![(t;x x0 ∈R x 0x(t;x ),t∈]- ∞,|x |] (t; ),t R | | x(t;x (t;0) ∈ 0 x](https://images.tandf.co.uk/common/jackets/crclarge/978041530/9780415308489.jpg "(t;x x0 ∈R x 0x(t;x ),t∈]- ∞,|x |] (t; ),t R | | x(t;x (t;0) ∈ 0 x")
ABSTRACT
In such a case the intersection in c taken
over all motions X passing through X
at t reduces to I
X
X that is
I
X
l
X
l
X
Besides Lagrange stability of X
X
guarantees boundedness of all motions
through X
X
In view of Denition we slightly generalise the denition of motion precom
pactness by LaSalle p Denition
Denition a A family F
fX X X
X
g F
F
X
of
generalised motions of the system is positively negatively precompact
relative to X if and only if every X F
is bounded for all t l
X
X
t l
X
X and has no positive negative limit point on the boundary
X of X
L
X
X X L
X
X X X F
respectively
b A set A
X
is positively negatively precompact relative to X and relative
to the system if and only if the family F
X
of generalised motions of
the system is positively negatively precompact relative to X for every X
A
respectively
c The expression relative to X is omitted if and only if X
n
d The expression relative to the system is omitted if and only if the
system is prespecied
e The expressions A family F
fX X X
X
g F
F
X
of
generalised motions and the family F
X
of generalised motions are
replaced by A generalised motion XtX
and the generalised mo
tion XtX
respectively if and only if the generalised motion XtX
is unique
relative
to X then l
X
X Analogously if XtX
is negatively precompact
relative to X then l
X
X Hence if XtX
is precompact relative
to X then I
X
X
Note Denition and Denition are equivalent as pointed out by
La Salle for the case of unique motions p
Denition Denition and the denitions of the stability domains Def
inition # Denition Section directly imply the following result that
concerns the system by noting that generalised motions are continuous in
t I
Denition and Denition
Theorem a If X is stable then its domain D
s
of stability is positively
precompact positively Lagrange stable
a If a set A is stable then its domain D
s
A of stability is positively precompact
positively Lagrange stable
b If X is attractive then its domain D
a
of attraction is positively precompact
positively Lagrange stable
b If a set A is attractive then its domain D
a
A of attraction is positively pre
compact positively Lagrange stable
c If X is asymptotically stable then D D
a
D
s
is positively precompact
positively Lagrange stable
c If a set A is asymptotically stable then DA D
a
A D
s
A is positively
precompact positively Lagrange stable
Example x of the system x x
Example Fig is completely
globally asymptotically stable
D
s
D
a
D
The whole space is positively precompact positively Lagrange stable
Example The set A
fX X
jx
j g is not positively precompact
positively Lagrange stable relative to the system of Example
Example The rst order system
dx
dt
x k sign x k
has generalised motions determined by
xtx
k sign x
x
k signx
exp t t x
or jx
j k
k sign x
x
k signx
exp t t t
t
jx
j k
