\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 35, 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 2009 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2009/35\hfil Stability of a family of second-order systems]
{Stability for a family of systems of differential equations with
sectionally continuous right-hand sides}
\author[A. d. J. Dominogs, G. de la Hera M., E. V\'{a}zquez S.\hfil
EJDE-2009/35\hfilneg]
{Ac\'acio da Concei\c{c}\~{a}o de Jesus Domingos,
Gilner de la Hera Mart\'{\i}nez, Efr\'{e}n V\'{a}zquez Silva}
\address{Ac\'acio da Concei\c{c}\~{a}o de Jesus Domingos \newline
University Agostinho Neto, Luanda, Angola}
\email{acacio\_dejesus@yahoo.com.br}
\address{Gilner de la Hera Mart\'{\i}nez \newline
Gilner de la Hera Martinez
University of Las Tunas, Cuba}
\address{Efr\'{e}n V\'{a}zquez Silva \newline
University of Informatics Sciences \\
Havana City, Cuba}
\email{vazquezsilva@uci.cu, ev2001es@yahoo.es}
\thanks{Submitted May 2, 2008. Published February 25, 2009.}
\subjclass[2000]{34K20, 34K25}
\keywords{Interconnecting systems; stability}
\begin{abstract}
In this work, we obtain necessary and sufficient conditions to guarantee
the asymptotic stability of the trivial solution for a family of
interconnected $2\times 2$ systems of differential equations
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\section{Introduction}
One of the most important questions in stability theory is the study of
families of systems of differential equations and differential inclusions.
Kharitonov \cite{k1} proved a necessary and sufficient condition for
stability when the coefficients belong the family
\begin{equation*}
F=\big\{\sum_{k=0}^{n}a_{k}\lambda ^{n-k}:a_{k}\in \lbrack \underline{a}_{k},
\overline{a}_{k}],\;k=0,\dots ,n\big\}\,.
\end{equation*}
Because uncertainties in a perturbation can be represented with matrices
whose entries are in certain intervals, it is important to study stability
for set of the form
\begin{equation*}
A=\big\{(a_{i,j})_{i,j}:b_{i,j}\leq a_{i,j}\leq c_{i,j},\;i,j=1,\dots ,n
\big\},
\end{equation*}
where the matrices $(b_{i,j})$ , $(c_{i,j})$ are stable; see for example
\cite{b2,c1}.
In \cite{l1} it is proved the existence of solutions for differential
inclusions of the form $x^{\prime }\in F(T(t)x)$, where $F$ is upper
semicontinuous multi-value function, such that $F(T(t)x)\subset \partial
V(x(t))$, $t\in \lbrack 0,T]$, $V$ is a convex and lower semicontinuous
function for which $(T(t)x)(s)=x(t+s)$. Now let us assume that there exists
a finite number of states, $f_{i}(x,t)$, $i=\overline{1,n}$, $t\geq 0$,
defined by the right-hand side, so that the corresponding equations becomes
\begin{equation*}
\dot{x}=\sum_{i=1}^{n}\alpha _{i}(t)f_{i}(x,t),
\end{equation*}
where the functions $\alpha _{i}(t)\geq 0$ are constant in some intervals,
and take the values zero or one in each instant of time. Furthermore assume
that $\sum_{i=1}^{n}\alpha _{i}(t)\equiv 1$. This kind of interconnecting
systems have uncertainties in the determination of the functions $f_{i}(x,t)$,
which characterize the different states of the system for those values of $
\tau \in (0,+\infty )$ that represent the moments of the jump from one state
to another state. These values of $\tau $ represent instants of jump for one
pair of functions $\alpha _{i}(t)$, $i=\overline{1,n}$. The problem about
studying the behavior of the solutions of the systems with uncertainty in
the parameters that determine it, is an important one because of their
applications in the Control Theory (see for example \cite{b1,h1,k1,p1}).
The problem to be studied is formulated in the next section. In section 3
the trajectories of the defined family of systems of differential equations
are classified as first and second type. After this, in sections 4 and 5, we
study the convergence towards the origin of coordinates for the trajectories
of first and second type. Finally, in section 6, the main results are proved
and some examples are given.
\section{Formulation of the problem}
Let us consider the real $2\time 2$ matrices
\begin{equation*}
A_1=
\begin{pmatrix}
a_{11}^1 & a_{12}^1 \\
a_{21}^1 & a_{22}^1
\end{pmatrix}, \quad A_2=
\begin{pmatrix}
a_{11}^2 & a_{12}^2 \\
a_{21}^2 & a_{22}^2
\end{pmatrix},
\end{equation*}
which are assumed to be stable; i.e., their eigenvalues have real negative
part. In $(a_{ij}^k)$ a letter $k$ identifies the matrices $A_1$ or $A_2$.
We denote by $N_{t}(A_1, A_2)$ the set of functions $t\to A(t)$, $t\in [0,
+\infty )$, where the matrix $A(t)=\alpha _1(t)A_1+\alpha_2(t)A_2$. The
functions $\alpha _1(t)$, $\alpha _2(t)$ acquire in each instant $t$ the
value $1$ or $0$, and $\alpha _1(t)+\alpha _2(t)\equiv 1$. In addition, the
set of jump moments of the functions $\alpha _{i}(t)$ , $i=1,2$, do not have
accumulation points in $\mathbb{R}$.
Let us consider a family of systems of differential equations
\begin{equation}
\dot{x}=A(t)x(t), \quad t\geq 0,\; A(t)\in N_{t}(A_1, A_2). \label{uno}
\end{equation}
The goal of this work is to establish necessary and sufficient conditions on
the matrices $A_1$and $A_2$ so that each system of the family \eqref{uno}
has a trivial asymptotically stable solution.
\section{Properties of the solutions to systems of differential equations}
The systems of differential equations
\begin{gather}
\dot{x}=A_1x(t), \label{dos} \\
\dot{x}=A_2x(t), \label{tres}
\end{gather}
belong to the family \eqref{uno} and are asymptotically stable. Clearly, the
trajectories of the systems of the family \eqref{uno} are formed by segments
of trajectories of systems \eqref{dos} and \eqref{tres}, respectively.
However, if we associate with each point $x$ of the phase plane the set $
F(x)=\{ A_1x, A_2x\} $, then all the trajectories of systems \eqref{uno} are
smooth sectionally continuous curves and they are such that, at each point $x
$ of these curves the tangent vector is one of the vectors $\nu _1(x)=A_1x$,
$\nu_2(x)=A_2x$; where the vectors $\nu _1(x)$ and $\nu _2(x)$ belong to the
set $F(x)$.
Thus, systems \eqref{dos} and \eqref{tres} belong to the family \eqref{uno},
and under conditions of the formulated problem, both systems are
asymptotically stable. Moreover, it is clear that the trajectories of the
systems of the family \eqref{uno} are formed by segments of trajectories of
systems \eqref{dos} and \eqref{tres}.
Associated with each point $x=(x_1, x_2)^{T}$ of the phase plane, we
consider the vector $x^{\perp }=(-x_2, x_1)^{T}$. It is seen that the vector
$x^{\perp }$ is orthogonal one to the radius vector $Ox$ and has the same
direction and orientation that the vector resulting by rotation of the
radius vector $x$ in positive direction and with an angle equal to $\pi/2$.
\begin{definition} \label{def1} \rm
Let be $\gamma $ a trajectory of some of the systems of family
\eqref{uno} and $x$ one of its point. It is said that the trajectory
$\gamma $ rotates in a positive direction (counter-clockwise) in the point
$x$ if $\langle \nu (x), x^{\perp }\rangle >0$, where $\nu (
x)$ is the phase velocity vector of the trajectory $\gamma $ at the
point $x$, and $\langle , \rangle $ denotes the usual
scalar product. Always, when $\langle \nu (x),x^{\perp }\rangle <0$,
it is said that the trajectory $\gamma $
rotates in a negative direction (clockwise) in the point $x$.
\end{definition}
\begin{lemma} \label{lem2}
Let be $x$ any point of the phase plane, then:
\begin{itemize}
\item[(i)] The family \eqref{uno} has at least one trajectory that rotates in
the positive sense at the point $x$ if and only if at least one of the
following inequalities occurs.
\begin{itemize}
\item[(a)] $\langle A_1x, x^{\perp }\rangle
=a_{21}^1x_1^2+(a_{22}^1-a_{11}^1)
x_1x_2-a_{12}^1x_2^2>0$,
\item[(b)] $\langle A_2x, x^{\perp }\rangle
=a_{21}^2x_1^2+(a_{22}^2-a_{11}^2)
x_1x_2-a_{12}^2x_2^2>0$.
\end{itemize}
\item[(ii)] The family \eqref{uno} has at least one trajectory that rotates
in the negative sense at the point $x$ if and only if similar condition to
(i) holds, which it is obtained changing the symbol $>$ by the symbol $<$ in
previous inequalities (a) and (b).
\end{itemize}
We note that the coefficients $a_{ij}^k$, are the entries of the matrices
$A_1$ and $A_2$, defined in Section 2.
\end{lemma}
\begin{proof}
Let be $x\in \mathbb{R}^2$ any point of the phase plane, which corresponds
to any of the systems of the family \eqref{uno}. Let us suppose that at this
point the trajectory $\gamma $ of such a system rotates in the positive
sense, then by Definition \ref{def1}, the inequality $\langle \nu (x),
x^{\perp}\rangle >0$ holds, where $\nu (x)$ is the phase velocity vector of
the trajectory $\gamma $ at the point $x$. But, how the tangent vector to
the trajectories of any system of the family \eqref{uno} belongs to the set $
F(x)$, then we have that $\nu (x) =A_1x$ or $\nu (x)=A_2x$. By substitution
of the vector $\nu (x)$ in the scalar product $\langle \nu (x), x^{\perp
}\rangle >0$, we obtain $\langle A_1x, x^{\perp }\rangle =a_{21}^1x_1^2+(
a_{22}^1-a_{11}^1)x_1x_2-a_{12}^1x_2^2>0$ and $\langle A_2x, x^{\perp
}\rangle =a_{21}^2x_1^2+(a_{22}^2-a_{11}^2) x_1x_2-a_{12}^2x_2^2>0$. The
proof of condition (ii) is similar.
\end{proof}
\begin{definition} \label{def2} \rm
We say that a trajectory $\gamma $ of a second order system of differential
equations, with homogenous right hand side is of the first type, if on the
phase plane there exists a radius vector starting at the origin of
coordinates, and there exists an instant $t_0>0$, such that the points of
$\gamma $ corresponding to the values $t\geq t_0$ do not belong to a such
radius vector. It is said that a trajectory $\gamma $ is of the second type,
when it is not of the first type.
\end{definition}
\begin{theorem} \label{thm4}
The family \eqref{uno} has trajectories of the second type if and only
if at least one of the following conditions holds:
\begin{itemize}
\item[(i)]
$a_{12}^1<0$ or $a_{12}^2<0$, % \label{cuatro}
and for each $k\in \mathbb{R}$,
\begin{equation}
-a_{21}^1-(a_{22}^1-a_{11}^1)k+a_{12}^1k^2<0\quad
\text{or}\quad
-a_{21}^2-(a_{22}^2-a_{11}^2)k+a_{12}^2k^2<0. \label{cinco}
\end{equation}
\item[(ii)]
$a_{12}^1>0$ or $a_{12}^2>0$,
and for each $k\in \mathbb{R}$,
\begin{equation}
-a_{21}^1-(a_{22}^1-a_{11}^1)k+a_{12}^1k^2>0\quad
\text{or}\quad
-a_{21}^2-(a_{22}^2-a_{11}^2)k+a_{12}^2k^2>0. \label{cinco5}
\end{equation}
\end{itemize}
\end{theorem}
\begin{proof}
We can see that condition (i) implies that for each point $x$ of the phase
plane it holds that $\max \{ \langle A_1x, x^{\bot }\rangle , \langle A_2x,
x^{\bot }\rangle\} >0$ and this condition implies that for each point of the
phase plane crosses a trajectory that rotates at this point in a positive
direction.
\noindent\textbf{Necessity:} Let us suppose that none of the conditions (i)
and (ii) is verified, then there exist two straight lines passing by the
origin of the phase plane, respectively $-a_{21}^1-(a_{22}^1-a_{11}^1)k$, $
-a_{21}^2-( a_{22}^2-a_{11}^2)k$; $k=\frac{x_1}{x_2}$, and such that on the
points of one of this line do not exist trajectory rotating in positive
direction, while on the points of the another line do not exist trajectory
rotating in negative direction. Thus these straight lines determine, in the
phase plane, an invariant angular section for the family \eqref{uno}, and
this fact implies that all the trajectories of the family \eqref{uno} are of
the first type.
\noindent\textbf{Sufficiency:} Let us suppose that condition (i) is verified
(the proof is similar when condition (ii) is verified). It is defined a
vector field by establishing a correspondence between each point $x\in
\mathbb{R}^2$ and the function $f(x)$ defined by
\begin{equation*}
f(x) =
\begin{cases}
A_1x & \text{if }\langle A_1x, x^{\bot }\rangle \geq \langle A_2x, x^{\bot
}\rangle \\
A_2x & \text{if }\langle A_2x, x^{\bot }\rangle \geq \langle A_1x, x^{\bot
}\rangle .
\end{cases}
\end{equation*}
Evidently $\langle f(x), x^{\bot}\rangle >0$, for each $x\in\mathbb{R}^2$.
That is to say, the trajectories of this vector field, which are
trajectories of the family \eqref{uno}, rotate at each of their points in
positive direction with strictly positive angular velocity. Then, for each
ray of the phase plane there exists a sequence $t_{n}$, $t_{n}\to \infty $,
such that the points of the trajectory which correspond to these instant of
time lay on such a ray; i.e., the trajectories of the vector field $x\to
f(x)$ are of the second type.
\end{proof}
\section{Convergence towards the origin for the first type trajectories}
In this Section we formulate some Lemmata that allows us to demonstrate a
Theorem \ref{thm13} which offers condition for the convergence towards the
origin of coordinates of all trajectories of the first type of the
considered family \eqref{uno}.
Let be $\nu \in\mathbb{R}^2$ any vector, and let be $\arg (\nu )$ the angle
formed by the vector $\nu $ with the semi-axis $x_1>0$; and let be $A_1$, $
A_2$ the matrices defined in Section 2. We denote by $\measuredangle
(A_1x_0, A_2x_0)$ the measure of the angle between the vectors $A_1x_0$ and $
A_2x_0$ such that $\arg (-x_0)$ belongs to the real segment for which the
extremes are $\arg (A_1x_0)$ , $\arg (A_2x_0)$ respectively.
\begin{lemma} \label{lem5}
To converge towards the origin of coordinates, the trajectories
of the first type it is necessary that the inequality $\measuredangle
(A_1x, A_2x)\leq 180^{\circ}$ hold for all
$x\in\mathbb{R}^2$.
\end{lemma}
\begin{proof}
Suppose that there exists $x_{0}\in \mathbb{R}^{2}$, such that $
\measuredangle (A_{1}x_{0},A_{2}x_{0})>180^{\circ }$, then it is clear that
there is an angle $\delta $ with vertex in the origin of coordinates, such
that $x_{0}$ lies on one of the limiting rays of $\delta $, and for each $
x\in \delta $ holds that $\measuredangle (A_{1}x,A_{2}x)>180^{\circ }$. Let
us denote by $\delta _{0}$ a limited region determined in $\delta $ by the
segment of trajectory $\chi $ of one of the systems \eqref{dos} or
\eqref{tres}, with initial condition $x_{0}$, besides the end of this
segment lies on the other side of the angle $\delta $. Then it can be
checked that any trajectory with initial condition $x_{0}\notin \delta _{0}$
and completely contained in $\delta $, has not common points with $
\mathop{\rm int}(\delta _{0})$ ($\mathop{\rm int}(M)$ denotes the interior
of the set $M$) because for this fact this trajectory must to cut the
trajectory $\chi $, and this implies that in the point of intersection $w$
holds $\measuredangle (A_{1}w,A_{2}w)\leq 180^{\circ }$. Let us take the
trajectory of the family of systems \eqref{uno} with initial condition $
2x_{0}$ completely contained in $\delta $ and formed by segments of
alternate trajectories of systems \eqref{dos} and \eqref{tres}, which have a
starting point on the one of boundary ray of the angle $\delta $, and the
end point on the other boundary ray of the angle $\delta $. Then this
trajectory has not common points with $\mathop{\rm int}(\delta _{0})$ and
therefore it does not converge to the origin of coordinates when $
t\rightarrow +\infty $.
\end{proof}
\begin{lemma} \label{lem6}
Let be $\gamma =\{ x(t):t\geq 0\} $
a trajectory of the first type of the family \eqref{uno}, then there exists
an angle $S$, with vertex in the origin of the phase plane, limited by
radius eigenvectors of the matrices $A_1$or $A_2$, such that in the
interior of $S$ functions $\ x\to \mathop{\rm sgn}\langle A_1x,
x^{\bot }\rangle $ , $x\to \mathop{\rm sgn}\langle A_2x,
x^{\bot }\rangle $ are constant, in addition, there exists $\tau >0$,
such that $x(t)\in S$ for $t\geq \tau $.
\end{lemma}
\begin{proof}
The straight lines defined by the eigenvectors of the matrices $A_1$ and $A_2
$ determine a division of the phase plane in sectors $S_{i}$, $i=1,\dots ,n$,
each of which is an angle with vertex at the origin of coordinates which
satisfies the property of the Lemma relatively to the angle $S$ related with
the functions $x\to \mathop{\rm sgn}\langle A_1x , x^{\bot }\rangle $, $x\to
\mathop{\rm sgn}\langle A_2x , x^{\bot }\rangle $. Since $\gamma $ is a
trajectory of the first type of the family \eqref{uno}, there are: a ray $L$
with starting point at the origin of coordinates and an instant $t_0>0$ such
that $x(t)\notin L$ for all $t>t_0$. If $L$ is within some of the angles $
S_{k}$ then instead of $S_{k}$ we consider the two angles determined within $
S_{k}$ by $L$. Let us denote these angles by $S_{k}^{\prime }$ and $
S_{k}^{\prime \prime }$. It turns out that if for one $\tilde{t}$ $>t_0$, $x(
\tilde{t})\in S_{j}$, for some $j\in \{ 1,\dots ,p\}$; $j\neq k$, then or $
x(t)\in S_{j} $ for all $t>$ $\tilde{t}$, or there exists $\bar{t}>\tilde{t}$
such that $x(t)\notin S_{j}$ for $t>\bar{t}$. This means, if the trajectory $
\gamma $ abandon any sector, it can not return to it. The same situation
presents with the angles $S_{k}^{\prime }$ and $S_{k}^{\prime \prime }$.
Thus, as there is only a finite number of these sectors, there must be a
sector $S=S_{l}$, for some $l\in \{ 1,\dots ,p\} $, and one instant $\tau
>t_0$, such that $x(t)\in S$ for $t>\tau $.
\end{proof}
\begin{lemma} \label{lem7}
Let $S$ be an angle with vertex in the origin of the phase plane, limited by
the radius eigenvectors of the matrices $A_1$or $A_2$, such that
$\mathop{\rm sgn}\langle A_1x, x^{\bot }\rangle =\mathop{\rm sgn}\langle
A_2x, x^{\bot }\rangle $, for all $x\in intS$. If any
trajectory of the family \eqref{uno} remains in the angle $S$ from an
instant $t$, then it converges towards the origin of coordinates.
\end{lemma}
\begin{proof}
Let be $\gamma $ a trajectory of the first type that satisfies the
hypotheses of the Lemma, then $\gamma $ rotates in all their points in the
same direction, so this trajectory approaching indefinitely to one ray $L=\{
\lambda y: \lambda \geq 0\} $. It is not possible that $\gamma $ not be a
limited one, neither $\gamma $ has more than one w-limit point on the ray
$L$, since the matrices $A_1$, $A_2$ are stable, thus the phase velocities
could not be directed in the sense of the vector $y$ which determines the
ray $L$. So, $\gamma $ converges to a single point $w$ (assuming $w\neq 0$).
Let us take $\varepsilon $ small enough such that the projection of the
phase velocity in the points of $\gamma $, belonging to the ball $
B(w,\varepsilon )$ over the bisector of the vectors $A_1w$ and $A_2w$, be
strictly positive. So, for arbitrarily large values of $t$, trajectory $
\gamma $ has points both inside and outside ball $B(w,\varepsilon )$ and so
there is a succession of points of $\gamma $ that are on the circle with
center $w$ and radius $\varepsilon $ where there should be another w-limit
point. This is a contradiction, because there is only one w-limit point.
\end{proof}
\begin{lemma} \label{lem8}
If for the all $x\in\mathbb{R}^2$ results
$\measuredangle (A_1x, A_2x)\leq 180^{\circ}$, then there exists one
or there not exists any straight line $d$
passing through the origin of coordinates and such that, for each $x\in d$
we have that $\measuredangle (A_1x, A_2x)=180^{\circ}$.
\end{lemma}
\begin{proof}
If \ $\mathop{\rm sgn}\langle A_1x, x^{\bot }\rangle =\mathop{\rm sgn}
\langle A_2x, x^{\bot }\rangle $ or $\langle A_{i}x, x^{\bot }\rangle =0$
for $i=1$ or $i=2$, then $\measuredangle (A_1x, A_2x)<180^{\circ}$. When $
\mathop{\rm sgn}\langle A_1x, x^{\bot }\rangle =-\mathop{\rm sgn}\langle
A_2x, x^{\bot }\rangle $ we can check that $\measuredangle (A_1x,
A_2x)<180^{\circ}$ if and only if $\mathop{\rm sgn}\langle (A_1x)^{\bot },
A_2x\rangle =-\mathop{\rm sgn}\langle A_1x, x^{\bot }\rangle $, and $
\measuredangle (A_1x, A_2x)=180^{\circ}$ if and only if $\langle
(A_1x)^{\bot }, A_2x\rangle =0$. If we calculated the scalar product $
\langle (A_1x)^{\bot }, A_2x\rangle $ we have
\begin{align*}
\langle (A_1x)^{\bot }, A_2x\rangle
&=(a_{21}^2a_{11}^1-a_{11}^2a_{21}^1)x_1^2+(
a_{22}^2a_{11}^1+a_{21}^2a_{12}^1 \\
&\quad -a_{11}^2a_{22}^1-a_{12}^2a_{21}^1)x_1x_2+(
a_{22}^2a_{12}^1-a_{12}^2a_{22}^1)x_2^2.
\end{align*}
From the coefficients of the previous quadratic form we see that it can not
happen that be $\measuredangle (A_1x, A_2x) =180^{\circ}$ for all $x\in
\mathbb{R}^2$ because to do so, there must be $\lambda \in\mathbb{R}_{-}$
such that $A_1=\lambda A_2$, but this contradicts the fact that the matrices
$A_1$, $A_2$ are stable. Suppose that there are two straight lines $d_1$, $
d_2$ passing through the origin of coordinates and such that for each $x\in
d_1$ or $x\in d_2$ holds that $\measuredangle (A_1x, A_2x)=180^{\circ}$;
i.e., $\langle (A_1x)^{\bot }, A_2x\rangle =0$. It is known from the theory
of quadratic forms that for any point $x\in d_1 $ or $x\in d_2$ there is a
neighborhood in which there exists points where the expression for $\langle
(A_1x)^{\bot }, A_2x\rangle $ takes positive sign and also points where this
expression takes negative sign and as for the points which are on $d_1$ and $
d_2$ the scalar product $\langle A_1x, x^{\bot}\rangle $ does not vanishes,
then there will be points near to $d_1$and $d_2$ on which $\mathop{\rm sgn}
\langle (A_1x)^{\bot }, A_2x\rangle =-\mathop{\rm sgn}\langle A_1x, x^{\bot
}\rangle $, on these points $\measuredangle (A_1x,A_2x)>180^{\circ}$, but
this contradicts the hypothesis of the Lemma.
\end{proof}
\begin{lemma} \label{lem9}
Let us suppose that for each $x\in \mathbb{R}^2$ follows that
$\measuredangle (A_1x, A_2x)\leq 180^{\circ}$. Let us consider for
each of the systems \eqref{dos} and \eqref{tres} a segment of
trajectories $\gamma _1$ and $\gamma _2$ respectively.
Suppose that the curves $\gamma _1$ and $\gamma _2$ intersect itself at
the point $w$. If at this point these trajectories rotate in opposite
directions, then after intersection, the future of each segment of the
trajectory will continue under the past of the other one.
\end{lemma}
\begin{proof}
If at the point $w$, where these trajectories intersect itself, the
inequality $\measuredangle (A_1w, A_2w)<180^{\circ}$ holds, then the
statement of the Lemma is true. Let us see that in the case when $
\measuredangle (A_1w, A_2w)=180^{\circ}$, the Lemma is also true. Suppose
the opposite, namely that there are two segment of trajectories $\chi _1$
and $\chi _2$ of the systems \eqref{dos} and \eqref{tres} which intersect
itself in the point $w$, $\measuredangle ( A_1w, A_2w)=180^{\circ}$ and in
addition the future of $\chi _1$ will be above to the past of $\chi _2$. If
we take a segment of the trajectory that is sufficiently close to $\chi _1$
and which be a solution of the same system that $\chi _1$, we have that this
segment will intersect $\chi _2$ at the point $w_{1 }$for which $
\measuredangle (A_1w_{1 }, A_2w_{1 })>180^{\circ}$, by virtue of Lemma
\ref{lem8}, and the future of this new segment should go above to the past of $
\chi _2$, resulting in a contradiction.
\end{proof}
\begin{lemma} \label{lem10}
Let $S$ be an angle in the phase plane with vertex in the origin of
coordinates, such that $\mathop{\rm sgn}\langle A_1x, x^{\bot
}\rangle =-\mathop{\rm sgn}\langle A_2x, x^{\bot }\rangle $
for all $x\in intS$. If a trajectory of the family of systems \eqref{uno}
remains, from an instant $t$, in the angle $S$, then this trajectory
converges to the origin of coordinates, i.e $x(t)\to 0$
when $t$ grows.
\end{lemma}
\begin{proof}
Let us begin by defining for each \ $x\in S$ one limited region in the plane
that is denoted by $S_{x}$. Let us consider the trajectories of systems
\eqref{dos} and \eqref{tres} with starting point $x$. Then $S_{x}$ will be
the plane region limited by these trajectories and the origin of coordinates
when these trajectories are completely contained in the angle $S$; in other
case $S_{x}$ will be the plane region limited by the segments of the
considered trajectories contained in the angle $S$ and the straight segments
which joint the exit points of these trajectories from de angle $S$ and the
origin of coordinates. By the construction of the region $S_{x}$ and the
affirmation of Lemma \ref{lem9}, we conclude that the region $S_{x}$
contains completely the semi - positive trajectories of the family
\eqref{uno} with starting point in $S_{x}$, besides, these trajectories do
not abandon angle $S$. Let be now $\gamma =\{ x(t):t\geq 0\} $ a trajectory
of the first type completely contained in $S$ from a certain moment $t_0$.
Suppose that $\gamma $ has more than one w-limit point, and let $w$ and $w_1$
be two of these points. It is easy to see that $w\in S_{x}$ and $w_1\in S_{x}
$, but this only occurs if there are two segments of trajectories, one of
system \eqref{dos} and another of system \eqref{tres}, such that, $w$ and $
w_1$ are its end points. We get a contradiction with the Lemma \ref{lem9}.
Suppose now that $w\neq 0$\ is the unique w-limit point of the trajectory $
\gamma $ and also suppose that $\measuredangle (A_1w, A_2w)<180^{\circ}$. It
is taken $\varepsilon $ small enough such that the projection of the phase
velocity at the point of $\gamma $ belonging to the ball $B(w,\varepsilon)$
on the bisector of the vectors $A_1w$ , $A_2w$\ be strictly positive. So,
for arbitrarily large values of $t$ the trajectory $\gamma $ has points both
inside and outside ball $B(w,\varepsilon )$, and so there is a succession of
points of $\gamma $ laying on the circle with center in $w$ and radius $
\varepsilon $ where there should be another w-limit point. Again we get a
contradiction. Suppose now that $w\neq 0$\ is the unique w-limit point of
the trajectory $\gamma $ and $\measuredangle (A_1w, A_2w)=180^{\circ}$. We
take $\varepsilon $ small enough such that the projection of the phase
velocity at the point of $\gamma $ belonging to the ball $B(w,\varepsilon )$
on the vector $A_1w$ has module greater than or equal to the number $\alpha
>0$. We know that $\gamma $ will remain in $B(w,\varepsilon )$ from certain
$t_0>0$. We also know that $\gamma $\ is formed by segments of the
trajectories of systems \eqref{dos} and \eqref{tres}. Let be $t_{i}$, $i\in
\mathbb{N}$, a succession of instants of change from one to another system.
As the succession $\{ t_{i}\} $ has no accumulation points in $\mathbb{R}$,
there exists a number $\mu >0$, such that, $| t_{i+1}-t_{i}| \geq \mu $.
Then follows that $| x(t_{i+1})-x(t_{i})| \geq \alpha (t_{i+1}-t_{i})\geq
\alpha \mu $ which contradicts the convergence to the origin for the
trajectory $\gamma $.
\end{proof}
\begin{lemma} \label{lem11}
A necessary and sufficient condition for
$\measuredangle (A_1x, A_2x)$$\leq 180^{\circ}$ to hold for each
$x\in\mathbb{R}^2$, is that the matrices
$C(\alpha )=\alpha A_1+(1-\alpha )A_2$, $\alpha \in [0, 1]$ be stable
or at most there exists one singular matrix $C(\alpha _0)$.
\end{lemma}
\begin{proof}
Vectors $A_1x$ and $A_2x$ form two angles such that the sum of their
amplitude is $360^{o}$ and the segment that links the extremes of $A_1x$ and
$A_2x$ is contained in the angle of lesser magnitude.
\noindent\textbf{Necessity:} Let be $x_0\in\mathbb{R}^2$ such that, $
\measuredangle (A_1x_0, A_2x_0)>180^{\circ}$. According to the definition of
the angle $\measuredangle (A_1x_0, A_2x_0)$, we have that for some $\lambda
>0$ the point $\lambda x_0$ is on the segment that connects the points $
A_1x_0$ and $A_2x_0$; i.e., $\alpha _0A_1x_0+(1-\alpha _0)$ $A_2x_0=\lambda
x_0$ for some $\alpha _0\in [0,1]$, and thus $C(\alpha _0)x_0=\lambda x_0$.
The latter equality means that $C(\alpha _0)$ is unstable.
\noindent\textbf{Sufficiency:} Let us contrary suppose that there exists $
\alpha _0\in [0, 1]$ such that, $C(\alpha _0)$ is unstable. Then $C(\alpha
_0)$ has an eigenvalue $\lambda $ such that, $\mathop{\rm Re}(\lambda )\geq 0
$. But the eigenvalues of $C(\alpha _0)$ are real because in other case
would be $\mathop{\rm Re}(\lambda )=\mathop{\rm tr}(C(\alpha _0))<0$. Then
there must be $x_0\in\mathbb{R}^2$, $x_0\neq 0$, such that, $C(\alpha
_0)x_0=\lambda x_0$, $\lambda \in\mathbb{R}_{+}$. It is means $\alpha
_0A_1x_0+(1-\alpha_0)$ $A_2x_0=\lambda x_0$, and this means that the vector $
\lambda x_0$ is on the segment formed by $A_1x_0$ and $A_2x_0$. From this
follows that $\measuredangle (A_1x_0, A_2x_0)>180^{\circ}$.
\end{proof}
\begin{lemma} \label{lem12}
The matrices $C(\alpha )=\alpha A_1+(1-\alpha )A_2$,
$\alpha \in [0, 1]$ are stable, except as most
for some $\alpha _0\in [0, 1]$. And the stability is
guaranteed if and only if $H\leq 2\sqrt{\det A_1\det A_2}$, where
$H=a_{12}^1a_{21}^2+a_{12}^2a_{21}^1-a_{11}^1a_{22}^2
-a_{11}^2a_{22}^1$.
\end{lemma}
\begin{proof}
The matrix $C(\alpha )$, $\alpha \in [0, 1]$ is stable if and only the
inequalities $\mathop{\rm tr}(C(\alpha ))<0$, $\det (C(\alpha ))>0$ hold.
But $\mathop{\rm tr}(C(\alpha ))=\mathop{\rm tr}(\alpha A_1+( 1-\alpha
)A_2)=\alpha \mathop{\rm tr}(A_1)+( 1-\alpha ) \mathop{\rm tr}(A_2)$ and $
\mathop{\rm tr}(A_1)<0$, $\mathop{\rm tr}(A_2)<0$. Thus $\mathop{\rm tr}
(C(\alpha ))<0$, for all $\alpha \in [0, 1]$. On the other hand
\begin{align*}
\det (C(\alpha ))&=\det (\alpha A_1+(1-\alpha )A_2) \\
&=(\det (A_1)+\det (A_2)+H)\alpha ^2 -(2\det (A_2)+H)\alpha +\det (A_2).
\end{align*}
If $(\det (A_1)+\det (A_2)+H) \leq 0$ then $\det (C(\alpha ))$, as a
function of $\alpha $, is a parable that opens down, or a straight line, but
as $\det(C(0))=\det (A_2)>0$ and $\det(C(1))=\det (A_1)>0$, then $
\det(C(\alpha ))>0$, for all $\alpha \in [0, 1]$. If $(\det (A_1)+\det
(A_2)+H)>0$ then $\det (C(\alpha ))$, as a function of $\alpha $, is a
parable that opens up, such that, for $\alpha =0$ and $\alpha =1$ this
parabola takes positive values. Soon $\det ( C(\alpha ))>0$, for all $\alpha
\in [0, 1]$, if and only if the vertex of the parable corresponds to a value
$\alpha _0\notin [0, 1]$, or in other case $\det (C(\alpha _0))\geq 0$, it
is means that holds the implication:
\begin{equation*}
-H\leq 2\det (A_1),\, -H\leq 2\det (A_2) \Longrightarrow H^2\leq 4\det
(A_1)\det (A_2).
\end{equation*}
If $H\leq 0$ and the left-hand side holds, by multiplying the inequalities
we obtain the right-hand side. In another way, if $H>0$ and the left-hand
side holds, then $H\leq 2\sqrt{\det A_1\det A_2}$.
\end{proof}
\begin{theorem} \label{thm13}
The trajectories of the first type of the family of systems \eqref{uno}
converge towards the origin of coordinates if and only if
\begin{equation}
H\leq 2\sqrt{\det A_1\det A_2}. \label{seis}
\end{equation}
\end{theorem}
\begin{proof}
Let be $A_1$, $A_2$ the matrices defined in Section 2.
\noindent\textbf{Necessity:} Let us consider that the trajectories of the
first type of the family of systems \eqref{uno} converge towards the origin
of coordinates. Then, by Lemma \ref{lem5}, the inequality $\measuredangle
(A_1x, A_2x)$ $\leq 180^{\circ}$ holds for each $x\in\mathbb{R}^2$. This
inequality, according to Lemma \ref{lem11}, is equivalent to the stability
of matrices $C(\alpha )=\alpha A_1+(1-\alpha)A_2$, $\alpha \in [0, 1]$. Now
we just apply Lemma \ref{lem12} and so we have $H\leq 2\sqrt{\det A_1\det
A_2}$.
\noindent\textbf{Sufficiency:} Let us now consider that $H\leq 2\sqrt{\det
A_1\det A_2}$ and suppose that the trajectories of the first type of the
family of systems \eqref{uno} do not converge towards the origin of
coordinates. Then, by Lemma \ref{lem5}, will be $\measuredangle (A_1x,
A_2x)>180^{\circ}$ for all $x\in\mathbb{R}^2$. Moreover, due to Lemma \ref
{lem12}, the matrices $C(\alpha )=\alpha A_1+(1-\alpha )A_2$ $_2$, $\alpha
\in [0, 1]$ are stable. But now, by Lemma \ref{lem11}, the inequality $
\measuredangle (A_1x, A_2x)$ $\leq 180^{\circ}$ holds for all $x\in \mathbb{R
}^2$. We obtain a contradiction with our assumption.
\end{proof}
\section{Convergence towards the origin for the second type trajectories}
In this section we analyze the convergence towards the origin of the
trajectories which rotate, at any point $x$ of the phase plane. In the
following we consider that the stable matrices $A_1$, $A_2$ satisfy
\eqref{seis}. This condition ensures, according to Theorem \ref{thm13}, the
convergence towards the origin of the first type trajectories of the family
\eqref{uno}
We want to establish additional conditions to ensure the convergence towards
the origin also for the trajectories of the second type.
Thus, taking into account the ideas developed by Baravanov \cite{b1}, we
introduce the so-called auxiliary systems:
\begin{gather}
\dot{x}=\arg \max_{f\in F(x),\,\langle f, x^{\bot }\rangle >0} \frac{\langle
f, x\rangle }{\| f\| }, \label{siete} \\
\dot{x}=\arg \max_{f\in F(x),\,\langle f, x^{\bot }\rangle <0} \frac{\langle
f, x\rangle }{\| f\| }, \label{ocho}
\end{gather}
where $F(x)=\{A_1x, A_2x\}$. We note that systems \eqref{siete} and
\eqref{ocho} make sense respectively, in the following regions of the plane:
\begin{gather*}
D^{+}=\{ x\in\mathbb{R}^2:\exists f\in F(x)\text{ such that } \langle f,
x^{\bot }\rangle >0\}, \\
D^{-}=\{ x\in\mathbb{R}^2:\exists f\in F(x)\text{ such that } \langle f,
x^{\bot }\rangle <0\}.
\end{gather*}
Given a pair of matrices $A_1$, $A_2$ it is possible that both systems
\eqref{siete} and \eqref{ocho} make sense in all the plane or in a
particular region of the plane. Although it is also possible that one of the
systems does not make sense, because one of the sets $D^{+}$ or $D^{-}$ may
be empty.
In addition, according to the definition of these systems and the results of
the section 3, we have that the trajectories of system \eqref{siete} rotate
in each of its points in positive direction, and in the case of system
\eqref{ocho}, its trajectories rotate in each point in negative direction.
Thus it is easy to determine expressions for systems \eqref{siete} and
\eqref{ocho} according to the matrices $A_1$, $A_2$ that determine each of
these systems. For this we must just resolve the extreme problems indicated
in the right hand-side of the expressions \eqref{siete} and \eqref{ocho},
which are simply ones, because the variable in each problem takes only two
values. Thus, it is clear that system \eqref{siete} takes the form
\begin{equation}
\begin{gathered} \dot{x}=V_1(x)x, \quad x\in D^{+}, \\ V_1(x)=\begin{cases}
A_1 & \text{if }\langle (A_1x)^{\bot }, A_2x\rangle \geq 0 \\ A_2 &\text{if
}\langle (A_1x)^{\bot }, A_2x\rangle <0, \end{cases} \end{gathered}
\label{nueve}
\end{equation}
while system \eqref{ocho} is given by the expression
\begin{equation}
\begin{gathered} \dot{x}=V_2(x)x, \quad x\in D^{-}, \\ V_2(x)=\begin{cases}
A_1 &\text{if }\langle (A_1x)^{\bot }, A_2x\rangle \leq 0 \\ A_2 &\text{if
}\langle (A_1x)^{\bot }, A_2x\rangle >0\,. \end{cases} \end{gathered}
\label{diez}
\end{equation}
Next, following the ideas of Barabanov, it is shown with the help of some
Lemmata that the trajectories of the second type of the family \eqref{uno}
converge towards the origin of coordinates, if and only if, the auxiliary
systems \eqref{nueve} and \eqref{diez} are asymptotically stable.
\begin{lemma} \label{lem14}
Let $\gamma =\{ x(t):t\geq 0\} $ be a trajectory of the second type of a
homogeneous second order system. Then
there exist $t_0>0$ and $\lambda >0$ ($\lambda $ is called characteristic
value) such that for all $k\in\mathbb{N}$, the equality
$x(t+k t_0)=\lambda ^kx(t)$ holds.
\end{lemma}
\begin{proof}
Let be $t_0$ the lowest of all real numbers $t>0$, such that $x(t_0)$ is
located on the ray that begins at the origin and contains $x(0)$, and take $
\lambda =\| x(t_0)\|/ \| x(0)\|$. We demonstrate the statement of the Lemma
by induction. Consider the trajectory $\gamma_{\lambda }=\{ \lambda
x(t):t\geq0\} $ for which its corresponding point, for the instant of time $
t=0$, is $x(t_0)$; i.e., $x(t_0)=\lambda x(0)$ and thus $x(t+t_0)=\lambda
x(t)$. This fact proves the affirmation of the Lemma for $k=1$. Let us
suppose that the statement of the Lemma is true for $k=n$; i.e., $
x(t+nt_0)=\lambda ^{n}x(t)$, and let us consider the trajectory $
\gamma_{\lambda , n+1}=\{ \lambda ^{n+1}x(t):t\geq 0\} $. The point of $
\gamma _{\lambda , n+1}$ corresponding to $t=0$ is $\lambda ^{n+1}x(0)$, but
due to the induction hypothesis we have that $\lambda ^{n+1}x(0)=\lambda
x(nt_0)$ is the point of \ $\gamma _{\lambda }$ corresponding to $t=nt_0$,
and thus $\lambda ^{n+1}x(0)=x(( n+1)t_0)$, by this reason its follows that $
x(t+( n+1)t_0)=\lambda ^{n+1}x(t)$.
\end{proof}
Now let us to consider the following sets:
\begin{gather*}
C^{+}(x)=\{ \lambda x(t): 0\leq \lambda \leq 1,\; x(t),\; t>0 \text{
solution of \eqref{nueve} and } x(0)=x\} , \\
C^{-}(x)=\{ \lambda x(t): 0\leq \lambda \leq 1,\; x(t),\; t>0\text{ solution
of \eqref{diez} and } x(0)=x\}.
\end{gather*}
Systems \eqref{siete} and \eqref{ocho} are homogeneous, by this reasons $
C^{+}(\alpha x)=\alpha C^{+}(x)$ and $C^{-}(\alpha x)=\alpha C^{-}(x)$ for
each $\alpha >0$.
\begin{lemma} \label{lem15}
Let be $\gamma =\{ x(t):t\geq 0\} $
a trajectory of the second type of the family \eqref{uno}.
Then there is $t_0\geq 0$ such that the segment of the trajectory
$ \{x(t):t\geq t_0\} $ is contained in one
of the sets $C^{+}(x(0))$ or $C^{-}(x(0))$.
\end{lemma}
\begin{proof}
If one of the systems \eqref{siete} or \eqref{ocho} is not asymptotically
stable, then one of the sets $C^{+}(x(0))$ or $C^{-}(x(0))$ coincides with
the whole plane and the Lemma is trivial. Consider the case when
\eqref{siete} and \eqref{ocho} are asymptotically stable. Let be $t_0$ such
that, the point $x( t_0)$ $\in L=\{ \lambda x(0):\lambda \geq 0\} $, and any
other point of $\gamma $ corresponding to $t>t_0$ until to complete its
first round around the origin, is on $L$. Without loss of generality let us
suppose that this round is given in a positive direction, then, by virtue of
Lemma \ref{lem9}, the definition of systems \eqref{siete} and \eqref{ocho}
and that two trajectories of the same system can not intersect itself, meets
up that the points of $\gamma $, for $t>t_0$, are in $C^{+}(x(0))$.
\end{proof}
\begin{lemma} \label{lem16}
For the convergence towards the origin of coordinates of the trajectories of
the second type of the family \eqref{uno} it is necessary and sufficient
that systems \eqref{siete} and \eqref{ocho} be asymptotically stable.
\end{lemma}
\begin{proof}
Let be $\gamma =\{ x(t):t\geq 0\} $ a trajectory of the second type and let
be $t_{n}$ the moment when $\gamma $ completes exactly n laps around the
origin (all the laps are considered, both happening in a positive direction
as those occurring in the negative sense). We consider the solutions $x^1(t)$
and $x^2(t)$, $t\geq 0$ of systems \eqref{siete} and \eqref{ocho}
respectively, that satisfy the initial condition $x^1(0)=x^2(0)=x(0)$. For
the definitions of systems \eqref{siete} and \eqref{ocho} and the assumption
that the family of systems \eqref{uno} has trajectories of the second type,
we have that at least one of the solutions $x^1(t)$, $x^2(t)$, $t\geq 0$, is
of the second type. Let us consider the characteristic values of these
solutions (defined in Lemma \ref{lem14}) in the case when they are of the
second type, and let be $\lambda $ the largest of them. By the form of the
sets $C^{+}(x(0))$, $C^{-}( x(0))$, and because of Lemmata \ref{lem14} and
\ref{lem15}, it can see that $\| x(t_1)\| \geq \lambda \|x(0)\| $;
similarly, by the form of $C^{+}(x(t_{i}))$, $C^{-}(x(t_{i}))$ and the
Lemmata \ref{lem14} and \ref{lem15}, the inequality $\| x(t_{i+1})\| \geq
\lambda \| x(t_{i})\| $ holds. From this fact, it follows that $\|
x(t_{i})\| \geq \lambda ^{i}\| x(0)\| $, and systems \eqref{siete} and
\eqref{ocho} are asymptotically stable, is $\lambda $ $<1$ and thus $
x(t_{i})\to 0$ when $i\to +\infty $. Then is verified that the sets $
C^{+}(x(t_{i}))$ and $C^{-}(x(t_{i}))$ tend to the set $\{ 0\} $ in the
Hausdorff metric when $i\to+\infty $, and how the points of the trajectory $
\gamma $, corresponding to the values $t>t_{i+1}$, belong to one of the sets
$C^{+}(x(t_{i}))$ , $C^{-}(x(t_{i}))$; we conclude that the trajectory $
\gamma $ converges towards the origin when $t\to \infty $.
\end{proof}
\subsection{Stability of auxiliary systems}
\begin{lemma} \label{lem17}
Suppose that the matrices $A_1$, $A_2$ are stable and satisfy
\eqref{seis}. If none of the systems \eqref{nueve} and \eqref{diez}
has trajectories of the second type or both have trajectories of the
second type, then the family \eqref{uno} is asymptotically stable.
\end{lemma}
\begin{proof}
In the first case all the trajectories of systems \eqref{nueve} and
\eqref{diez} are of the first type and as these are trajectories of the
family of systems \eqref{uno}, they must converge to the origin. In the
second case, it is clear that systems \eqref{nueve} and \eqref{diez}
coincide with systems \eqref{dos} and \eqref{tres} and thus both are
asymptotically stable. Now there remains to apply Lemma \ref{lem16}.
\end{proof}
Let us consider the more complex case, in which one of the systems
\eqref{nueve} or \eqref{diez} has trajectories of the second type while the
other does not have. Clearly, in this case we have to investigate only the
asymptotic stability of the system that has trajectories of the second type.
So, consider that system \eqref{nueve} has trajectories of the second type,
while system \eqref{diez} does not have trajectories of the second type. Due
to the homogeneity of system \eqref{nueve}, for its asymptotic stability we
will verify the convergence towards the origin of coordinates only for one
trajectory of this system.
Thus, we consider the trajectory $\gamma =\{ x(t):t\geq 0\} $ that satisfies
$x_1(0)=-1$; $x_2(0)=0$. Because this is a trajectory of the second type,
there exist $t>0$ such that $x_2(t)=0$ and $x_1(t)>0$. Let $T$ be the lowest
of these $t$. Further we consider the trajectory $\gamma _{T}=\{
-x_1(T)x(t):t\geq 0\} $\ of \eqref{nueve}. The point of this trajectory
corresponding to $t=0$ is $x(T)$, in which $x(t+T)=-x_1(T)x(t)$. From this
equality follows $x_1(2T)=-x_1^2(T)$ and $x_2( 2T)=0$; and as a consequence
of Lemma \ref{lem14}, we have that $\gamma $ converges towards the origin of
coordinates if and only if $x_1(T)<1$. So, we proved the following lemma.
\begin{lemma} \label{lem18}
For the asymptotic stability of system \eqref{nueve}, when it has
trajectories of the second type, and the solution solution
$x(t)$, $t\in [0, T]$, satisfies boundary conditions
$x_1(0)=-1$; $x_2(0)=0$; $x_2(T)=0$; $x_2(t)\neq 0$,
$t\in (0, T)$; is necessary and sufficient that $x_1(T)<1$.
\end{lemma}
For the effective implementation of Lemma \ref{lem18} it is convenient to
obtain an expression for $x_1(T)$ as a function of the elements that define
system \eqref{nueve}; i.e., the matrices $A_1$, $A_2$.
Let be $w_{ij}(x)$, $i$, $j=1$, $2$, the elements of the matrix $V_1(x)$;
i.e.,
\begin{equation*}
V_1(x)=
\begin{pmatrix}
w_{11}(x) & w_{12}(x) \\
w_{21}(x) & w_{22}(x)
\end{pmatrix}.
\end{equation*}
System \eqref{nueve} is rewritten as
\begin{gather*}
\dot{x_1}=w_{11}(x)x_1+w_{12}(x) x_2 \\
\dot{x_2}=w_{21}(x)x_1+w_{22}(x) x_2.
\end{gather*}
We multiply both equations of this system in order to obtain
\begin{equation*}
\dot{x_1}(w_{21}(x)x_1+w_{22}( x)x_2)=\dot{x_2}(w_{11}(x) x_1+w_{12}(x)x_2).
\end{equation*}
If in the region of the phase plane that is obtained by eliminating the axes
of coordinates, we make the change of variables $z=\frac{x_1}{x_2}$, we
obtain
\begin{equation}
\frac{dx_1}{x_1}=\frac{(\phi _{11}(z) z+\phi _{12}(z)) dz}{z(-\phi _{21}(z)
z^2+(\phi _{11}(z)-\phi _{22}(z) ) z+\phi _{12}(z))}, \label{once}
\end{equation}
where $\phi _{i j}(z)=w_{i j}(x)$, $i $, $j=1$, $2$. The coefficients $\phi
_{i j}(z)$ are well defined as the function $V_1(x)$ is homogeneous. We form
the matrix
\begin{equation*}
\Phi (z)=
\begin{pmatrix}
\phi _{11}(z) & \phi _{12}(z) \\
\phi _{21}(z) & \phi _{22}(z)
\end{pmatrix},
\end{equation*}
then by the relationship between the matrices $\Phi (z)$ and $V_1(x)$, and
from the expression of $V_1(x)$ in \eqref{nueve} it is obtained that
\begin{equation*}
\Phi (z)=
\begin{cases}
A_1 & \mathit{if } \langle (A_1(z,1)^{T})^{\bot }, A_2(z, 1)^{T}\rangle \geq
0 \\
A_2 & \mathit{if } \langle (A_1(z, 1)^{T})^{\bot }, A_2(z, 1)^{T}\rangle <0.
\end{cases}
\end{equation*}
Let us consider the trajectory $\gamma =\{ x(t):t\geq 0\} $ of system
\eqref{nueve} referred in Lemma \ref{lem18}. On this trajectory we take the
points: $P_1(-1+\varepsilon , x_2(-1+\varepsilon ))$; $P_2(-\delta ,
x_2(-\delta ))$; $P_{3}(\delta , x_2(\delta ))$;
$P_{4}(x_1(T) -\varepsilon, x_2(x_1(T)-\varepsilon ))$,
which appear on the trajectory always when $t$ grows in the same order of their sub - indexes, besides $\delta >0$ and $
\varepsilon >0$. To the sections $P_1P_2$ and $P_{3}P_{4}$ of trajectories
of \eqref{nueve} there correspond integral curves of the equation (\ref{once}
) and by direct integration of this equation between the considered extreme
points, we obtain the equalities
\begin{gather*}
\int_{-1+\varepsilon }^{-\delta } \frac{dx_1}{x_1}= \int_{\frac{
-1+\varepsilon }{x_2(-1+\varepsilon )}} ^{\frac{-\delta }{x_2(-\delta )}}
\frac{(\phi _{11}(z) z+\phi _{12}(z))\,dz}{z(-\phi _{21}(z) z^2+(\phi
_{11}(z)-\phi _{22}(z)) z+\phi _{12}(z))}; \\
\int_{\delta} ^{x_1(T)-\varepsilon } \frac{dx_1}{x_1} = \int_{\frac{\delta }{
x_2(\delta )}} ^{\frac{x_1(T)-\varepsilon }{x_2(x_1(T)-\varepsilon )}} \frac{
(\phi _{11}(z) z+\phi _{12}(z))\,dz}{z(-\phi _{21}(z) z^2+(\phi
_{11}(z)-\phi _{22}(z)) z+\phi _{12}(z))}.
\end{gather*}
Adding these expressions we obtain
\begin{align*}
&\int_{-1+\varepsilon }^{-\delta} \frac{dx_1}{x_1}+
\int_{\delta}^{x_1(T)-\varepsilon } \frac{dx_1}{x_1} \\
&=\int_{\frac{-1+\varepsilon }{x_2(-1+\varepsilon )}} ^{\frac{-\delta }{
x_2(-\delta )}} \frac{(\phi _{11}(z) z+\phi _{12}(z)) dz}{z(-\phi _{21}(z)
z^2 +(\phi_{11}(z)-\phi _{22}(z)) z+\phi_{12}(z))} \\
&\quad +\int_{\frac{\delta }{x_2(\delta )}} ^{\frac{x_1(T)-\varepsilon }{
x_2(x_1(T)-\varepsilon )}} \frac{(\phi _{11}(z) z+\phi _{12}(z)) dz}{z(-\phi
_{21}(z) z^2 +(\phi _{11}(z)-\phi _{22}(z)) z+\phi _{12}(z))}.
\end{align*}
As
\begin{align*}
&\frac{(\phi _{11}(z) z+\phi _{12}(z)) }{z(-\phi _{21}(z) z^2+( \phi
_{11}(z)-\phi _{22}(z)) z+\phi _{12}(z))} \\
&=\frac{1}{z}-\frac{1}{2}\frac{-2\phi _{21}(z) z+(\phi _{11}(z)-\phi
_{22}(z))}{(-\phi _{21}(z) z^2+(\phi _{11}(z)-\phi _{22}(z)) z+\phi _{12}(z))
} \\
&\quad +\frac{1}{2}\frac{(\phi _{11}(z)+\phi _{22}( z))}{(-\phi _{21}(z)
z^2+( \phi _{11}(z)-\phi _{22}(z)) z+\phi _{12}(z))},
\end{align*}
we have
\begin{equation}
\begin{aligned} &\ln \big(\frac{x(T)-\varepsilon }{1-\varepsilon }\big)\\
&=\int_{\frac{-1+\varepsilon }{x_2(-1+\varepsilon )}} ^{\frac{-\delta
}{x_2(-\delta )}} \Big(\frac{ 1}{z}-\frac{1}{2}\frac{-2\phi _{21}(z) z+(\phi
_{11}(z)-\phi _{22}(z))}{(-\phi _{21}(z) z^2+(\phi _{11}(z)-\phi _{22}(z))
z+\phi _{12}(z))} \Big)dz\\ &\quad +\int_{\frac{\delta }{x_2(\delta )}}
^{\frac{x_1(T)-\varepsilon }{x_2(x_1(T)-\varepsilon )}}
\Big(\frac{1}{z}-\frac{1}{2}\frac{-2\phi _{21}(z) z+(\phi _{11}(z)-\phi
_{22}(z))}{(-\phi _{21}(z) z^2+(\phi _{11}(z)-\phi _{22}(z)) z+\phi
_{12}(z))}\Big)dz \\ &\quad +\frac{1}{2} \int_{\frac{-1+\varepsilon
}{x_2(-1+\varepsilon)}} ^{\frac{-\delta }{x_2(-\delta )}} \Big(\frac{(\phi
_{11}(z)+\phi _{22}(z) )}{(-\phi _{21}(z) z^2+(\phi _{11}(z)-\phi _{22}(z))
z+\phi_{12}(z))}\Big)dz \\ &\quad +\frac{1}{2}\int_{\frac{\delta
}{x_2(\delta )}} ^{\frac{x_1(T)-\varepsilon }{x_2(x_1(T)-\varepsilon )}}
\Big(\frac{(\phi _{11}( z)+\phi _{22}(z))}{(-\phi _{21}( z) z^2+(\phi
_{11}(z)-\phi _{22}( z)) z+\phi _{12}(z))}\Big)dz. \end{aligned}
\label{doce}
\end{equation}
By direct calculations we can prove that a primitive of the first two
integrals in the right hand-side is
\begin{equation}
\ln \big| \frac{z}{| -\phi _{21}(z) z^2+(\phi _{11}(z)-\phi _{22}(z)) z+\phi
_{12}(z)| ^{1/2}}\big|, \label{trece}
\end{equation}
which converges when $z\to \pm \infty $, because the function $\phi _{21}(z)$
is a constant and different from zero one for $z$ sufficiently large.
Furthermore, the denominator of the last two integrals is different from
zero for all $z\in\mathbb{R}$, this ensures that for these integrals we can
apply the criteria of comparison.
The lower limit of integration in the third integral in (\ref{doce}) tends
to $-\infty $ when $\varepsilon \to 0$. While the upper limit of the fourth
integral in (\ref{doce}) tends to $+\infty $, and as the expression
\begin{equation*}
\frac{z^2}{-\phi _{21}(z) z^2+(\phi _{11}(z)-\phi _{22}(z)) z+\phi _{12}(z)}
\end{equation*}
converges when $z\to \pm \infty $ to non-zero numbers, we conclude that,
when we pass to the limit in (\ref{doce}) with $\varepsilon \to 0$, the
considered integrals are converging. As integrands in the last two integrals
of the expression (\ref{doce}) are continuous functions in all $\mathbb{R}$,
passing to the limit when $\delta \to 0$ we obtain
\begin{equation}
\begin{aligned} \ln (x_1(T)) &=\lim_{\delta \to 0}\Big\{\int_{-\infty
}^{-\delta} \Big(\frac{1}{z}-\frac{1}{2}\frac{-2\phi _{21}( z) z+(\phi
_{11}(z)-\phi _{22}( z))}{(-\phi _{21}(z) z^2+( \phi _{11}(z)-\phi _{22}(z))
z+\phi _{12}(z))}\Big)dz \\ &\quad+ \int_\delta^ {+\infty }
\Big(\frac{1}{z}- \frac{1}{2}\frac{-2\phi _{21}(z) z+(\phi _{11}(z)-\phi
_{22}(z))}{(-\phi _{21}(z) z^2+(\phi _{11}(z)-\phi _{22}(z)) z+\phi
_{12}(z))} \Big)dz\Big\} \\ &\quad +\frac{1}{2} \int_{-\delta}^\delta
\Big(\frac{(\phi _{11}(z)+\phi _{22}(z))}{( -\phi _{21}(z) z^2+(\phi
_{11}(z) -\phi _{22}(z)) z+\phi _{12}(z))}\Big)dz. \label{catorce}
\end{aligned}
\end{equation}
Using the primitive (\ref{trece}) of two first integrals in (\ref{doce}),
evaluating them and passing to the limit when $\delta \to 0$, we obtain
\begin{equation}
\ln (x_1(T))=\ln Q+\int_{-\infty}^{+ \infty } \big(\frac{\mathop{\rm tr}
\Phi (z)}{g(z)}\big)dz, \label{quince}
\end{equation}
where
\begin{gather*}
g(z)=-\phi _{21}(z) z^2+(\phi_{11}(z)-\phi _{22}(z)) z+\phi_{12}(z), \\
h(z)=\langle (A_1(z, 1)^{T})^{\bot }, A_2(z, 1)^{T}\rangle \\
Q=
\begin{cases}
\frac{g(z_1- 0)g(z_2- 0)}{g(z_1+ 0)g(z+ 0)} & \text{if $h(z)$ has twow real
roots} z_1, z_2 \\[3pt]
\frac{g(z_1- 0)g(+ \infty )}{g(z_1+ 0)g(- \infty )} & \text{if $h(z)$ has a
single real root} z_1 \\
1 & \text{if $h(z)$ does not have real roots},
\end{cases}
\end{gather*}
Let us denote $I^{+}=\ln Q+\int{+ \infty }^{+\infty } \big(\frac{\mathop{\rm
tr}\Phi (z)}{g(z)}\big)dz$.
\begin{lemma} \label{lem19}
A necessary and sufficient condition for the asymptotic stability of
\eqref{nueve}, when it has trajectories of the second type, is
$I^{+}<0$.
\end{lemma}
The assertion of the above lemma is a direct consequence of Lemma \ref{lem18}
and \eqref{quince}.
In the case where only system \eqref{diez} has trajectories of the second
type, we associate to this system a number which is denoted by $I^{-}$. This
number is obtained by adding a negative sign $(-)$ to the right hand-side in
the expression (\ref{quince}) and substituting $\Phi(z)$ by
\begin{equation*}
\Phi ^{-}(z)=
\begin{cases}
A_2 & \text{if } \langle (A_1(z,1)^{T})^{\bot }, A_2(z, 1) ^{T}\rangle \geq 0
\\
A_1 & \text{if } \langle (A_1(z,1)^{T})^{\bot }, A_2(z, 1) ^{T}\rangle <0.
\end{cases}
\end{equation*}
In this case a similar result to Lemma \ref{lem19} is valid:
\begin{lemma} \label{lem20}
A necessary and sufficient condition for the asymptotic stability of
\eqref{diez}, when it has trajectories of the second type, is
$I^{-}<0$.
\end{lemma}
The assertion of the above lemma is a direct consequence of Lemma \ref{lem18}
and (\ref{quince}), when a negative sign is added to the right hand-side in
this expression and the function $\Phi (z)$ is replaced by $\Phi ^{-}(z)$.
\section{Main Result}
\begin{theorem} \label{thm21}
For the systems of the family \eqref{uno} to have trivial
asymptotically stable solutions, it is necessary and sufficient that:
\begin{itemize}
\item[(i)] $\mathop{\rm tr}(A_{i})<0$, $\det (A_{i})>0$, $i=1$, $2$;
\item[(ii)] $a_{12}^1a_{21}^2+a_{12}^2a_{21}^1-a_{11}^1a_{22}^2
-a_{11}^2a_{22}^1\leq 2 \sqrt{\det A_1\det A_2}$;
\item[(iii)] One of the following two conditions holds:
\begin{itemize}
\item[(a)] if $a_{12}^1>0$ or $a_{12}^2>0$ and for each
$k\in\mathbb{R}$,
$-a_{21}^1-(a_{22}^1-a_{11}^1)k+a_{12}^1k^2>0$,
$i=1 $ or $2$, then $I^{+}<0$;
\item[(b)] if $a_{12}^1<0$ or $a_{12}^2<0$ and for each
$k\in\mathbb{R}$,
$-a_{21}^1-(a_{22}^1-a_{11}^1)k+a_{12}^1k^2<0$, $i=1$ or $2$,
then $I^{ -}<0$.
\end{itemize}
\end{itemize}
\end{theorem}
\begin{proof}
Let $A_1$, $A_2$ be stable matrices. As we saw in section 3, the
trajectories of the systems that integrate the family \eqref{uno} are
segments of the trajectories of systems \eqref{dos} and \eqref{tres}, and
these systems are asymptotically stable.
\noindent \textbf{Necessity:} Suppose that the family \eqref{uno} is
asymptotically stable. Then condition (i) is guaranteed because asymptotic
stability of systems \eqref{dos} and \eqref{tres}, as this condition
equivalents to the stability of matrices $A_1$, $A_2$. The asymptotic
stability of the family \eqref{uno} means that the trajectories of all
systems of the family converge towards the origin of coordinates. Then for
the trajectories of the first type, due to Theorem \ref{thm13}, condition
(ii) holds. While, for the trajectories of the second type, condition (iii)
holds due to Lemmata \ref{lem19}, \ref{lem20}.
\noindent\textbf{Sufficiency:} Suppose now that conditions (i), (ii), (iii)
hold. Then, due to condition (i), the trajectories of systems \eqref{dos}
and \eqref{tres} converge towards the origin of coordinates. The
trajectories of the first type of family \eqref{uno} also converge towards
the origin of coordinates because of condition (ii) and
Theorem \ref{thm13}. The same happens with the trajectories of the second type because of
condition (iii) and the Lemmata \ref{lem19}, \ref{lem20}. Thus, the trivial
solution of the family of systems \eqref{uno} is asymptotically stable.
\end{proof}
\subsection{Examples}
In this section we present some examples of families of systems of
differential equations whose stability of the trivial solution is
determined. In each case the matrices $A_1$, $A_2$ are stable, and based on
this fact, the different conditions of the previous theorem hold.
\begin{example} \label{exa22} \rm
Let $A_1=\begin{pmatrix}
1 & 2 \\
-2 & -2 \end{pmatrix}$,
$A_2=\begin{pmatrix}
-1 & 1 \\
-1 & 0 \end{pmatrix}$. In this case conditions (i), (ii), (iii)(a)
of Theorem \ref{thm21} hold, therefore it is necessary to calculate $I^{+}$
to know its sign.
After the required calculations we obtain $I^{+}=-0.8150$; now we can say
that the family of systems of differential equations \eqref{uno} determined
by the pair of matrices $A_1$, $A_2$ is asymptotically stable.
\end{example}
\begin{example} \label{exa23} \rm
Let $A_1=\begin{pmatrix}
1 & -1 \\
2 & -\frac{3}{2} \end{pmatrix}$,
$A_2=\begin{pmatrix}
0 & -1 \\
3 & -3 \end{pmatrix}$.
In this case conditions (i), (ii), (iii)(b) of Theorem \ref{thm21} hold,
therefore it is necessary to calculate $I^{-}$ to know its
sign. After the required calculations we obtain $I^{ -}=\sqrt{12}\pi $;
now we can say that the family of systems of differential equations
\eqref{uno} determined by the pair of matrices $A_1$, $A_2$ is unstable.
\end{example}
\begin{example} \label{exa24} \rm
Let $A_1=\begin{pmatrix}
0 & -1 \\
3 & -3 \end{pmatrix}$,
$A_2=\begin{pmatrix}
-1 & 1 \\
-1 & 0\end{pmatrix}$.
Again the matrices $A_1$, $A_2$ are asymptotically stable and
for them, conditions (i), (ii), (iii)(a)(b), of Theorem \ref{thm21} hold. For
this reason, both systems \eqref{nueve} and \eqref{diez} have trajectories
of the second type, thus, by Lemma \ref{lem17}, the corresponding family of
systems of differential equations is asymptotically stable.
\end{example}
\begin{example} \label{exa25} \rm
Let $A_1=\begin{pmatrix}
-\frac{35}{10} & 5 \\
3 & -3 \end{pmatrix}$,
$A_2=\begin{pmatrix}
0 & -1 \\
3 & -3 \end{pmatrix}$.
In this example, the pair of matrices $A_1$, $A_2$ satisfies
condition (i) of Theorem \ref{thm21}, but condition (ii) of
this theorem is no longer satisfied. By this reason, it is no longer
guaranteed the asymptotic stability of the corresponding family of
systems of differential equations.
\end{example}
\begin{example} \label{exa26} \rm
Let $A_1=\begin{pmatrix}
-1 & 0 \\
1 & -1 \end{pmatrix}$,
$A_2=\begin{pmatrix}
-1 & 1 \\
-1 & 0\end{pmatrix}$.
In this case conditions (i), (ii), (iii)(a) of Theorem \ref{thm21} hold,
thus we need to determine the sign of $I^{+}$. However, the function
$g(z)=-z^2$ has the real root $0$; i.e., the integral appearing
in the expression for $I^{ +}$ is an improperly mixed integral, which
tends to $-\infty $. Thus the family of systems of differential equations
determined by the pair of matrices $A_1$, $A_2$ is asymptotically stable.
\end{example}
\subsection*{Conclusion}
In this work we found the necessary and sufficient conditions for the
asymptotic stability of the family of systems of differential equations
under review. These conditions are given explicitly depending on the
coefficients of the matrices that determine each family. To continue this
line of research, it would be interesting to study the same problem, when
uncertainty is present not only in the moments of change, but in both
schemes of connection; i.e., in the element of matrices $A_1$, $A_2$
determining a new family of systems.
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\end{document}