f} on page 1 hold;
\item  The parameter functions $\alpha(t)$,
$\beta(t)$ and $\gamma(t)$ of the systems
\eqref{rg-1}-\eqref{rg-2} and \eqref{rg-masod} fulfills the
following conditions:
\begin{itemize}
\item[(a)] $\alpha(t)$ is a positive, upper bounded
and continuously differentiable and $\beta(t)$ is a negative lower
bounded function on $[t_0,\infty)$ ;
\item[(b)] $\gamma(t)$ is a monotone non-increasing,
continuously differentiable function
 on $[t_0,\infty)$ and
  $\inf_{t\ge t_0}\gamma(t)> 1$ ;
 \end{itemize}
 \item  For the  damping parameter $\lambda(t)$ the
following assumptions hold:
\begin{itemize}
\item[(a)] $\lambda(t)$ is a positive continuously
differentiable monotone decreasing  function on $[t_0,\infty)$ and
convex for all $t\ge t_1$;
\item[(b)] $\alpha(t)\lambda(t)$ is a monotone non-increasing  function;
\item[(c)] $\lim_{t\to\infty}\lambda(t)=
 \lim_{t\to\infty}\dot\lambda(t)=0$,
 \[
 \lim_{t\to\infty}\frac{\dot\alpha(t)}{\alpha^2(t)\lambda(t)}=
 \lim_{t\to\infty}\frac{\dot\lambda(t)}{\alpha(t)\lambda^2(t)}=
 \lim_{t\to\infty}\frac{\dot\lambda(t)}{\alpha^2(t)\lambda(t)}=0;
 \]
\item[(d)] $\int_{t_0}^{\infty}\alpha(t)\lambda(t)
=\infty$.
\end{itemize} \end{enumerate}
 Then
\begin{enumerate}
\item the trajectories of \eqref{rg-1}-\eqref{rg-2}, respectively of
\eqref{rg-masod} exist and unique on the whole half-line
$[t_0,\infty)$ with any initial point \eqref{kezd-first};

\item the  RS-FR model given by \eqref{rg-1}-\eqref{rg-2} (or
\eqref{rg-masod}) is minimizing; i.e.,
\[
\lim_{t\to\infty}f(\mathbf{x}(t))=f(\mathbf{x}_*)=\inf_{\mathbf{x}\in
\mathbb{R}^n}f(\mathbf{x});
\]

\item  the trajectories converge to the minimal
norm solution; i.e., if $\;\mathbf{x}_*$ satisfies the condition
$\inf_{\mathbf{x}\in X_*}\|\mathbf{x}\|=\|\mathbf{x}_*\|$,
then $\lim_{t\to\infty}\|\mathbf{x}(t)-\mathbf{x}_*\|=0$;

\item $\lim_{t\to\infty}\|\alpha(t){\mathbf{p}}(t)\|
=\lim_{t\to\infty}\|\dot{\mathbf{x}}(t)\|=0$;

\item the minimal norm solution $\mathbf{x}_*$
 is a stationary in limit minimum point; i.e.,
$\lim_{t\to\infty}\|\ddot{\mathbf{x}}(t)\|=0$.
\end{enumerate}
\end{proposition}

\begin{proof}
The existence and uniqueness of the trajectories on the whole
$[t_0,\infty)$ follows from the convexity of the function $f(x)$ and
the local Lipschitz continuity of the gradient $\nabla
f(\mathbf{x})$.

 For every fixed $t_0<\tau<\infty$  the function
$F(\mathbf{x},\tau)$
defined by \eqref{c-T} is a strongly convex function, therefore it has
a unique minimum point $\mathbf{x}_{\tau}^*$. Let $\mathbf{x}_*$ be
the optimum point of the function $f$ with minimal norm on $X_*$. It
follows from the Lemma \ref{t-reg} that
$\lim_{\tau\to\infty}\|\mathbf{x}_\tau^*-\mathbf{x}_*\|=0$.

We will show that
$\lim_{\tau\to\infty}\|\mathbf{x}(\tau)-\mathbf{x}_*\|=0$. To
do this it is sufficient to prove that
$\lim_{\tau\to\infty}\|\mathbf{x}(\tau)-\mathbf{x}_{\tau}^*\|=0$
since $\|\mathbf{x}(\tau)-\mathbf{x}_*\|\leq
\|\mathbf{x}(\tau)-\mathbf{x}_{\tau}^*\|
+\|\mathbf{x}_{\tau}^*-\mathbf{x}_*\|$.

Let us introduce the parametric function
\begin{align*}
g(t,\tau)&= \frac{1}{2}\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*+\alpha(t){\mathbf{p}}(t)\|^2+\\
&\quad+\frac{1}{4}\alpha(t)\lambda(t)\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*\|^2
+\frac{1}{2}(\gamma(t)-1)\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*\|^2
\end{align*}
for fixed $\tau\ge t_0$.

It follows from the conditions 2(a), 2(b) and 3(a) that
$g(t,\tau)\ge 0$ for all $t_0\le t\le \tau$.  For the derivative of
$g(t,\tau)$ we have
\begin{align*}
 \frac{{\rm d}}{{\rm d}t}g(t,\tau)
&= -\alpha(t)\big\langle\nabla_{\mathbf{x}}{F(\mathbf{x}(t)+\alpha(t){\mathbf{p}}(t),t),\mathbf{x}(t)-\mathbf{x}_\tau^*+\alpha(t){\mathbf{p}}(t)\big\rangle}+\\
&\quad +\frac{1}{4}\frac{{\rm d}}{{\rm
d}t}(\alpha(t)\lambda(t))\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*\|^2
+\frac{1}{2}\alpha(t)\lambda(t)\big\langle \mathbf{x}(t)-\mathbf{x}_\tau^*,\alpha(t){\mathbf{p}}(t)\big\rangle\\
&\quad +(1-\gamma(t))\|\alpha(t){\mathbf{p}}(t)\|^2+
\frac{1}{2}\dot\gamma(t)\|\mathbf{x}(t)-\mathbf{x}_\tau^*\|^2
\end{align*}
for all $t_0\le t\le \tau$.

Omitting the  negative terms and taking into consideration that
$F(\mathbf{x}(t),t)$ is  strongly convex in its first variable with
the convexity modulus $\frac{1}{2}\lambda(t)$ for every $t\ge t_0$,
monotone decreasing in the second variable and $\mathbf{x}_\tau^*$
is the minimum point of $F(\mathbf{x},\tau)$ we have
\begin{align*}
\frac{{\rm d}}{{\rm d}t}g(t,\tau)
&\le \alpha(t)\big(F(\mathbf{x}_\tau^*,t)-F(\mathbf{x}(t)+\alpha(t){\mathbf{p}}(t),t)\big)-\\
&\quad -\frac{1}{2}\alpha(t)\lambda(t)\|\mathbf{x}(t)-\mathbf{x}_\tau^*
+\alpha(t){\mathbf{p}}(t)\|^2+
\frac{1}{2}\alpha(t)\lambda(t)\big\langle \mathbf{x}(t)
-\mathbf{x}_\tau^*,\alpha(t){\mathbf{p}}(t)\big\rangle \\
&=\alpha(t)\Big(
\underbrace{F(\mathbf{x}_\tau^*,t)-F(\mathbf{x}_\tau^*,\tau)}_
{\scriptstyle{=-\frac{1}{2}(\lambda(\tau)-\lambda(t))\|\mathbf{x}_\tau^*\|^2}}+
\underbrace{F(\mathbf{x}_\tau^*,\tau)-F(\mathbf{x}(t)+\alpha(t){\mathbf{p}}(t),\tau)}_{\scriptstyle{\le
0}}\\
&\quad +\underbrace{F(\mathbf{x}(t)+\alpha(t){\mathbf{p}}(t),\tau)
-F(\mathbf{x}(t)+\alpha(t){\mathbf{p}}(t),t)}_
{\scriptstyle{=\frac{1}{2}(\lambda(\tau)-\lambda(t))\|\mathbf{x}(t)
+\alpha(t){\mathbf{p}}(t)\|^2\le 0}} \Big)\\
&\quad -\frac{1}{2}\alpha(t)\lambda(t)\|\mathbf{x}(t)-\mathbf{x}_\tau^*+\alpha(t){\mathbf{p}}(t)\|^2+
\frac{1}{2}\alpha(t)\lambda(t)\big\langle
\mathbf{x}(t)-\mathbf{x}_\tau^*,\alpha(t){\mathbf{p}}(t)\big\rangle
\\
&\le-\frac{1}{2}\alpha(t)\lambda(t)\|\mathbf{x}(t)-\mathbf{x}_\tau^*+\alpha(t){\mathbf{p}}(t)\|^2+
\frac{1}{2}\alpha(t)\lambda(t)\big\langle
\mathbf{x}(t)-\mathbf{x}_\tau^*,\alpha(t){\mathbf{p}}(t)\big\rangle\\
&\quad -\frac{1}{2}\alpha(t)\big(\lambda(\tau)-\lambda(t)\big)
\|\mathbf{x}_\tau^*\|^2.
\end{align*}

 Under the assumption 3(a)   the inequalities
\[
\lambda(\tau)-\lambda(t)\ge
\dot\lambda(t)(\tau-t), \quad \dot\lambda(t)<0
\]
hold for all $t_0\le t\leq \tau$. Moreover, let us observe that
$\|\mathbf{x}_\tau^*\|$ is uniformly bounded since
\[
f(\mathbf{x}_*)+\frac{1}{2}\lambda(\tau)\|\mathbf{x}_*\|^2
\ge f(\mathbf{x}_\tau^*)+\frac{1}{2}\lambda(\tau)
 \|\mathbf{x}_\tau^*\|^2
\ge f(\mathbf{x}_*)+\frac{1}{2}\lambda(\tau)\|\mathbf{x}_\tau^*\|^2,
\]
from where $\|\mathbf{x}_\tau^*\|\le \|\mathbf{x}_*\|=K$.

Decomposing
$-\frac{1}{2}\alpha(t)\lambda(t)\|\mathbf{x}(t)+\alpha(t){\mathbf{p}}(t)-\mathbf{x}_\tau^*\|^2$
into two equal terms and omitting the negative term
$-\frac{1}{4}\alpha(t)\lambda(t)\|\alpha(t){\mathbf{p}}(t)\|^2$  we
have that
\begin{align*}
\frac{{\rm d}}{{\rm d}t}g(t,\tau)
&\le -\frac{1}{4}\alpha(t)\lambda(t)\|\mathbf{x}(t)
 +\alpha(t){\mathbf{p}}(t)-\mathbf{x}_\tau^*\|^2\\
&\quad -\frac{1}{4}\alpha(t)\lambda(t)\|\mathbf{x}-\mathbf{x}_{\tau}^*\|^2
-\frac{1}{2}\alpha(t)\big(\lambda(t)-\lambda(\tau)\big)
 \|\mathbf{x}_{\tau}^*\|^2\\
&= -A(t)\frac{1}{2}\|\mathbf{x}(t)+\alpha(t){\mathbf{p}}(t)
 -\mathbf{x}_\tau^*\|^2
-B(t)\frac{1}{4}\alpha(t)\lambda(t)\|\mathbf{x}-\mathbf{x}_{\tau}^*\|^2\\
&\quad -C(t)\frac{1}{2}(\gamma(t)-1)\|\mathbf{x}
 -\mathbf{x}_{\tau}^*\|^2 -\frac{1}{2}\alpha(t)\dot\lambda(t)
K^2(\tau-t),
\end{align*}
where $A(t)=\frac{1}{2}\alpha(t)\lambda(t)$, $B(t)=\frac{1}{2}$
and $C(t)=\frac{1}{4(\gamma(t)-1)}\alpha(t)\lambda(t)$
 Since $\gamma(t)$ is monotone nonincreasing, therefore
$C(t)\ge
\frac{\alpha(t)\lambda(t)}{4(\gamma(t_0)-1)}=C_1\alpha(t)\lambda(t)$.
Otherwise, $\alpha(t)\lambda(t)$ is decreasing and tends to zero, so
there exists $T\ge t_0$ such that $A(t)\le \frac{1}{2}$ and
$C_1(t)\le \frac{1}{2}$ for every $t\ge T$. Consequently, there
exists $K_1>0$, depending only on $\gamma(t_0)$ such that
\[
\frac{{\rm d}}{{\rm d}t}g(t,\tau)\le
-K_1\alpha(t)\lambda(t)g(t,\tau)-\frac{1}{2}\alpha(t)
\dot\lambda(t)K^2(\tau-t).
\]
Conditions 3(c) and 3(d) ensure that $g(t,\tau)$ satisfies the
conditions of Lemma \ref{s-lemma} and hence
$\lim_{\tau\to\infty}g(\tau,\tau)=0$.

Since $g(t,\tau)$ is a sum of non-negative functions every member of
the sum tends to $0$. This together with condition 2(b) proves
the validity of
\[
\lim_{\tau\to\infty}\|\mathbf{x}(\tau)-\mathbf{x}_\tau^*\|=0
\quad\text{and}\quad
\lim_{\tau\to\infty}\|\mathbf{x}(\tau)-\mathbf{x}_\tau^*
+\alpha(\tau){\mathbf{p}}(\tau)\|=0.
\]
It follows from the triangle inequality that
\[
\|\alpha(\tau)\mathbf{p}(\tau)\|\le\|\mathbf{x}(\tau)-\mathbf{x}_\tau^*+\alpha(\tau){\mathbf{p}}(\tau)\|+
\|\mathbf{x}(\tau)-\mathbf{x}_\tau^*\|\to 0\] which proves the limit
\[\lim_{\tau\to\infty}\|\dot{\mathbf{x}}(\tau)\|=
\lim_{\tau\to\infty}\|\alpha(\tau)\mathbf{p}(\tau)\|=0.
\]

Since
\[
0\le \|\ddot {\mathbf{x}}(t)\|
\le \alpha(t)\big(\|\nabla
f(\mathbf{x}(t)+\alpha(t)\mathbf{p}(t))\|
+\lambda(t)\|\mathbf{x}(t)+\alpha(t)\mathbf{p}(t)\|\big)
+ \gamma(t)\cdot \|\dot{\mathbf{x}}(t)\|,
\]
the gradient $\nabla f(\mathbf{x})$ is continuous and the conditions
2(a), 2(b) and 3(c) hold, therefore
 $\|\ddot {\mathbf{x}}(t)\|\to 0$.

Finally, using the continuity of the function $f$ the limit
\[
 \lim_{\tau\to\infty}f(\mathbf{x}(\tau))=f(\mathbf{x}_*)
\]
 holds, too.
The last statement is trivial from the definition.
\end{proof}


\subsection{Regularized simplified model}

Approximating $\nabla_{\mathbf{x}} F(x(t)+\alpha(t)p(t),t)$ by
$\nabla_{\mathbf{x}} F(x(t),t)$
 the \emph{regularized simplified model} (shortly  RS-FR model)
to solve the problem \eqref{problem} can be given by the following
first order system of differential equations:
\begin{gather}
\dot{\mathbf{x}}=\alpha(t){\mathbf{p}} \label{rs-1}\\
\dot{{\mathbf{p}}}=
-\nabla_{\mathbf{x}}F(\mathbf{x},t)+\beta(t){\mathbf{p}}\label{rs-2}
\end{gather}
with the initial values \eqref{kezd-first},
 where $F(\mathbf{x},t)$ is given by \eqref{c-T} in which the damping
 parameter
$\lambda(t)$ is a positive monotone decreasing function.

The equivalent SODE system  is as follows:
\begin{equation}
\ddot{\mathbf{x}}+\gamma(t)\dot{
\mathbf{x}}+\alpha(t)\nabla_{\mathbf{x}}F(\mathbf{x},t)=0.
\label{rs-masod}
\end{equation}
with the initial values \eqref{kezd-second}, where $\gamma(t)$ is
given by \eqref{gamma}.

The convergence of the trajectories of this SODE to a minimum point of
the function $f(\mathbf{x})$ has been  analyzed in detail in papers
of \cite{attouch-charnecki} and \cite{cabot} when both $\alpha(t)$
and $\gamma(t)$ are constant functions. Now we formulate a theorem
on the convergence of its trajectories to the stationary in limit
minimal norm solution with function parameters and prove it by
constructing an appropriate Lyapunov-like function for the
RS-FR model given by \eqref{rs-1}-\eqref{rs-2}, respectively by
\eqref{rs-masod}.


\begin{proposition}\label{rs}
   Let the following assumptions hold:
   \begin{enumerate}
\item In the minimization problem \eqref{problem} $ f$ is defined and
continuously differentiable convex function on $\mathbb{R}^n$ and
its gradient $\nabla f$ is local Lipschitz continuous and the
conditions given in \eqref{condi-f} on page 1 hold;

\item The parameter  functions $\alpha(t)$ and $\beta(t)$ satisfy the
following conditions
\begin{itemize}
\item[(a)] $\alpha(t)$ is a positive upper bounded and
$\beta(t)$ is a negative lower bounded con\-tinuously differentiable
function on $[t_0,\infty)$; both $\alpha(t)$ and $\beta(t)$ are
continuously differentiable on $[t_0,\infty)$ and
$\frac{\dot\alpha(t)}{\alpha(t)}$ is bounded on $[t_0,\infty)$;
\item[(b)] there exists $t_1\ge t_0$ such that $\alpha(t)+\beta(t)<0$
and $\frac{\beta(t)}{\alpha(t)}$ is nondecreasing on
$[t_1,\infty)$;
\end{itemize}
\item Let the damping parameter $\lambda(t)$ satisfy
the following conditions
\begin{itemize}
\item[(a)] $\lambda(t)$ is a positive, continuously
differentiable monotone decreasing con\-vex function on
$[t_0,\infty)$;
\item[(b)]
$\lim_{t\to\infty}\lambda(t)=
\lim_{t\to\infty}\dot\lambda(t)=\lim_{t\to\infty}\frac{\dot\lambda(t)}{\lambda^2(t)}=0,$
\item[(c)]
$\int_{t_0}^{\infty}\lambda (t){\rm d}t=\infty$;
\item[(d)]$\alpha (t)+\beta (t) \le
-\frac{1}{2}\lambda(t)$ for every $t_1\le t$;
\item[(e)]
$ -\frac{\dot\alpha(t)}{\alpha(t)}- \frac{\alpha(t)}{2} \le
-\frac{1}{4}\lambda(t)$ for all $t_1\le t$.
\end{itemize}
\end{enumerate}
 Then
\begin{enumerate}
\item {the trajectories of \eqref{rs-1}-\eqref{rs-2}, respectively of
\eqref{s-h} exist and unique on the whole half-line $[t_0,\infty)$
with any initial point \eqref{kezd-first} (resp.
\eqref{kezd-second});}
\item  the  RS-FR model given by
\eqref{rs-1}-\eqref{rs-2} (or \ref{rs-masod}) is minimizing; i.e.,
\[
\lim_{t\to\infty}f(\mathbf{x}(t))=f(\mathbf{x}_*)=\inf_{\mathbf{x}\in
\mathbb{R}^n}f(\mathbf{x});
\]
\item  the trajectories converge to the minimal
norm solution; i.e., if $\mathbf{x}_*$ satisfies the condition
$\inf_{\mathbf{x}\in X_*}\|\mathbf{x}\|=\|\mathbf{x}_*\|$,
then $\lim_{t\to\infty}\|\mathbf{x}(t)-\mathbf{x}_*\|=0$;

\item $\lim_{t\to\infty}\|{\mathbf{p}}(t)\|=\lim_{t\to\infty}\|\dot{
\mathbf{x}}(t)\|=0$;

\item  the minimal norm solution $\mathbf{x}_*$ is a
stationary in limit minimum point; i.e.,
$\lim_{t\to\infty}\|\ddot{\mathbf{x}}(t)\|=0$.
\end{enumerate}
\end{proposition}


\begin{proof}
 Analogously to the proof of the Proposition \ref{rg}
 it is sufficient to prove that
\[
\lim_{\tau\to\infty}\|\mathbf{x}(\tau)-\mathbf{x}_{\tau}^*\|=0.
\]
Let us introduce the function
\begin{align*}
g(t,\tau)&=\frac{2}{\alpha(t)} \Big(F(\mathbf{x}(t),t)
 -F(\mathbf{x}_\tau^*,\tau) \Big)
+\frac{1}{2}h(t)\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*\|^2\\
&\quad +\frac{1}{2}\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*
+{\mathbf{p}}(t)\|^2+\frac{1}{2}\|{\mathbf{p}}(t)\|^2,
\end{align*}
where $h(t)=-1-\frac{\beta(t)}{\alpha(t)}>0$. This function is
defined  for all $t\in [t_0,\tau]$, for every fixed $\tau<\infty$ and
$g(t,\tau) \ge 0$, in these intervals since $\lambda(t)$ is monotone
decreasing.

For all $t_0\le t\le \tau$ the derivative of $g(t,\tau)$ by $t$ with
a fixed $\tau$ is
\begin{align*}
\frac{{\rm d}}{{\rm d}t}g(t,\tau)
&=\frac{-2\dot\alpha(t)}{\alpha^2(t)}
\Big(F(\mathbf{x}(t),t)-F(\mathbf{x}_{\tau}^*,\tau)\Big)
+\frac{\dot\lambda(t)}{\alpha(t)}\|\mathbf{x}(t)\|^2\\
&\quad + \frac{1}{2}\dot
h(t)\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*\|^2+\big(\alpha(t)
+\beta(t)+h(t)\alpha(t)\big)\big\langle
\mathbf{x}(t)-\mathbf{x}_\tau^*,{\mathbf{p}}(t)\big\rangle\\
&\quad +\big(\alpha(t)+2\beta(t)\big)\|{\mathbf{p}}(t)\|^2
-\big\langle\nabla_{\mathbf{x}}F(\mathbf{x}(t),t),\mathbf{x}(t)
-\mathbf{x}_{\tau}^*\big\rangle.
\end{align*}
Taking into consideration   the conditions 2,
3(a) and 3(c),  we obtain
\begin{align*}
\frac{{\rm d}}{{\rm d}t}g(t,\tau)
&\le \frac{-2\dot\alpha(t)}{\alpha^2(t)}
\Big(F(\mathbf{x}(t),t)-F(\mathbf{x}_{\tau}^*,\tau)\Big)
-\frac{1}{2}\lambda(t)\|{\mathbf{p}}(t)\|^2 \\
&\quad -\big\langle\nabla_{\mathbf{x}}F(\mathbf{x}(t),t),\mathbf{x}(t)
-\mathbf{x}_{\tau}^*\big\rangle,
\end{align*}
for all $t_1\le t\le \tau$.

 Since $F(\mathbf{x},t)$ is a strongly
convex function in the variable $\mathbf{x}$ for all $t\ge t_0$ and
its convexity modulus is $\frac{1}{2}\lambda(t)$ for every
$t_0\le t$, therefore for all $t\ge t_1$, we have the inequality
\begin{align*}
&-\big\langle \nabla_{\mathbf{x}}F(\mathbf{x}(t),t),
\mathbf{x}(t)-\mathbf{x}_{\tau}^*\big\rangle \\
&\leq -\big(F(\mathbf{x}(t),t)-F(\mathbf{x}_\tau^*,t)\big)
 -\frac{1}{2}\lambda(t)\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*\|^2\\
&= -\big(F(\mathbf{x}(t),t)-F(\mathbf{x}_\tau^*,\tau)\big)
-\big(F(\mathbf{x}_\tau^*,\tau)-F(\mathbf{x}_\tau^*,t)\big)-
\frac{1}{2}\lambda(t)\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*\|^2\\
&\quad \le -\big(F(\mathbf{x}(t),t)-F(\mathbf{x}_\tau^*,\tau)\big)
 -\frac{1}{2}\big(\lambda(\tau)-\lambda(t)\big)\|\mathbf{x}_\tau^*\|^2-
\frac{1}{2}\lambda(t)\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*\|^2.
\end{align*}
 Substituting this inequality into the estimation of
$\frac{\rm d}{{\rm d}t}g(t,\tau)$  we can obtain the inequality
\begin{align*}
\frac{{\rm d}}{{\rm d}t}g(t,\tau)
&\leq \Big(-\frac{2\dot\alpha(t)}{\alpha^2(t)}-1\Big)
\Big(F(\mathbf{x}(t),t)- F(\mathbf{x}_\tau^*,\tau)\Big)\\
&\quad -\frac{1}{2}\lambda(t)\|\mathbf{x}(t)-\mathbf{x}_{\tau}^*\|^2 -
\frac{1}{2}\lambda(t)\|{\mathbf{p}}(t)\|^2
-\frac{1}{2}\big(\lambda(\tau)-\lambda(t)\big)\|\mathbf{x}_{\tau}^*\|^2.
\end{align*}
for all $t_1\le t\le \tau$.
Since the inequality
\[
-\|\mathbf{x}(t)-\mathbf{x}_\tau^*\|^2-\|{\mathbf{p}}\|^2\le -
\frac{1}{2}\|\mathbf{x}(t)-\mathbf{x}_\tau^*\|^2
-\frac{1}{2}\|{\mathbf{p}}(t)\|^2-
\frac{1}{4}\|\mathbf{x}(t)-\mathbf{x}_\tau^*+{\mathbf{p}}(t)\|^2
\]
and the conditions 3(c)-3(d) of the
proposition hold,  with the coefficients
\[
A(t)=\frac{\dot\alpha(t)}{\alpha(t)}+\frac{\alpha(t)}{2},\quad
B(t)=\frac{\lambda(t)}{2h(t)}, \quad
C(t)=\frac{1}{2}\lambda(t),\quad
D(t)=\frac{1}{4}\lambda(t)
\]
we obtain, for all $t_1\le t\le \tau$,
\begin{align*}
\frac{{\rm d}}{{\rm d}t}g(t,\tau)
& \leq -A(t)\cdot\frac{2}{\alpha(t)}\Big(F(\mathbf{x}(t),t)-
F(\mathbf{x}_\tau^*,\tau)\Big) -B(t)\frac{1}{2}h(t)\|\mathbf{x}(t)
-\mathbf{x}_{\tau}^*\|^2 \\
&\quad - C(t)\frac{1}{2}\|{\mathbf{p}}(t)\|^2
-D(t)\frac{1}{2}\|\mathbf{x}(t)-\mathbf{x}_\tau^*+{\mathbf{p}}(t)\|^2
-\frac{1}{2}\big(\lambda(\tau)-\lambda(t)\big)\|\mathbf{x}_{\tau}^*\|^2.
\end{align*}
 It is obvious that $-C(t)\le -D(t)$, and
from the condition 3(e) we have that $-A(t)\le -D(t)$, too. After a
short calculation we can obtain that
\[
-B(t)\le -D(t) \quad\text{if }h(t)\le 2 \quad
\text{and} \quad -B(t)\ge -D(t) \quad \text{if }  h(t)\ge 2.
\]
Since $h(t)$ is nonincreasing, there are two cases:

\noindent\textbf{Case 1.} $h(t)\ge 2$ (or equivalently
$3\alpha(t)+\beta(t)\le 0$) for all $t\ge t_1$. In this case
$-B(t)=\max(-A(t),-B(t),-C(t),-D(t)$ for all $t\ge t_1$. It means
that
\[
\frac{{\rm d}}{{\rm d}t}g(t,\tau) \le -B(t)g(t,\tau)-
\frac{1}{2}\big(\lambda(\tau)-\lambda(t)\big)
\|\mathbf{x}_{\tau}^*\|^2
\]
for all $t_1\le t\le \tau$.
 Using the definition of $B(t)$
and the the fact, that $h(t_1)\ge h(t)$ for all $t\ge t_1$ we can
give the following upper bound:
\[
-B(t)=-\frac{\lambda(t)}{2h(t)}\le
-\frac{\lambda(t)}{2h(t_1)},
\]
consequently,
\[
\frac{{\rm d}}{{\rm d}t}g(t,\tau) \leq -\frac{\lambda(t)}{2h(t_1)}g(t,\tau)-
\frac{1}{2}\big(\lambda(\tau)-\lambda(t)\big)
\|\mathbf{x}_{\tau}^*\|^2
\]
for all $t_1\le t\le \tau$.

\noindent\textbf{Case 2.}
There exists $t_2\ge t_1$ such that $h(t)\le 2$ (or
equivalently $3\alpha(t)+\beta(t)\ge 0$) for all $t\ge t_2$. Then
$-D(t)=\max(-A(t),-B(t),-C(t),-D(t)$ for all $t\ge t_2$, therefore
\[
\frac{{\rm d}}{{\rm d}t}g(t,\tau) \leq -\frac{1}{4}\lambda(t)g(t,\tau)-
\frac{1}{2}\big(\lambda(\tau)-\lambda(t)\big)\|\mathbf{x}_{\tau}^*\|^2
\]
for all $t_2\le t\le \tau$.

The estimation of the last term in both cases can be done as in the
proof of Proposition \ref{rg}.
 So, in both cases there exists
a positive constant $K_1$ and time $T\ge t_0$ such that   the
inequality
\[
\frac{{\rm d}}{{\rm d}t}g(t,\tau)\leq
-K_1\lambda (t)g(t,\tau)-\frac{1}{2}K^2\dot\lambda(t)(\tau-t)
\]
holds for all $T\le t\le \tau$.
To complete the proof one can follow  the proof of the Proposition
\ref{rg}.
 \end{proof}


\section{Analysis and comparison of the methods}

\subsection{Existence of parameters}

 For both models one can give the triplet of parameter
functions $(\alpha(t),\beta(t),\lambda(t))$ such that conditions of
the propositions are satisfied. Namely,
\begin{itemize}
\item[(A)] for the   RG-FR model
\begin{itemize}
\item[(a)] if
\begin{gather*}
\alpha(t)=\alpha_0,\\
\gamma(t)=-\beta(t)=-\beta_0-{B}{(1+t)^{-b}},\\
\lambda(t)={L}{(1+t)^{-\ell}},
\end{gather*}
 then the conditions of
the proposition \ref{rg} are fulfilled if either
\[
b=0,\quad \alpha_0>0,\quad \beta_0+B<-1,\quad 0<\ell< 1,\quad
 L> 0,
\]
or
\[
b>0,\quad \alpha_0>0,\quad \beta_0<-1,\quad B<0,\quad 0<\ell< 1,\quad
 L> 0;
\]
\item[(b)] if
\begin{gather*}
\lambda(t)=\alpha(t)={\alpha_0}{(1+t)^{-a}}, \\
-\beta(t)=-\beta_0-{B}{(1+t)^{-1}} ,
\end{gather*}
 then the conditions of the proposition \ref{rg} are fulfilled if
\[
\alpha_0>0,\quad  \beta_0<-1, \quad\frac{1}{2}>a\geq B>0.
\]
\end{itemize}
\item[(B)] for the  RS-FR model
\begin{itemize}
\item[(a)]
if
\begin{gather*}
\alpha(t)=\alpha_0,\\
\gamma(t)=-\beta(t)=-\beta_0-{B}{(1+t)^{-\ell}},\\
\lambda(t)={L}{(1+t)^{-\ell}},
\end{gather*}
then the conditions of the proposition \ref{rs} are fulfilled
if
 \[ 0<\alpha_0,\quad \beta_0<-\alpha_0,\quad
B<0,\quad  0<\ell< 1,\quad L> 0,
\]
\item[(b)] if
\begin{gather*}
\alpha(t)=\alpha_0(1+t)^{-\ell},\\
\beta(t)=-\beta_0(1+t)^{-\ell}, \\
\lambda(t)=L(1+t)^{-\ell}
\end{gather*}
then the conditions of the proposition \ref{rs} are fulfilled if
\[
\alpha_0>0,\quad \beta_0>0,\quad L>0,\quad 0<\ell<1,\quad
 2(\alpha_0-\beta_0)<-L,\quad L<2\alpha_0.
\]
\end{itemize}
\end{itemize}

More families of parameters satisfying the conditions of the
proposition can be obtained by the technique given in
\cite{hajba-2}.

\begin{figure}[thp]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1}
\end{center}
\caption{Trajectory of the continuous method for the RS-FR model}
\label{fig1}
\end{figure}

\begin{figure}[thp]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig2}
\end{center}
\caption{Trajectories of the continuous method for the RG-FR model}
\label{fig2}
\end{figure}

\subsection{Comparison of the generalized and simplified
models}\label{example}

Let us  illustrate the behaviour of the trajectories of the given
models on a numerical example.
Let us minimize the the function
\[
f(x,y,z)=(x+y-3)^2+(x+z-3)^2.
\]
where $f$ is a convex function and the
minimum points of $f$ lie on the line
\[
x=3-t,\quad y=t, \quad z=t.
\]
The minimum point with the minimum norm is $(2,1,1)$.
We have solved the RS-FR model and the  RG-FR model
with the third ordered Runge-Kutta method with different
parameter functions and with two
different initial points $x_0=(1,2,4)$ and $x_0=(4,2,-3)$ and step
size $h=0.1$. The results can be seen in Figures \ref{fig1}-ref{fig2}.

On Figure \ref{fig1} we have drawn the minimizing lines
obtained by the discrete Fletcher-Reeves algorithm which show that
starting from different initial points the obtained minimum points
could be different.

On the other hand both the generalized and simplified models
converge to the unique minimal norm solution. However we can see,
that the shape of the trajectories are quite different in the two
models, especially the  RG-FR model gives a ``smoother'' trajectory.
Since in the  RS-FR model $\nabla_{\mathbf{x}} F(x(t)+\alpha(t)p(t),t)$
 is approximated by
$\nabla_{\mathbf{x}} F(x(t),t)$ we can expect that the  RG-FR model
 converges faster to the minimum point but the  RS-FR model
could be easier to solve numerically.

\subsection{Comparison of the heavy ball with friction and the
simplified models}

Let us consider the system
\begin{equation}
\ddot {\mathbf{x}}+\gamma\dot{
\mathbf{x}}+\nabla
f(\mathbf{x})+\lambda(t)\mathbf{x}=0.\label{at-c}
\end{equation}
where $\gamma$ is a constant. This equation is known as the
regularized version of the heavy ball system with friction model and
has been  studied  in papers \cite{attouch-charnecki} and
\cite{cabot}.  If we assume that $\alpha(t)\equiv 1$ and
$\gamma(t)\equiv \gamma$ in the  RS-FR-model \eqref{rs-masod},
then the regularized heavy ball with friction model can be
considered as a special case of it. In this special case our
proposition turns into the following result.

\begin{corollary}\label{cor-s}
Under  assumption 1. of Proposition \ref{rs},
the trajectories of the SODE system \eqref{at-c} exist
and unique on the whole half-line $[t_0,\infty)$ with any initial
point and the following limits hold
\[
\lim_{t\to\infty}f(\mathbf{x}(t))=f(\mathbf{x}_*)=\inf_{\mathbf{x}\in
\mathbb{R}^n}f(\mathbf{x}), \quad
\lim_{t\to\infty}\|\mathbf{x}(t)-\mathbf{x}_*\|=0,
\]
where $\mathbf{x}_*$ is the minimal norm solution of \eqref{problem}
and
\[
\lim_{t\to\infty}\|\dot {\mathbf{x}}(t)\|=0,\quad
\lim_{t\to\infty}\|\ddot {\mathbf{x}}(t)\|=0,
\]
if the following four conditions hold:
\begin{enumerate}
\item $ \gamma>1$;
\item    $\lambda(t)$ is positive monotone decreasing continuously
differentiable convex function  on $[t_0,\infty)$;
\item $\lim_{t\to\infty}\lambda(t)=
\lim_{t\to\infty}\dot\lambda(t)=\lim_{t\to\infty}\frac{\dot\lambda(t)}{\lambda^2(t)}=0;$
\item $\int_{t_0}^{\infty}\lambda (t){\rm d}t=\infty$.
\end{enumerate}
\end{corollary}

According to the convergence conditions of the theorems in
\cite{attouch-charnecki} and \cite{cabot} the condition 1, the
third term of the condition 3 and the convexity of $\lambda(t)$ can
be omitted.  However we wanted to give common conditions which
guarantee the convergence of the trajectories without doing
difference between the  cases when the coefficient of
$\dot{\mathbf{x}}$ is  a positive constant or a function. So, on one
hand our result is  weaker and on the other hand it is stronger then
the results of \cite{attouch-charnecki} and \cite{cabot}.

Otherwise in our models (not only in the simplified but in the
generalized one, too) there is a function parameter $\alpha(t)$ in
the coefficient of the gradient of the function. It is true, that
applying a time-transformation $t=z(s)$ this function parameter
turns into constant 1 if we get the transformation  from the
differential equation
\[
\frac{{\rm d}z(s)}{{\rm d}s}=\frac{1}{\sqrt{\alpha(z(s))}},
\]
but the transformed $\gamma(z(s))$ will be constant only for a
special function of $\gamma(t)$. So, the heavy ball with friction
model with constant $\gamma$ in general can not be obtained
from our model by time-transformation.

The discrete Fletcher-Reeves iteration has  two parameters.
Therefore we have insisted on such models which has two
corresponding function parameters, too.

The Fletcher-Reeves iteration has some very favorable properties
which have not been investigated in this paper. It would be interesting
to know which properties preserved in the proposed continuous
GM-FR and SM-FR models. This is the subject  of our further
research.


\begin{thebibliography}{28}

\bibitem{alvarez} Alvarez, F.;
\emph{On the minimizing property of a second order
dissipative system in Hilbert space}. SIAM J. on Control and
Optimization, 38(4)(2000):1102--1119.

\bibitem{antipin} {Antipin A. S.;}
\emph{Minimization of convex functions on convex
sets by means of differential equations}. Differential Equations
30(9)(1994):1365--1375. Translated from:   Antipin A. S.,
\emph{Minimizatsiya vypuklykh funktsi\u{i} na vy\-puk\-lykh
mnozhestvakh s pomosh'yu differentsialp'nykh uravneni\u{i}},
Differentsialp1nye uravneniya, 30(9)(1994):1475--1486.

\bibitem{archetti} {Archetti F.}, {Szeg\H o G. P.};
\emph{Global optimization algorithms}, in: Dixon L.C.W.,
Spedicato G.P. (eds), \emph{Nonlinear
Optimization: Theory and Algorithms}, Birkhauser, 1980, 429--469.

\bibitem{alvarez-attouch} {Attouch H.};
\emph{An inertial proximal method for maximal
monotone operators via discretization of a nonlinear oscillator with
damping}. Set-Valued Analysis, 9(2001): 3--11.

\bibitem{attouch-cominetti} {Attouch H.}, {Cominetti R.};
\emph{A dynamical approach to convex
minimization coupling approximation with the steepest descent
method}. J. Differential Equations, 128(2)(1996):519--540.

\bibitem{attouch-charnecki} {Attouch H.}, {Czarnecki M-O.};
 \emph{Asymptotic control and
stabilization of nonlinear oscillators with non isolated
equilibria}. J. Differential Equations, 179(1)(2002): 278--310.

\bibitem{attouch-goudou-redont} {Attouch H.}, {Goudou X.}, {Redont P.};
\emph{The heavy ball with friction method.~ {I.} {The continuous
dynamical system}}. Comm. Contemp. Math., 2(1)(2000):1--34.

\bibitem{botsaris} Botsaris C. A.; \emph{Differential
gradient method}, J. Math. Anal. Appl., 63(1978):177--198.

\bibitem{hoo} Branin F. H., Hoo S. K.;
\emph{A method for finding multiple
extrema of a function of $N$ variables}, Proceedings of the
Conference on Numerical Methods for Nonlinear Optimization,
University of Dundee, Scotland, June 28-July 1, 1971, in: Lootsma,
F. (ed.), \emph{Numerical Methods of Nonlinear Optimization},
Academic Press, London, 1972.

\bibitem{brown} Brown A. A., Bartholomew-Biggs M. C.;
\emph{Some effective methods for
unconstrained optimization based on the solution of systems of
ordinary differential equations}, J. Optim. Theory and Appl.,
62(2)(1988): 211--224.

\bibitem{cabot} {Cabot A.};
 \emph{Inertial gradient-like dynamical system
controlled by a stabilizing term}. J. Optim. Theory Appl.,
120(2004): 275--303.

\bibitem{cabot-engler-gadat-a} {Cabot A.}, {Engler H.}, {Gadat S.};
\emph{On the long time behavior of second or\-der differential
equations with asymptotically small dissipation}. Trans. Amer. Math.
Soc., 361(11)(2009): 5983--6017.

\bibitem{cabot-engler-gadat-b} {Cabot A.}, {Engler H.}, {Gadat S.};
\emph{Second-order differential equations with asymptotically small
dissipation and piecewise flat potentials}. Conf. Seventh
Mississippi State - UAB Conference on Differential Equations and
Computational Simulations, Electronic Journal of Differential
Equations, Conference 17(2009): 33--38.

\bibitem{cominetti-peypouquet-sorin} {Cominetti R.}, {Peypouquet J.},
{Sorin S.},  \emph{Strong asymptotic convergence of evolution
equations governed by maximal monotone operators with Tikhonov
regularization}. J. Differential Equations,
245(12)(2008):3753--3763.

\bibitem{evt} {Evtushenko Yu. G.}, {Zhadan V. G.};
\emph{Application of the method of Lyapunov functions to the study
of convergence of
numerical methods}. USSR Comput. Math. Math. Phys.,
15(1)(1975):96--108. Translated from:   Evtushenko Yu. G.,
Zhadan V.G., \emph{Primeneniya metoda funktsi\u{i} Lyapunova dlya
issledovaniya skhodimosti chislennykh metodov}, Zh. vychisl. mat. i
mat. fiz., {15}(1)(1975): 101-112.

\bibitem{flam} {Fl{\aa}m S. D.};
\emph{Solving convex programming by means of
ordinary differential equations}. Mathematics of Operations Research
17(2)(1992):290--302

\bibitem{flr} {Fletcher R.}, {Reeves C. M.};
\emph{Function minimization by
conjugate gradients}. Comput. J., 7(1964): 149--154.

\bibitem{goudou-munier} {Goudou X.}, {Munier J.};
\emph{The gradient and heavy ball with
friction dynamical systems: the quasiconvex case}. Math. Program.
Ser. B, 116(1-2)(2009): 173--191.

\bibitem{hajba-1} {Hajba T.};
 \emph{Optimization methods modelled by second order
differential equation}. Annales Univ. Sci. Budapest, Sect. Comp.
26(2006): 145--158.

\bibitem{hajba-2} {Hajba T.};
 \emph{A m\'asodrend\H u differenci\'alegyenlettel
modellezett optimaliz\'al\'ok megengedett param\'etereinek
elem\-z\'ese, (Parameter analysis of optimization methods modelled
by second order differential equation)}. Alkalmazott Matematikai
Lapok 25(2008): 61--79, (in Hungarian).

\bibitem{hauser-nedic} {Hauser R.}, {Nedi\'c J.};
\emph{The continuous Newton-Raphson
method can look ahead}, SIAM J. Optim. 15(3)(2005): 915--925.

\bibitem{km1}{Kovacs M.};
 \emph{Continuous analog of gradient-type iterative
regularization}. J. Mosc. Univ. Comput. Math. Cybern.
3(1979):37--44.
Translated from:   Kovach M.,
\emph{Nepreryvny\u{i} analog iterativno\u{i}
regulyari\-zatsii gradientnogo tipa}. Vestnik Mosk. un-ta, Ser. XV.
  Vychisl. mat. i kibern., {3} (1979): 36-42.

\bibitem{km2}{Kovacs M.};
 \emph{Some convergence theorems on nonstationary
minimization precesses}. Math. Operationsforschung u Statist.,
15(2)(1984): 203--210.

\bibitem{levenberg} {Levenberg K.};
 \emph{A method for the solution of certain
non-linear problems in least squares}. Quaterly of Applied
Mathematics, 2(1944): 164--168.

\bibitem{marquard} {Marquard D. V.};
 \emph{An algorithm for least squares estimation
of non-linear parameters}. SIAM Journal, 11(1963):431--441.

\bibitem{nedic} {Nedi\'c A.};
 \emph{The continuous projection-gradient method of
the fourth order}. Yugosl. J. Oper. Res., 5(1)(1995):27--38.

\bibitem{rouche} {Rouche N.}, {Habets P.}, {Laloy M.};
 \emph{Stability Theory by Liapunov's Direct Method}.
Springer Verlag, 1977.

\bibitem{t-dokl} {Tikhonov A. N.}
\emph{Solution of incorrectly formulated problems
and the regularization method}. Soviet Math. Dokl.
4(1963):1035--1038.  Translated from:   Tikhonov A.N., 
\emph{O reshenii nekorrektno postavlennykh zadach i metode
  regulyariza\-tsii}, Dokl. Akad. Nauk SSSR, {153.1}(1963): 49--52.

\bibitem{banach} {Tikhonov A. N.}, {Vasilev F. P.};
\emph{Methods for the solution of ill-posed extremal problems}.
In: Banach Center Publications, Vol. { 3}, PWN, Warszawa,
1978, 297--342,
(in Russian:   Tikhonov A. N., Vasilp1ev F. P.: Metody
  resheniya nekorrektnykh ekstremal'nykh zadach).

\bibitem{vasilev-nedic-2} {Vasil'ev F. P.}, {Nedich A.};
\emph{Regularized continuous
third-order gradient projection method}. Differential equation
30(12)(1994): 1869--1877.  Translated from:   Vasil'ev F.P.,
Nedich A.,  Regulyarizovanny\u{i} nepreryvny\u{i} metod tret'ego
poryadka proektsii gradienta, Differentsialp1nye uravneniya,
30(12)(1994), 2033-2042.

\bibitem{vasilev-amochkina-nedic} {Vasil'ev F. P.}, {Amochkina T. V.},
 {Nedi\'c A.};
\emph{On a regularized variant of the second-order continuous
gradient projection method}. J. Mosc. Univ. Comput.
  Math. Cybern., 3(1995):33--39.
Tanslated from:   Vasil'ev F. P., Amochkina T. V.,
  Nedich A., \emph{Ob odnom regulyarizovannom variente nepreryvnogo metoda
  proek\-tsii gradienta vtorogo poryadka}, Vestnik Mosk. un-ta, Ser. XV.
  Vychisl. mat. i kibern., 3(1995), 39-46.

\bibitem{vasilev-nedic-3} {Vasiljev F. P.}, {Nedi\'c A.};
\emph{A regularized continuous
projection-gradient method of the fourth order}. Yugosl. J. Oper.
Res., 5(2)(1995):195--209.

\bibitem{venec} {Venec V. I.}, {Rybashov M. V.};
\emph{The method of {L}yapunov
function in the study of continuous algorithms of mathematical
programming}. USSR Comput. Math. Math. Phys 17(1977):64--73.
Translated from:   V.I. Venets, M.V. Rybashov,
\emph{Metod funktsi\u{i} {L}yapunova v issledovanii nepreryvnykh
algoritmov matematicheskogo programmiro\-vaniya}. Zh. vychisl. mat. i mat.
fiz., {17}(3)(1977): 622-633.

\bibitem{zhang} Zhang L.-H., Kelley C. T, Liao
L.-Z.;
\emph{A continuous Newton-type method for unconstrained
optimization}, Pacific Journal of Optimization, 4(2)(2008):259-277.

\end{thebibliography}

\end{document}