ABSTRACT
Figure a The domain D
a
A of attraction of the set A fx x
x
x
g
b The domain D
s
A of stability of the set A Example
i it has the stability domain D
s
and the attraction domain D
a
and
ii D D
s
D
a
b The state X of the system has the strict domain of asymptotic
stability the strict asymptotic stability domain denoted by D
c
if and only if
it has both the strict stability domain D
sc
and the strict attraction domain D
ac
and D
c
D
sc
D
ac
Comment If the state X of the system has both D
s
# the stability
domain and D
a
# the attraction domain then they are neighbourhoods of it Hence
D # the asymptotic stability domain is also a neighbourhood of X It can
be closed can be open which will be illustrated via examples It need not be
connected in general However D
c
is a connected neighbourhood of X due to
such property of D
sc
and D
ac
and D
c
D
ac
D
sc
Example Let the second order system analyzed in Example and
Example be further considered
dX
dt
jx
j jx
j X
It was shown that X of the system has both the stability domain D
s
D
s
X X
jx
j jx
j
D
sc
D
s
and the attraction domain D
a
D
s
D
stability domain D
D D
s
D
a
X X
jx
j jx
j
D
a
D D
c
In this example D equals D
a
and they are subsets of D
s
They are exactly the
interior
D
s
of D
s
D
a
D
D
s
in this case
Example The zero state of the second order nonlinear system
dX
dt
kXk
kXk
kXk
kXk
kXk
kXk
X
has the stability domain D
s
Example
D
s
&
B
X X
kXk
D
sc
D
s
and the attraction domain D
a
Fig
D
a
B
X X
kXk
D
ac
D
a
Hence its asymptotic stability domain D
D D
s
D
a
B
X X
kXk
D
c
D
equals D
a
and they are subsets of D
s
but they are also proper subsets of the
interior
D
s
of D
s
Example In the case of the system dened by its motions
XtX
e
t
X
tX
X
i kX
k
kX
k
kX
k
i kX
k
the zero state has both the strict stability domain D
sc
Example D
sc
B
fX kXk g D
s
n
D
sc
D
s
D
sc
D
s
and the strict attraction domain
D
ac
n
D
a
D
ac
Hence it has also the strict asymptotic stability domain D
c
D
c
D
sc
D
ac
B
n
B
fX kXk g which is compact hyperball B
and equal to D
sc
They are in this case subsets of D
ac
D
c
D
ac
Notice that D
ac
is open and D
c
is closed in this example Besides D
s
D
a
D
n
D
c
D D
c
D
Hence the asymptotic stability domain D equals the whole state space
n
but
the strict asymptotic stability domain D
c
does not The state X is globally
stable
stability domain denoted by DA if and only if both
i it has the stability domain D
s
A and the attraction domain D
a
A
and
ii DA D
s
A D
a
A
b A set A
n
of states of the system has the strict domain of asymp
totic stability the strict asymptotic stability domain denoted by D
c
A if and
only if it has both the strict stability domain D
sc
A and the strict attraction
domain D
ac
A and D
c
A D
sc
A D
ac
A
Example The set A
A fX X
jx
j jx
j g
of states of the system
dX
dt
jx
j jx
j jx
j jx
j X
has both the stability domain D
s
A Example
D
s
A fX X
jx
j jx
j g D
sc
A D
s
A
and the attraction domain D
a
A Example
D
a
A fX X
jx
j jx
j g
D
s
A D
ac
A D
a
A
Hence its asymptotic stability domainDA equals D
a
A that is the interior
D
s
A
of D
s
A Notice that X is unstable thus it does not have the stability domain
and the asymptotic stability domain
Comment If X is the unique equilibrium state of the system and
is stable then as soon as X
D
s
the motion XtX
is either not dened on
the whole time axis
or it is not bounded However such a statement cannot
be applied either to D
a
and D if X is also attractive as soon as D
a
D
s
or
to D
sc
if D
sc
D
s
Denitions of exponential stability domains
In order to explain the relative sense of the denitions of exponential stability
domains we shall rst consider the system of Example once more
Example The motions of the system
dX
jx
j jx
j X
kXtX
k kX
k exp t t
only for
X
S
fX X
jx
j jx
j g
and for
with
We can accept only Once has been accepted then is completely
determined by The bigger the larger S
and the smaller In other
words the larger set S
over which the motions obey the exponential estimate
kXtX
k kX
k exp t t
the smaller rate of the exponential convergence of the motions to the origin If
we wish to nd in the limiting case ie
S
S
fX X
jx
j jx
j g
then we conclude that does not exist The same holds for the interior
S
fX X
jx
j jx
j g of S
The consequence of this is that
there does not exist the maximum set S
or
S
for which we can nd obeying
the exponential estimate because and max does not exist although
sup # exists Therefore we cannot speak of the largest or the maxi
mum set of system states over which the exponential estimate holds for some
and Instead we can look for the largest set of system states
over which the exponential estimate holds with respect to given and
given
Denition a The state X of the system has the domain D
e
of exponential stability with respect to if and only if both
i D
e
is a neighbourhood of X
and
ii the exponential estimate
kXtX
k kX
k exp t t
holds provided only that X
D
e
where and
b The state X of the system has the strict domain D
ec
of expo
nential stability the strict exponential stability domain with respect to
if and only if D
ec
is the largest connected neighbourhood of X which
dX
dt
jx
j jx
j X
has D
e
for any and it equals S
D
e
S
fX X
jx
j jx
j g
D
ec
D
e
However for the set S
S
fX X
jx
j jx
j g
we cannot nd and for which
S
satises Denition
despite
S
being the asymptotic stability domain of X
S
D This is clear
because
S
S
which means that we should nd min This minimum does
not exist and inf The asymptotic stability domain D
S
is not the exponential stability domain with respect to any
In fact
D
e
D
and
D
e
S
fX X
jx
j jx
j g
Denition a A set A
n
of states of the system has the domain
D
e
A of exponential stability with respect to if and only if both
i D
e
A is a neighbourhood of the set A
and
ii the exponential estimate
XtX
A X
A exp t for all t
holds provided only that X
D
e
A where and
b A set A
n
of states of the system has the strict domain D
ec
A
of exponential stability the strict exponential stability domain with respect to
if and only if D
ec
A is the largest connected neighbourhood of A
which is subset of D
e
A D
ec
A D
e
A
N
Let the system be of the Lurie form Section
dX
dt
AX Bfw a
w CX Dfw b
In order to explain the need for the study of domains of asymptotic stability
on N
i
we present the following simple example
Example Let n and
dX
dt
sin X
This system has innitely many equilibrium points located at X k where k
is any integer Hence X obviously is not asymptotically stable in the large
However it is asymptotically stable with the domain of asymptotic stability D
Over S D the nonlinearity f fX sin X belongs to the family
N
LMS for L andM If the system is embedded into the class
of Lurie systems then we can speak only about asymptotic stability of X
for a particular f or for any f N
LMS or for any f N
LS This
means that we can look only for the asymptotic stability domain for a particular f
eg fX sin X or for every f N
LMS or for every f N
LS
Let D
f
denote the asymptotic stability domain of X of the Lurie
system for a particular nonlinearity f
Denition The state X of the system has the strict asymptotic
stability domain on N
i
LMS which is denoted by D
i
LMS D
ic
LMS
if and only if
a it has the strict asymptotic stability domain D
f
D
f
c
for every f
N
i
L M S
and
b D
i
LMS D
f
f N
i
LMS is a neighbourhood of X
D
ic
LMS D
f
c
f N
i
LMS is a connected neighbourhood
of X respectively
This denition was introduced in It can be extended to sets as follows
Denition A set A
n
of the states of the system has the strict
asymptotic stability domain on N
i
LMS which is denoted by D
i
LMAS
D
ic
LMAS if and only if both
a it has the strict asymptotic stability domain D
f
D
f
c
for every f
N
i
L M S
b D
i
LMAS D
f
f N
i
LMAS is a neighbourhood of the
set A D
ic
D
f
c
f N
i
LMAS is a connected neighbourhood of
the set A respectively
Domains of practical stability properties
Denitions of domains of practical stability
By following and Section we accept the following denition for the
system
dX
dt
fX i f
n
l
n
Denition The system has the domain of practical stability with respect
to fX
A
Ig which is denoted by D
ps
X
A
I if and only if both
a its motions obey
XtX
i X
A
for every t i
I
provided only that X
D
ps
X
A
I
and
b the interior
D
ps
X
A
I of D
ps
X
A
I is nonempty
When X
A
and I are prespecied then we may replace D
ps
X
A
I by D
ps
Denition A set A of states of the system has the domain of practical
stability with respect to fX
A
Ig which is denoted by D
ps
X
A
IA if and only
if both
a the system motions obey
XtX
i X
A
for every t i
I
provided only that X
D
ps
X
A
IA
and
b D
ps
X
A
IA is a neighbourhood of the set A
When X
A
and I are known then we maywrite D
ps
A instead of D
ps
X
A
IA
Comment If we are interested in the practical stability domain of a state X
then we can apply Denition by settling A fX
g
with settling time
As for practical stability domains we rst introduce the notion of the domain of
practical contraction with settling time for the system see Section
Denition The system has the domain of practical contraction with the
settling time
s
with respect to fX
F
Ig which is denoted by D
pc
s
X
F
I if
and only if both
a its motions obey
XtX
i X
F
for every t i T
s
I
provided only that X
D
pc
s
X
F
I
and
b the interior
D
pc
s
X
F
I of D
pc
s
X
F
I is nonempty
When
s
X
F
and I are prespecied then we may write D
pc
instead
of D
pc
s
X
F
I
Denition A set A of states of the system has the domain of practical
contraction with the settling time
s
with respect to fX
F
Ig which is denoted
by D
pc
s
X
F
IA if and only if both
a the system motions obey
XtX
i X
F
for every t i T
s
I
provided only that X
D
pc
s
X
F
IA
and
b D
pc
s
X
F
IA is a neighbourhood of the set A
When
s
X
F
and I are known then we may replace D
pc
s
X
F
IA
by D
pc
A
Comment If we are interested in the domain of practical contraction of a
state X
then we may use Denition with A fX
g
Denitions of domains of practical stability
with settling time
In view of the preceding denition and the notion of practical stability with settling
sta
bility with the settling time
s
with respect to fX
A
X
F
Ig which is denoted by
D
p
s
X
A
X
F
I if and only if
a both
XtX
i X
A
for every t i
I
and
XtX
i X
F
for every t i T
s
I
hold provided only X
D
p
s
X
A
X
F
I
and
b the interior
D
p
s
X
A
X
F
I of D
p
s
X
A
X
F
I is nonempty
When
s
X
A
X
F
and I are given then we may write D
p
instead
of D
p
s
X
A
X
F
I
For the set A we deduce from Denition and Denition the
following
Denition A set A of states of the system has the domain of practical
contractive stability with the settling time
s
with respect to fX
A
X
F
Ig which
is denoted by D
p
s
X
A
X
F
IA if and only if
a both
XtX
i X
A
for every t i
I
and
XtX
i X
F
for every t i
s
I
hold provided only that X
D
p
s
X
A
X
F
IA
and
b D
p
s
X
A
X
F
IA is a neighbourhood of the set A
When
s
X
A
X
F
and I are known and xed then we may write D
p
A in the
sense of D
p
s
X
A
X
F
IA
Chapter
Qualitative features of
stability domains properties
Introductory comments
Denition of a motion
If a physical technical system is described by a rst order vector dierential
equation in a
dq
dt
rq W r
n
l
n
q
n
a
and by an algebraic vector equation in b
y gq W g
n
l
m
y
m
b
then the former describes its internal dynamics that via b determines its output
behaviour under the inuence of an input vector function W
Let y
d
m
denote a specic output response of the system which
is of a particular interest which is aimed to and therefore called a desired output
vector function of the system
Denition The system is in the nominal desired regime with respect
to y
d
if and only if yt y
d
t A pair q
W
is nominal desired with respect
to y
d
if and only if the system is in the nominal regime with respect to the
same y
d
Theorem In order for a pair q
W
to be nominal denoted by q
N
W
N
with
both and
dq
t
dt
rq
t W
t a
gq
t W
t y
d
t b
rrr
Proof The statement of Theorem follows directly from and Denition
A nominal motion q
N
with respect to a desired output y
d
of the system is an
unperturbed motion in Lyapunovs terminology Section
Assumption A nominal pair q
N
W
N
with respect to y
d
is known and it is
time invariant
From now on it is accepted that the Assumption holds This means that q
N
W
N
is elementwise constant solution to
Let the following change of variables ie translation of coordinate systems be
dened
X q q
N
i W W
N
fX i rq
N
X W
N
i rq
N
W
N
It is easy to show that after subtracting a from a and using we
derive
dX
dt
fX i f X I
n
Notice that the function f obeys
f
as soon as it is dened as above by
In case it then we simplify the notation as follows
fX fX
so that reduces to see Section
dX
dt
fX f X
n
In the sequel we shall study either properties of systems described by or
by Once we determine a solution X to or to for a given i ie
for i respectively then we easily determine q q
N
X W W
N
i and
y gq W This justies the study of or since Assumption holds
For its validity we have to solve the equations for a given y
d
Denition A function X I
X
I X is a solution a motion of the
X
i I
ii X is continuous and dierentiable in t I
for all i I
iii X identically satises the equation on I
I that is
d
dt
XtX
i f XtX
i it
and
iv X fulls the initial condition
X X
i X
I fi i g for the system Hence we shall use XtX
in the sense
of XtX
XtX
XtX
In general X
is a subset of
n
In special cases X
n
The time interval I
I
depends on X
X
I
I
X
l
l
It is the maximal time interval over which X is dened with respect to X
Since X is the domain of denition of f or of f with respect to X then
the condition iii of Denition can be satised only when XtX
i X
With XI
we denote the set of all vectors XtX
over t I
XI
fX
t I
XtX
Xg and XT X
designates the set of all vectors XtX
over t T for T I
T XT X
fX t T XtX
Xg The set
XI
is called a trajectory of the system through X
and XT X
is its arc
over T
Existence of motions
There are various theorems on existence of motions of the system Their
common feature is that they provide su!cient conditions rather than necessary and
su!cient conditions for existence of motions Here will be presented the classical
results by referring to
Theorem In order for the system to have a motion through X
X at
t it is sucient that there exists a compact closed and bounded neighbourhood
N X
of X
N X
X such that the function f is continuous on N X
fX CN X
Then the motion is dened in the time interval where
X
N X
max kfX k X N X
rrr
Nemytskii and Stepanov Theorem established the following result by
a compact
nonempty subset X
c
of the interior N X
of the neighbourhood N X
on which
the conditions of Theorem are fullled then the motion X may be continued for
is dened on the whole interval
rrr
This theorem is very important for discovering a link between stability of X
and the existence of solutions which is stated as follows
Theorem a If X is stable then XtX
exists on
for every X
D
s
b If X is attractive and XtX
is continuous in t X
kX
k
for every X
D
a
then XtX
exists on
for every X
D
a
c If X is asymptotically stable then XtX
exists on
for every X
D
d If X is exponentially stable then XtX
exists on
for every
X
D
e
rrr
Proof Theorem and
a the denition of D
s
Denition Section imply directly the statement
under a
b the denition of D
a
Denition Section yield directly the statement
under b by noting that kX
k in X
in the same denition
c the statement c follows from a and b due to the denition ofD Denition
Section
d The statement under d is a direct corollary to Theorem in view of the
denition of D
e
Denition Section
Several other criteria for the existence of motion follow The next one is due to
Zubov p
Theorem If the function f is dened and continuous on
n
fX C
n
with bounded norm on
n
sup kfXk X
n
then motions of the system are dened on for every X
n
rrr
In case the function f is continuous on
n
but does not obey the condition
then Zubov proposed p a time scale transformation by introducing a
new time variable
d dt kfX k
so that the system takes a new form
dX
fX
Since the function
kfk
obeys all the conditions of Theorem for fX
C
n
then the motions of the system are dened on for every X
n
Zubov stated p that geometrically the integral curves of the systems
and will coincide where integral curve is synonym to motion
and to solution
Existence and uniqueness of motions
Denition a A motion X of a system is backwardtime unique through X
if and only if it and any other motion X
of the same system through X
obey
X
tX
X tX
for all in an interval over which both X
X
and XX
are dened
and for all t
for which both X
tX
and X tX
are
dened
b A motion X of a system is forwardtime unique through X
if and only if it
and any other motion X
of the same system through X
obey
X
tX
X tX
for all in an interval over which both X
X
and XX
are dened
and for all t
for which both X
tX
and X tX
are
dened
c A motion X of a system is unique through X
if and only if it is both backward
time unique through X
and forwardtime unique through X
In order to illustrate the preceding denition several examples follow
Example The rst order system
dx
dt
x
has solutions x
through x
at t determined by
x
t
t
t
t
t
t
f g
and the trivial solution
