Funkcialaj Ekvacioj, 8 (1966), 79-90
Eventual Properties and Quasi-asymptotic
Stability of a Non-compact Set
Dedicated to Professor A. Kobori on his sixtieth birthday
By Taro YOSHIZAWA
(Tohoku University)
1.
Introduction
Stability theory for compact sets does not differ essentially from that of the
stability of equilibrium states. However, the extensions of stability theory to
that for non-compact sets are no longer trivial. The author has discussed the
stability and the boundedness of solutions concerning non-compact sets [4], [5].
Recently, LaSalle has given a theorem on stability of non-compact manifolds
which appears to have applications to the design of control systems with adjustable feedback [1]. He considered the system
$¥left¥{¥begin{array}{l}x^{¥prime}=f(t,x,y)¥¥y^{¥prime}=g(t,x,y),¥end{array}¥right.$
where the vector represents the error in control and the vector
represents
parameters such as control parameters. His problem is that we would like to
have the error approach 0, and about control parameters , we are less concerned as long as they remain bounded.
In this case, the set $x=0$ is a noncompact set. This is a kind of stabilities of the non-compact set $x=0$ .
In this paper, first of all, we shall discuss the eventual boundedness of
solutions and the eventual stability of a set. To make considerations simple,
we shall discuss the eventual properties in simple forms. In the second part,
concerning LaSalle’s problem, we shall discuss the quasi-asymptotic stability of
a non-compact set.
For the definitions of stability of a set and eventual properties, refer to [4].
In this article, we shall denote by $C_{0}(x)$ the family of functions which
satisfy locally the Lipschitz condition with respect to . In the case where the
Lipschitz constant of $f(t, x)$ with respect to
is independent of and depends
only on each compact set of , we shall denote it by
.
$x$
$y$
$x$
$y$
$x$
$x$
$x$
2.
$t$
$f(t, x)¥in¥overline{C}_{0}(x)$
Eventual properties
We shall discuss the eventual boundedness of solutions and the eventual
stability. We can prove some theorems on eventual boundedness or on eventual
80
T. YOSHIZAWA
$I$
†
stability concerning non-compact sets. However, to make our considerations
simple, we shall consider the usual eventual properties. In case we discuss the
eventual properties concerning non-compact sets, we have to assum that the
solutions exist in the future, or we have to see the existence of solutions by
$¥underline{¥mathrm{p}}$
additional conditions.
Consider a system of differential equations
(1)
$x^{¥prime}=F(t, x)$
$(’=¥frac{d}{dt})$
,
where is an -vector and $F(t, x)$ is defined and continuous on ¥ ¥ ,
is the Euclidean -space).
(
Theorem 1. We assume that there exists a continuous Liapunov function
¥
$V(t, x)$ defined on $t¥in L||x||¥geqq R_{0}$ (
is the interval ¥
and $R_{0}>0$ may
be sufficientfy large), which satisfies the foflowing conditions:
(i) $a(||x||)¥leqq V(t, x)¥leqq b(||x||)$ , where $a(r)$ is continuous, increasing, positive and
as
, and $b(r)$ is continuous and increasing,
,
(ii)
$x$
$R^{n}$
$ 0 leqq t< infty$
$¥mathrm{w}$
$R^{n}$
$x$
$¥in$
$¥mathrm{n}$
$/$
$ a(r)-¥succ¥infty$
$ 0 leqq t< infty$
$ r-¥succ¥infty$
$V(t, x)¥in¥overline{C}_{0}(x)$
(iii)
.
$V_{(1)}^{f}(t, x)=¥varlimsup_{h¥rightarrow 0^{+}}¥frac{1}{h}¥{V(t+h, x+hF(t, x))-V(t, x)¥}¥leqq 0$
Then, the solutions of
(2)
$x^{¥prime}=F(t, x)+G(t, x)$
are eventually uniform-bounded, where $G(t, x)$ is continuous on
for any bounded continuous function $x(t)$ on ,
$t¥in I$
,
$x¥in R^{n}$
and
$I$
$¥int_{0}^{¥infty}||G(t, x(t))||dt<¥infty$
Proaf.
such that
.
Correspoding to any a $>R_{0}$ , we have
for $t¥in I$ and
$ R_{0}¥leqq||x||¥leqq¥alpha$ .
Choose a
so that $2b(¥alpha)<a(¥beta)$ ¥ ¥ . This
possible, because
as
. Let $L(¥beta)$ be the Lipschitz constant of
$V(t, x)$ for
¥
. If we set
,
is inte$V(t, x)¥underline{-¥leq}b(¥alpha)$
$¥beta(¥alpha)$
$ a(r)¥rightarrow¥infty$
$¥check{1}¥mathrm{S}$
$ r¥rightarrow¥infty$
$¥alpha¥leqq||x||¥leqq¥theta$
Let
$T(¥alpha)>0$
such that if
$¥int_{0}^{¥infty}g(t, ¥beta)dt<¥frac{b(¥alpha)}{L(¥beta)}$
be a solution of (2) through
.
Suppose that we have
exist
and
such that
,
that for
, we have
$x(t;x_{0}, t_{0})$
$t_{2}$
$t_{3}$
$t_{0}¥leqq t_{2}<t_{3}¥leqq t_{1}$
$t_{0}¥geqq T(¥alpha)$
$(t_{0}, x_{0})$
such that
at some
$||x(t_{2}; x_{0}, t_{0})||=¥alpha$
$t_{2}<t<t_{¥theta}$
$¥alpha<||x(t;x_{0}, t_{0})||<¥beta$
beta)$
, we have
.
$||x(t_{1} ; x_{0}, t_{0})|-|=¥beta$
$t_{0}¥geqq T(¥alpha)$
$g(t,
$g(t, ¥beta)=¥max_{||x||¥leqq ¥mathrm{P}}||G(t, x)||$
grable, and hence, there exists a
(3)
$x$
$( beta> alpha)$
,
$t_{1}$
.
a and
Then, there
$||x_{0}||¥leqq$
$||x(t_{3}; x_{0}, t_{0})||=¥beta$
and)
.
Since we have
$V_{(2)}^{¥prime}(t, x)¥leqq L(¥beta)g(t, ¥beta)$
$¥downarrow itii$
’
81
Eventual Properties and Quasi-asymptotic Stability of a Non-compact Set
in the domain
$ t¥in L¥alpha¥leqq||x||¥leqq¥theta$
, we have
$V(t_{3}, x(t_{¥mathfrak{Z}}; x_{0}, t_{0}))¥leqq V(t_{2}, x(t_{2};x_{0}, t_{0}))+¥int_{t_{2}}^{t_{3}}L(¥beta)g(t, ¥beta)dt$
.
From this and (3), it follows that
$V(t_{3}, x(t_{3}; x_{0}, t_{0}))¥leqq 2b(¥alpha)<a(¥beta)$
,
¥
. This is a contradiction. Therefore, we can
whence we have
.
, we have $||x(t;x_{0}, t_{0})||<¥beta$ for all
see that if
and
This shows that the solutions of (2) are eventually uniform-bounded.
By the same argument as in the proof of Theorem 1, we have the followresult.
Theorem 2. We assume that there exists a continuous Liapunov function
$V(t, x)$ defined on $t¥in L||x||¥geqq R_{0}$ (
may be sufficientfy large), which satisfies
the following conditions:
(i) $a(||x||)¥leqq V(t, x)¥leqq b(||x||)$ , where $a(r)$ and $b(r)$ are the same as in
Theorem 1,
$V(t,
x)¥in C_{0}(x)$ ,
(ii)
$||x(t_{3};x_{0}, t_{0})||< beta$
$t¥geqq t_{0}$
$t_{0}¥geqq T(¥alpha)$
$||x_{0}||¥leqq¥alpha$
$¥mathrm{i}¥mathrm{n}¥mathrm{g}$
$R_{0}$
(iii)
$V_{(1)}^{¥prime}(t, x)¥leqq h(t)q(t, x)$
, where
$¥int_{0}^{¥infty}|h(t)|dt<¥infty$
, and
$q(t, x)$
is conti
$¥leftarrow$
and is bounded for bounded .
Then, the solutions of (1) are eventually uniform-bounded.
Now we shall discuss the eventual stability, and hence, we assume that
$F(t, x)$ in (1) and $G(t, x)$ in (2) are defined and continuous on $ 0¥leqq t<¥infty$ , $||x||$
$¥leqq H$
, where $H$ is a positive constant.
Theorem 3. We assume that there exists a continuous Liapunov function
$V(t, x)$ defined on $t¥in I$ , $||x||¥leqq H^{*}$ $(¥leqq H)$ , which satisfies the following conditions:
(i) $a(||x||)¥leqq V(t, x)¥leqq b(||x||)$ , where $a(r)$ is continuous, increasing,
as
positive definite and $b(r)$ is continuous, increasing and
,
nuous on
,
$t¥in I$
$||x||¥geqq R_{0}$
$x$
$b(r)¥rightarrow 0$
$r¥rightarrow 0$
for
(ii)
$V(t, x)¥in¥overline{C}_{0}(x)$
(iii)
$V_{(1)}^{¥prime}(t, x)¥leqq 0$
,
.
Then, the set $M=I¥times¥{0¥}$ is an eventually uniform-stable set
any continuous function $x(t)$ such that $||x(t)||¥leqq H^{*}for$ $aff$
of (2), where
$t¥geqq 0$
$¥int_{0}^{¥infty}||G(t, x(t))||dt<¥infty$
Proof. For a given
. Let $L$ be Lipschitz
$¥epsilon>0$
$.(¥delta<¥epsilon)$
’.Fhen, since
$g(t)$
,
$¥epsilon<H^{*}$
$¥mathrm{c}¥mathrm{o}¥mathrm{n}¥dot{¥mathrm{s}}$
, choose
tant of
$V(t, x)$
is integrable, there exists a
$a$
,
.
$¥delta(¥mathrm{e})>0$
and let
$T(¥epsilon)>0$
so that
be
$g(t)$
$2b(¥delta)<a(¥epsilon)$
$¥max_{||x||¥leqq H^{*}}||G(t, x)||$
such that if
$t_{0}¥geqq T(¥epsilon)$
.
,
T. YOSHIZAWA
$¥mathfrak{B}2$
-we have
$¥int_{t_{0}}^{¥infty}g(t)dt<¥frac{b(¥delta)}{L}$
Let
.
such that
be a solution of (2) through
¥
Then, as long as
, we have
$x(t;x_{0}, t_{0})$
$ t||x_{0}||<¥delta$
.
$(f_{0}, X_{0})$
$t_{0}¥geqq T(¥epsilon)$
and
$||x(t;x_{0}, t_{0})||< epsilon$
$V^{¥prime}(t, x(t;x_{0}, t_{0}))¥leqq Lg(t)$
Therefore, as long as
$||x(t;x_{0}, t_{0})||<¥epsilon$
.
,
$V(t, x(t;x_{0}, t_{0}))¥leqq V(t_{0}, x_{0})+L¥int_{t_{0}}^{t}g(t)dt$
$¥leqq 2b(¥delta)$
$<a(¥epsilon)$
-whence we have
$||x(t;x_{0}, t_{0})||<¥epsilon$
for all
,
$t¥geqq¥tau_{0}$
.
This shows that
$M$
is an even-
tually uniform-stable set of (2).
By the same argument as in the proof of Theorem 3, we have the following theorem.
Theorem 4. We assume that there exists a continuous Liapunov function
$V(t, x)$ defined on $t¥in L||x||¥leqq H^{*}$ $(¥leqq H)$ , which
satisfies the following condi-
.tions:
(i)
$a(||x||)¥leqq V(t, x)¥leqq b(||x||)$
, where
$a(r)$
and
$b(r)$
are the same as in
Theorem 3,
,
(ii)
$V(t, x)¥in C_{0}(x)$
(iii)
$V_{(1)}^{¥prime}(t, x)¥leqq h(t)q(t, x)$
, where
$ f_{0}^{¥infty}|h(t)|dt<¥infty$
and
$q(t, x)$
is bounded.
Then, the set $M=I¥times¥{0¥}$ is an eventually uniform-stable set of (1).
Theorem 5. Under the same assumptions as in Theorem 3 (or Theorem 4),
$M=I¥times¥{0¥}$ is an invariant set
of (2) (or (1)), then the zero solution is uni-
if
form-stable.
This theorem is an immediate result of Theorems above and Theorem 2 in [5]
Next we shall discuss the asymptotic stability. Here, we consider the eventual asymptotic stability of the set $M=I¥times¥{0¥}$ . We can apply the same argu-ment to the eventual asymptotic stability of a non-compact set.
Theorem 6. We assume that there exists a continuous Liapunov function
$V(t, x)$ defined on $t¥in L||x||¥leqq H^{*}$ $(¥leqq H)$ , which
satisfies the conditions (i),
in Theorem 4 and
$¥backslash (¥mathrm{i}¥mathrm{i})$
(iii)’
$V_{(1)}^{¥prime}(t, x)+V^{*}(t, x)¥rightarrow 0$
as
$0<¥lambda¥leqq||x||¥leqq H^{*}$
(for any )
$¥lambda$
is continuous and there exists a continuous
such that $V^{*}(t, x)¥geqq c(||x||)$ .
function
for
Then, the set $M$ is an eventually uniform-asymptotically stabfe set of (1).
Moreover, if $M$ is invariant, $M$ is a uniform-asymptotically stable set of
$ t¥rightarrow¥infty$
, where
uniformly on
$c(r)>0$
$V^{*}(t, x)$
$r>0$
$(1)_{¥sim}$
$¥_$
Eventual Properties and Quasi-asymptotic Stability
of a Non-compact Set
Proof. For a given
,
, there exists
¥
By
.
the assumptions, on the domain ¥ ¥
and a $T(¥epsilon)>0$ such that
and that if
$¥epsilon>0$
$¥epsilon¥leqq H^{*}$
$a(¥epsilon)$
$a$
$V_{(1)}^{J}(t, x)+V^{*}(t, x)¥leqq¥frac{¥uparrow¥cdot(¥epsilon)}{2}$
.
$b(¥delta)<$
$¥delta(¥epsilon)>0$
$ delta leqq||x|| leqq H^{*}$
$V^{*}(t, x)¥geqq¥gamma(¥epsilon)$
such that
, there exist
, we have
83
$a$
$¥gamma(¥epsilon)>0$
$t¥geqq T(¥epsilon)$
Thus, we have
(4)
$V_{(1)}^{¥prime}(t, x)¥leqq-¥frac{¥gamma(¥epsilon)}{2}$
in the domain
,
and
, we have
a solution of (1) through
stable set of (1). For
we can assume that
(1) such that
,
$t¥geqq T(¥epsilon)$
.
, )
From this, it follows that if
$||x¥{t;x_{0}$
, where $x(t;x_{0}, t_{0})$ is
for all
. This shows that $M$ is an eventually uniform, we set
and $T_{0}=T(H^{*})$ . For any ,
and
. Consider a solution $x(t;x_{0}, t_{0})$ of
,
. On the domain
,
$t_{0}¥geqq T(¥epsilon)$
$¥delta¥leqq||x||¥underline{¥underline{¥leq}}H^{*}$
$||x_{0}||<¥delta(¥epsilon)$
$t_{0}$
$||<¥epsilon$
$t¥geqq t_{0}$
$(t_{0}, x_{0})$
$¥epsilon=H^{*}$
$¥delta_{0}=¥delta(H^{*})$
$T(¥epsilon)¥geqq T_{0}$
$¥delta(¥epsilon)¥leqq¥delta_{0}$
$t_{0}¥geqq T(¥epsilon)$
$¥epsilon$
$¥delta(¥epsilon)¥leqq||x||¥leqq H^{*}$
$||x_{0}||<¥delta_{0}$
we have (4) and hence, we can see that if
$t>t_{0}+T_{1}(¥epsilon)$
,
$t¥geqq T(¥epsilon)$
$T_{1}(¥mathrm{e})=¥frac{2}{¥gamma(¥epsilon)}(b(H^{*})-$
, we have
. This
es the eventual uniform-asymptotic
stability.
In case $M$ is invariant, $M$ is a uniform-stable set of (1) by Theorem 2 in
[5]. For a solution
of (1) such that
is a posi, where
tive constant such that if
and
, we have $||x(t;x_{0}, t_{0})||<H^{*}$ for
¥
¥
all
, if
, we have
for
, and if ¥
¥
, we can see that
for $t>t_{0}+T(¥epsilon)+T_{1}(¥epsilon)$ , because
$||x(T(¥epsilon);x_{0}, t_{0})||<H^{*}$ .
Thus, we can see that $M$ is a uniform-asymptotically
stable set of (1).
Remark. Massera has proved the equiasymptotic stability of the trivial
solution under the same conditions as those in Theorem 6 (cf. Theorem 5 in [2]).
To discuss the case in the large, we assume that $F(t, x)$ in (1) is defined
and continuous on $t¥in I$, ¥
.
Theorem 7. We assume that there exists a continuous Liapunov function
¥
$V(t, x)$ defined on $t¥in I$,
which satisfies the following conditions:
$a_{
¥
backslash
}^{(}||x||)
¥
leqq
V(t,
x)
¥
leqq
b(||x||)$ ,
(i)
where $a(r)$ is continuous, increasing, positive definite and
as
, and $b(r)$ is continuous, increasing and
as
,
$a(¥delta))$
$||x(t;x_{0}^{-},t_{0})||<¥mathrm{e}$
$¥mathrm{p}¥mathrm{r}¥mathrm{o}¥overline{¥mathrm{v}}$
$x(t;x_{0}, t_{0})$
$||x_{0}||<¥delta_{1}$
$||x_{0}||<¥delta_{1}$
$t_{0}¥geqq T(¥epsilon)$
$t¥geqq t_{0}$
$T(¥epsilon)$
$¥delta_{1}$
$t_{0}¥in I$
$||x(t;x_{0}, t_{0})||< epsilon$
$t>t_{0}+T_{1}( epsilon)$
$0 leqq t_{0}<$
$||x(t;x_{0}, t_{0})||< epsilon$
$x in R^{n}$
$x in R^{n}$
$ a(r)¥rightarrow¥infty$
$b(r)¥rightarrow 0$
$ r¥rightarrow¥infty$
$r¥rightarrow 0$
(ii)
$V(t, x)¥in C_{0}(x)$ ,
(iii)
$V_{(1)}^{¥prime}(t, x)+V^{*}(t, x)¥rightarrow 0$
uniformly on
$ 0<¥lambda¥leqq||x||¥leqq¥mu$
,
for any
$¥lambda$
and
, as
, where $V^{*}(t, x)$ is continuous and there exists a continuous function $c(r)>0$ for $r>0$ such that $V^{*}(t, x)¥geqq c(||x||)$ .
Then, the set $M$ is an eventually uniform-asymptotically stabfe set of (1)
in the large.
$¥mu$
$ t¥rightarrow¥infty$
84
T. YOSHIZAWA
Proof. First of all, we shall see the eventual boundedness of solutions of
(1). For any $¥alpha>0$ , there exists a
such that ¥ ¥ ¥ , because
as
. By (iii), there exist a $K(¥alpha)>0$ and an $S(¥alpha)>0$ such that
$V^{*}(t, x)¥geqq K(¥alpha)$ and $V_{(1)}^{J}(t, x)+V^{*}(t, x)¥leqq¥frac{K(¥alpha)}{2}$
for
and ,
$¥infty$
$ a(r)¥rightarrow$
$b( alpha)< alpha( beta)$
$¥beta(¥alpha)$
$ r¥rightarrow¥infty$
$t¥geqq S(¥alpha)$
$¥beta(¥alpha)$
.
From this, it follows that
$V_{(1)}^{¥prime}(t, x)¥leqq-¥frac{K(¥alpha)}{2}$
$¥alpha¥leqq||x||¥leqq$
$x$
on the domain
$t¥geqq S(¥alpha)$
,
.
Therefore, by the standard argument, we can see that if
and
, we have
for all
, which shows
that the solutions of (1) are eventually uniform-bounded.
By Theorem 6, the set $M$ is an eventually uniform-stable set of (1), that
is, for any
, there exist a
and a $T(¥epsilon)>0$ such that if
¥
and
, we have
.
for all
Let
be a solution of (1) such that
. For
and
such that
, there exist a
such
and an
that
and that if
, we have
$¥alpha¥leqq||x||¥leqq¥beta(¥alpha)$
$t_{0}¥geqq S(¥alpha)$
$||x_{0}||¥leqq¥alpha$
$||x(t;x_{0}, t_{0})||¥leqq¥beta(¥alpha)$
$¥epsilon>0$
$¥delta(¥epsilon)>0$
$||x_{0}||<¥delta(¥epsilon)$
$t_{0}¥geqq T(¥epsilon)$
$||x(t;x_{0}, t_{0})||< epsilon$
$t¥geqq t_{0}$
$x(t;x_{0}, t_{0})$
$t_{0}¥geqq S(¥alpha)$
$¥delta(¥epsilon)¥leqq||x||¥leqq¥beta(¥alpha)$
$x$
$t¥geqq S_{1}(¥epsilon, ¥alpha)(S_{1}(¥mathrm{e}, ¥alpha)¥geqq T(¥mathrm{e}))$
$V_{(1)}^{¥prime}(t, x)+V^{*}(t, x)¥leqq¥frac{¥gamma(¥epsilon,¥alpha)}{2}$
.
$||x(t;x_{0}, t_{0})||<¥mathrm{e}$
In case
for all
or
$V_{(1)}^{¥prime}(t, x)¥leqq-¥frac{¥gamma(¥epsilon,¥alpha)}{2}$
$t_{¥Phi}¥geqq S_{1}(¥epsilon, ¥alpha)$
We can assume that
.
, we can find a
. In fact, we can set
$T^{*}(¥epsilon, ¥alpha)>0$
such that
$t>t_{0}+T^{*}(¥epsilon, ¥alpha)$
$T^{*}(¥mathrm{e}, ¥alpha)=¥frac{2}{¥gamma(¥mathrm{e},¥alpha)}¥{b(¥beta(¥alpha))-a(¥delta(¥mathrm{e}))¥}$
In case
$||x_{0}||¥leqq¥alpha$
$S_{1}(¥epsilon, ¥alpha)>0$
$¥gamma(¥epsilon, ¥alpha)>0$
$V^{*}(t, x)¥geqq¥gamma(¥epsilon, ¥alpha)$
$S_{1}(¥epsilon, ¥alpha)>S(¥mathrm{o}¥mathrm{e})$
$t¥geqq t_{0}$
.
¥
¥
, ciearly we have $||x(t;x_{0}, t_{0})||<¥epsilon$ for
. Thus, we can find a
and
such that if
, we have
for all
. This shows that
$M$ is an eventually quasi-uniform-asymptotically stable set of (1) in the large.
The proof is completed.
$S(¥alpha)¥leqq t_{0}<S_{1}(¥epsilon, oe)$
$t>t_{0}+S_{1}( epsilon,
$S(¥alpha)+T^{*}(¥epsilon, ¥alpha)$
$T_{1}(¥epsilon, ¥alpha)>0$
$t_{0}¥geqq S(¥alpha)$
3.
$||x(t;x_{¥phi}, t_{0})||<¥mathrm{e}$
alpha)-$
$||x_{0}||¥leqq¥alpha$
$t>t_{0}+T_{1}(¥epsilon, ¥alpha)$
Stability of a non-compact set
Consider a system of differential equations
$x^{¥prime}=F(t, x)$ ,
(5)
where is an -vector and $F(t, x)$ is defined and continuous on ¥ , ¥
.
We shall discuss the stability of a set $M=I¥times N$, where $N$ is a non-empty set
in
. Here, a function $W(t, x)$ is said to be positive definite with respect to
$N$, if $W(t, x)¥geqq 0$ for all
, and if for any compact set
and for any
$x
¥
in-U(
¥epsilon, N)$ , we
$x
¥
in
K$
$c(
¥
epsilon,
K)>0$
, there exists a constant
such that if
and
have $W(t, x)¥geqq c(¥epsilon, K)$ , where
is the -neighborhood of $N$ in
. We
shall denote by $d(x, N)$ the distance between and $N$.
First of all, we shall
prove the following theorem by the similar argumehts to those in the
evious
theorems.
$x$
$t in I$
$¥mathrm{n}$
$x in R^{n}$
$R^{n}$
$t$
$K¥subset R^{n}$
$x$
$¥epsilon>0$
$U(¥epsilon, N)$
$¥epsilon$
$R^{n}$
$x$
$¥dot{¥mathrm{s}}$
$¥mathrm{p}¥dot{¥mathrm{r}}$
of a Non-compact Set
Eventual Properties and Quasi-asymptotic Stability
85
Theorem 8. We assume that the solutions of (5) are uniform-bounded.
Moreover, we assume that there exists a continuous Liapunov function $V(t, x)$
, which satisfies the following conditions:
defined on $t¥in I$, ¥
(i) $a(d(x, N))¥leqq V(t, x)¥leqq b(d(x, N), ||x||)$ , where $a(r)$ is continuous,
$a(r)>0$ for
and $b(r, s)$ is continuous, increasing and
,
as
$x in R^{n}$
$r¥neq 0$
$b(r, s)¥rightarrow 0$
$r¥rightarrow 0$
(ii)
(iii)
$V(t, x)¥in C_{0}(x)$ ,
uniformly on
$V_{(5)}^{¥prime}(t, x)+V^{*}(t, x)¥rightarrow 0$
, where
for any , , as
tive definite with respect to $N$.
$¥lambda$
$ t¥rightarrow¥infty$
$¥alpha$
$¥mu$
$ 0<¥lambda¥leqq d(x, N)¥leqq¥mu$
,
$||x||¥leqq¥alpha$
is continuous and is posi-
$V^{*}(t, x)$
Then, the set $M$ is an eventually t-uniform-asymptotically stable set of (5)
uniform-asymptoticalfy
in the large. Moreover, if
is invariant, $M$ is a
is the closure of $M$.
stable set of (5) in the large, where
$¥overline{M}$
$t-$
$¥overline{M}$
Proof. First of all, we shall see that the solutions of (5) are -uniform-bounded. Since the solutions of (5) are uniform-bounded, for a given a $>0$
, we have $||x(t;x_{0}, t_{0})||¥leqq$
there exists a ¥ ¥
such that if
and
. Since
, where
for all
is a solution of (5) through
$N$ is not empty, if we choose
contains
large enough, the region
$N$
a point of . Therefore, we have
$¥mathrm{M}$
$ beta( alpha)>0$
$¥beta(¥alpha)$
$||x_{0}||¥leqq¥alpha$
$t_{0}¥in I$
$x(t;x_{0}, t_{0})$
$t¥geqq t_{0}$
$(t_{0}, x_{0})$
$||x||¥leqq¥theta$
$¥beta(¥mathrm{a})$
$d(x(t;x_{¥theta}, t_{0}),N)¥leqq 2¥beta(¥mathrm{a})$
.
This shows the-uniform-M-boundedness.
. For a given
Consider a solution
of (5) such that
such that $||x||¥leqq$
, there exists a
such that
. For
such
and
, there exist a
and a
$x(t;x_{0},t_{¥theta})$
$¥epsilon>0$
$b(¥delta, ¥beta(¥alpha))<a(¥epsilon)$
$¥delta(¥epsilon, ¥alpha)$
$¥delta(¥epsilon, ¥alpha)¥leqq d(x, N)¥leqq¥epsilon$
$¥beta(¥alpha)$
that
$||x_{¥theta}||¥leqq¥alpha$
$V^{*}(t, x)¥geqq¥gamma_{1}(¥epsilon, ¥alpha)$
$x$
and that if
$t¥geqq T_{1}(¥epsilon, ¥alpha)$
, we have
From this, it follows that if
and
, ¥
,
for all
, which shows that the set
$ N)< epsilon$
$t_{0})$
$t¥geqq t_{0}$
uniform-stable set of (5).
Consider a solution
and
such that
$x(t;x_{0}, t_{0})$
$||x||¥leqq¥beta(¥alpha)$
and a
$T_{2}(¥epsilon, ¥alpha)>0$
of (5) such that
$t¥geqq T_{2}(¥epsilon, ¥alpha)$
Thus, we can find a
. In fact, we can set
such that
$T(¥epsilon, ¥alpha)>0$
, we haved ( $x(t$ ;
is an eventually
$M$
$ 0¥leqq t_{0}<¥infty$
$¥delta(¥epsilon, ¥alpha)¥leqq d(x, N)¥leqq 2¥beta(¥alpha)$
such that if
.
$V_{(6)}^{¥prime}(t, x)¥leqq-¥frac{¥gamma_{1}(¥epsilon,¥alpha)}{2}$
$d(x_{0}, N)<¥delta(¥mathrm{e}, ¥alpha)$
$t_{0}¥geqq T_{1}(¥epsilon, ¥alpha)$
$x_{0}$
$T_{1}(¥epsilon, ¥mathrm{a})>0$
$¥gamma_{1}(¥epsilon, ¥alpha)>0$
$t-$
,
$||x_{0}||¥leqq¥alpha$
, there exist a
, we have
.
For
$x$
$¥gamma(¥epsilon, ¥alpha)>0$
$V_{(5)}^{¥prime}(t, x)¥leqq-¥frac{¥gamma(¥epsilon,¥alpha)}{2}$
$d(x(t;x_{0}, t_{0}), N)<¥mathrm{e}$
for all
.
$t>t_{0}+$
$T(¥epsilon, ¥mathrm{a})$
,
$T(¥epsilon, ¥alpha)=¥max(T_{1}(¥epsilon, ¥alpha), T_{2}(¥epsilon, ¥alpha))+_{¥alpha¥overline{)}}^{¥frac{2}{¥gamma(¥epsilon}},¥{b(2¥beta(¥alpha), ¥beta(¥alpha))-a^{*}(¥epsilon, ¥alpha)¥}$
where
$a^{*}(¥epsilon, ¥alpha)=¥delta(¥epsilon,a)¥leqq r¥leqq 2¥beta(¥alpha¥backslash ¥min_{¥prime}a(r)$
.
This means that
$M$
is a
$¥mathrm{q}¥mathrm{u}¥mathrm{a}¥mathrm{s}¥mathrm{i}-2-¥mathrm{u}¥mathrm{n}¥mathrm{i}¥mathrm{f}¥mathrm{o}¥mathrm{r}¥mathrm{m}-$
86
T. YOSHIZAWA
asymptotically stable set of (5) in the large.
is invariant, by Lemma 8 in [4], we can see that $M$ is a t-uniformIf
stable set of (5) and hence, the set $M$ is a t-uniform-asymptotically stable set
of (5) in the large.
Now we consider a system in the product space
$¥overline{M}$
(6)
$¥left¥{¥begin{array}{l}x^{¥prime}=F(t,x,y)¥¥y,=G(t,x,y),¥end{array}¥right.$
where is an -vector,
on $ 0¥leqq t<¥infty$ , $||x||<¥infty$ ,
$M$ be a set such that
Theorem 9. Let ¥
$x$
$¥mathrm{n}$
$y$
is an
$¥mathrm{m}$
-vector and $F(t, x, y)$ , $G(t, x, y)$ are continuous
and let
. Let $N$ be a non-empty set in
$||y||<¥infty$
$I¥times N¥times R^{m}$
$R^{n}$
.
$ {x(t;x_{0}, y_{0}, t_{0}), y(t;x_{0}, y_{0}, t_{0})¥}$
of (6) through
be a solution
, that is, $||x(t;x_{0}, y_{0}, t_{0})||^{2}+||y(t;x_{0}, y_{0}, t_{0})||^{2}$
which is bounded by ¥
. We assume that $F(t, x, y)$ is bounded for bounded , and
for $alf$
,
that there exists a continuous Liapunov function $V(t, x, y)$ defined on
$||x||^{2}+||y||^{2}¥leqq¥alpha^{2}$ , which
satisfies the foffowing conditions:
$ alpha>0$
$(t_{0}, x_{0}, y_{0})$
$x$
$t¥geqq t_{0}$
$¥leqq¥alpha^{2}$
$y$
$ t_{0}¥leqq t<¥infty$
(i)
(ii)
$V(t, x, y)¥geqq 0$
,
$V(t, x, y)¥in C_{0}(x, y)$
,
$V_{(6)}^{¥prime}(t, x, y)=¥varlimsup^{¥underline{1}}¥{V(t+h, x+hF(t, x, y), y+hG(t, x, y))-V(t, x, y)¥}$
(iii)
$h¥rightarrow 0^{+h}$
$¥leqq-W(x, y)$ ,
where
the set
Then,
$W(x, y)$
$N¥times R^{m}$
definite with
is continuous and is positive
respect to
.
$x(t;x_{0}, y_{0}, ¥tau_{0})¥rightarrow N$
as
$ t¥rightarrow¥infty$
.
Proof. Suppose that $x(t;x_{0}, y_{0}, t_{0})$ does not approach $N$ as
as
such that
and a sequence
there exist an
$¥epsilon>0$
$¥{t_{k}¥}$
$ t_{k}¥rightarrow¥infty$
$ d(x(t_{k};x_{0}, y_{0}, t_{0}), N)>¥epsilon$
Since the solution is bounded by ,
by a positive constant , and hence, we have
.
is bounded
.
for all
$F(t, x(t;x_{0}, y_{0}, t_{0}), y(t;x_{0}, y_{0}, t_{0}))$
$¥alpha$
$L$
From this, we can see that
. Then,
and that
$ t¥rightarrow¥infty$
$ k¥rightarrow¥infty$
$||x^{¥prime}(t;x_{0}, y_{0}, t_{0})||¥leqq L$
$d(x(t;x_{0}, y_{0}, t_{0}), N)>¥frac{¥epsilon}{2}$
$t_{k}-¥frac{¥epsilon}{4L}-¥leqq t¥leqq t_{h}+¥frac{¥epsilon}{4L}$
$t¥geqq t_{0}$
on the intervals
.
We can assume that these intervals do not overlap and that
large. By (iii), on these intervals
$t_{1}$
is sufficiently
$V^{¥prime}(t, x(t;x_{0}, y_{0}, t_{0}), y(t;x_{0}, y_{0}, t_{0}))¥leqq-c$
for some
$V$
$c>0$
(
and on other intervals
$t_{k}+¥frac{¥epsilon}{4L}$
,
$x$
(
$t_{k}+¥frac{¥epsilon}{4L};x_{0}$
$¥leqq-ck¥frac{¥epsilon}{2L}¥rightarrow-¥infty$
,
$y_{0}$
,
$t_{0}$
as
$V^{¥prime}¥leqq 0$
),
.
Therefore, we have
$y(t_{k}+¥frac{¥epsilon}{4L};x_{0},$
$ k¥rightarrow¥infty$
,
$y_{0}$
,
$t_{0}))-V(t_{0}, x_{0}, y_{0})$
87
Eventual Properties and Quasi-asymptotic Stability of a Non-compact Set
.
as
. Thus, we can see that
Corollary. We assume that there exists a continuous Liapunov function
$V(t, x, y)$
which satisfies the same conditions as in Theodefined on
$rem$
for $t¥in I$, $x¥in R^{n}$ and $y¥in R^{m}$ . If $F(t, x, y)$ is bounded for bounded , ,
. Moreover, if every
every bounded solution of(6)approaches ¥
as
solution of (6) is bounded, the set $M$ isa quasi-asymptoticaffy stable set of (6)
in the large.
By the same arguments, we can prove the following theorem for a system
which contradicts
$x(t;x_{0}, y_{0}, t_{0})¥rightarrow N$
$V¥geqq 0$
$ t¥rightarrow¥infty$
$I¥times R^{n}¥times R^{m}$
$x$
$9$
$N times R^{m}$
(7)
$¥mathrm{y}$
$ t¥rightarrow¥infty$
$¥left¥{¥begin{array}{l}x^{¥prime}=F(t,x,y)+H(t,x,y)¥¥y^{¥prime}=G(t,x,y)¥end{array}¥right.$
where the assumptions on $F(t, x, y)$ and $G(t, x, y)$ are the same as in (6) and
$H(t, x, y)$ is continuous on
and for any continuous bounded function
$(x(t), y(t))$ , $H(t, x(t), y(t))$ is integrable.
Theorem 10. Let $¥{x(t;x_{0}, y_{0}, t_{0}), y(t;x_{0}, y_{0}, t_{0})¥}$ be a solution of (7) through
which is bounded by $¥alpha>0$ . Under the same assumptions as in Theo.
as
$rem$
, if $V_{(7)}^{¥prime}(t, x, y)¥leqq-W(x, y)$ , then
. Then,
Proof. Suppose that
does not approach $N$ as
for some
we have similar intervals
$I¥times R^{n}¥times R^{m}$
$(t_{0}, x_{0}, y_{0})$
$x(t;x_{0}, y_{0}, t_{0})¥rightarrow N$
$9$
$ t¥rightarrow¥infty$
$ t¥rightarrow¥infty$
$x(t;x_{0}, y_{0}, t_{0})$
$¥epsilon>0$
(8)
$t_{k}¥leqq t¥leqq t_{k}+¥frac{¥epsilon}{4L}$
to those in the proo{of Theorem 9.
large, we have
Since we can assume that
$¥int_{t_{k}}^{t_{h}+¥frac{¥mathrm{e}}{4L}}||H(s, x(s;x_{Q}, y_{0}, t_{0}), y(s;x_{0}, y_{0}, t_{0}))||ds<¥frac{¥epsilon}{4}$
Therefore, on the intervals (8), we have
$t_{1}$
is sufficiently
.
$d(x(t;x_{0}, y_{0}, t_{0}), N)>¥frac{¥epsilon}{2}$
.
By the same
argument as in the proof of Theorem 9, we have a contradiction.
can see that
Corollary 1.
rem9,
if
as
.
Under the same assumptions as those in Corollary
$x(t;x_{0}, y_{0}, t_{0})¥rightarrow N$
Thus, we
$ t¥rightarrow¥infty$
$V_{(7)}^{¥prime}(t, x, y)¥leqq-W(x, y)$
and
if $F(t, x, y)$
is bounded
of
Theo-
for bounded
$x$
, ,
$¥mathrm{y}$
. Moreover, if
as
then, every bounded solution of (7) approaches
every solution of (7) is bounded, the set $M$ is a quasi-asymptoticalfy stable set
of (7) in the large.
Corollary 2. We assume that there exists a continuous Liapunov function
$V(t, x, y)$ defined on
which satisfies the foffowing conditions:
(i) $V(t, x, y)¥geqq 0$ ,
$N¥times R^{¥mathrm{z}n}$
$ t¥rightarrow¥infty$
$I¥times R^{rt}¥times R^{m}$
(ii)
$V(t, x, y)¥in¥overline{C}_{0}(x, y)$
(iii)
$V_{(6)}^{¥prime}(t, x, y)¥leqq-W(x, y)$
,
, where
$W(x, y)$
is continuous and is positive
T. YOSHIZAWA
88
definite with
If $F(t, x, y)$
proaches
respect to
is bounded
as
$N¥times R^{m}$
$N¥times R^{m}$
for bounded
$ t¥rightarrow¥infty$
$x$
,
.
then every bounded solution
$y$
of (7) ap-
.
be a bounded solution of (7). Then, there exists
such that $||x(t)||^{2}+||y(t)||^{2}¥leqq¥alpha^{2}$ . Let $g(t)$ be such that
and $g(t)$ is integrable. In the domain $||x||^{2}+$
$¥max_{||x||^{2}+|y||^{2}¥leqq a^{2}}|||H(t, x, y)||=g(t)$ ,
Proof. Let
a positive constant
$(x(t), y(t))$
$||y||^{2}¥leqq¥alpha^{2}$
$¥alpha$
, consider a Liapunov function
such that
$U(t, x, y)$
$U(t, x, y)=e-¥int_{0}^{t}g(t)dt(V(t, x, y)+L)$
where $L$ is the Lipschitz constant of
arly, $U(t, x, y)$ is defined on
Since we have
in the domain
$V$
and
$I¥times R^{n}¥times R^{m}$
,
$||x||^{2}+||y||^{2}¥leqq¥alpha^{2}$
$U(t, x, y)>0$ ,
.
Cle-
$U(t, x, y)¥in C_{0}(x, y)$
.
$U_{(7)}^{¥prime}(t, x, y)=e-¥int_{0}^{t}g(t)dt¥{-g(t)V(t, x, y) -Lg(t)-W(x, y)+Lg(t)¥}$
$¥leqq-e-¥int^{¥infty}..g(t)dtW(x, y)$
,
, by Theorem 10.
as
we can see that
Remark. In Corollary of Theorem 9 and Corollary 1 of Theorem 10, if
, every solution of (6) and every
uniformly as
solution of (7) is bounded (cf. [3]).
Now we shall observe the following which is a little generalization of
LaSalle’s Theorem in [1], Consider a system
$x(t)¥rightarrow N$
$ t¥rightarrow¥infty$
$||x||^{2}+||y||^{2}¥rightarrow¥infty$
$ V(t, x, y)¥rightarrow¥infty$
(8)
$¥left¥{¥begin{array}{l}x^{¥prime}=F(t,x,y)+H(t,x,y)¥¥y,=G(t,x,y)¥end{array}¥right.$
where is an -vector, is an -vector and $F(t, x, y)$ , $G(t, x, y)$ , $H(t, x, y)$ are
, and $H(t, x, y)$ is integrable
and ¥
,
defined and continuous on
for any continuous bounded function $x=x(t)$ , $y=y(t)$ . The following assumptions will be made:
,
(a) $F(t, x, y)$ is bounded for bounded , and all
(b) there exists a continuous Liapunov function $V(t, x, y)$ defined on
which satisfies the following conditions:
(i) $a(||x||^{2}+||y||^{2})¥leqq V(t, x, y)¥leqq b(||x||^{2}+||y||^{2})$ , where $a(r)$ is con, and
as
tinuous, increasing, positive definite and
,
$b(r)$ is continuous, increasing and
as
(ii) $V(t, x, y)¥in C_{0}(x, y)$ ,
(iii) $V_{(8)}^{¥prime}(t, x, y)¥leqq-W(x)+h(t)q(t, x, y)$ , where $W(x)$ is continuous,
$x$
$y$
$¥mathrm{w}$
$¥mathrm{w}$
$t¥in I$
$x¥in R^{n}$
$y in R^{m}$
$x$
$t¥geqq 0$
$y$
$ I¥times$
$R^{n}¥times R^{m}$
$ a(r)¥rightarrow¥infty$
$b(r)¥rightarrow 0$
positive definite,
$¥int_{0}^{¥infty}|h(t)|dt<¥infty$
, and
$ r¥rightarrow¥infty$
$r¥rightarrow 0$
$q(t, x, y)$
is continuous
Eventual Properties and Quasi-asymptotic Stability of a Non-compact Set
and is bounded for bounded , and all
.
$x=0$
$y=0$
Then,
,
is eventually uniform-stable, and for any
exists a $T(¥alpha)>0$ such that $||x(t_{0})||^{2}+||y(t_{0})||^{2}<¥alpha^{2}$ for some
that $y(t)$ is bounded and
as
.
In addition, if we have the following condition
(c) for some $R_{0}>0$ , if $||x||^{2}+||y||^{2}¥geqq R_{0^{2}}$ , we have
$x$
$t¥geqq 0$
$y$
$¥alpha>0$
$t_{0}¥geqq T(¥alpha)$
$x(t)¥rightarrow 0$
$¥varphi(u)$
, there
implies
$ t¥rightarrow¥infty$
$|q(t, x, y)|¥leqq¥varphi(V(t, x, y))$
where
89
,
is a continuous positive function and
$¥int^{¥infty}¥frac{du}{¥varphi(u)}=¥infty$
,
then, all solutions $y(t)$ are bounded and all
.
as
By Theorem 4, it is clear that $x=0$ , $y=0$ is eventually uniform-stable, and
by Theorem 2, we can see that the solutions of (8) are eventually uniformbounded. Therefore, for any $¥alpha>0$ , there exist a $T(¥alpha)>0$ and a $¥beta(¥alpha)>0$ such
¥
¥
that if
and
, we have $||x(t)||^{2}+||y(t)||^{2}¥leqq¥beta^{2}$.
Since $q(t, x, y)$ is bounded for bounded , , there exists a positive constant
$L$
¥
¥
such that if
, we have $|q(t, x, y)|¥leqq L$ . On the domain $t¥in I$,
¥
¥
, consider a continuous function
$x(t)¥rightarrow 0$
$t_{0}¥geqq T(¥alpha)$
$ t¥rightarrow¥infty$
$||x(t_{0})||^{2}+||y(t_{0})||^{2} leqq alpha^{2}$
$x$
$y$
$||x||^{2}+||y||^{2} leqq beta^{2}$
$||x||^{2}+||y||^{2} leqq beta^{2}$
$-¥int_{0}t|h(t)|dt$
$(V(t, x, y)+L)$ .
$U(t, x, y)=e$
Then, we have
$U_{(8)}^{¥prime}(t, x, y)¥leqq e-¥int_{0}^{t}|h(t)|dt$
$¥{-|h(t)| V-L|h(t)|-W(x)+L|h(t)|¥}$
$-J_{0}r_{|h(t)^{1}dt}¥infty$
$¥leqq-e$
$W(x)$ .
Thus, by Theorem 10, we can see that
as
In case we have the condition (c), on the domain
consider a continuous function
$x(t)¥rightarrow 0$
$ t¥rightarrow¥infty$
.
$t¥in L||x||^{2}+||y||^{2}¥geqq R_{0^{2}}$
$U(t, x, y)=¥exp(-¥int_{0}^{t}|h(t)|dt+¥int_{V_{0}}^{V}¥frac{du}{¥varphi(u)})$
,
.
Then, we have
$U_{(8)}^{¥prime}(t, x, y)¥leqq¥exp(-¥int_{0}^{t}|h(t)|dt+¥int_{V_{0}}^{V}¥frac{du}{¥varphi(u)})$
$¥mathrm{x}¥{-|h(t)|+¥frac{1}{¥varphi(V)}(-W(x)+|h(t)||q(t, x, y)|)¥}¥leqq 0$,
and hence, we can see that the solutions of (8) are uniform-bounded (cf. [3]).
Thus, by the same consideration as the above, we can see that
as
$x(t)¥rightarrow 0$
$ t¥rightarrow¥infty$
.
93
T. YOSHIZAWA
References
[1] LaSalle, J. P., Recent advances in Liapunov stability theory, SIAM Review, 6
(1964), 1-11.
[2] Massera, J. L., On Liapounoff’sconditions of stability, Annals of Math., 50 (1949),
705-721.
[3] Yoshizawa, T., Liapunov’sfunction and boundedness of solutions, Funkcialaj
Ekvacioj, 2 (1959), 95-142.
[4] Yoshizawa, T., Stability of sets and perturbed system, Funkcialaj Ekvacioi, 5
(1962), 31-69.
[5] Yoshizaw a, T., Some notes on stability of sets and perturbed system, Funkcialaj
Ekvacioj, 6 (1964), 1?11.
(Ricevita la 24-an de aprilo, 1965)