ABSTRACT
Figure Motions of the system x sat x Example
Xt sX
exp s t X
s ln jX
j t
XtXsX
exp t XsX
exp t exp s X
exp t s X
for t ln jX
j
but
X tXsX
ln jXsX
j t signX sX
ln jX
j t s signX
for t ln jX
j
Hence
X tXsX
Xt sX
t ln jX
j
This illustrates that the motions do not possess Property Hence the system is
not a dynamical system Denition
Comment The function f of the system
X satX fX satX is
dierentiable in X f g However the motions are dierentiable in
time t for all t and every X
This means they are dierentiable also
at time t
when jXt
X
j which further means satXt
X
signX
X
However at that moment f is not dierentiable with respect to
X Xt
X
This illustrates that dierentiability of f on
n
is not necessary
n
in
sys
tems
Denition of generalised dynamical systems
Theorems # suggest generalisation of Denition
Denition A system is a generalised dynamical system if and only if its gen
eralised motions X have the properties P
Denition P
and P
Denition
where
P
Continuous Dependence Every generalised motion X is continuous in t
for every X
n
and in X
n
for every t that is
XtX
C
t
n
C
n
Properties of generalised dynamical systems
Note Theorems # hold for generalised dynamical systems provided
I
X
n
Note The system considered in Example Section is not generalised
dynamical system because X is not continuous in every X
n
for every t
XtX
C
n
Stability properties and invariance properties
of sets
At rst we generalise Theorem by Bhatia and Szeg)o p to systems
that need not be dynamical systems in the sense of the classical mathematical
Denition
In what follows we consider the system that has generalised motions
XtX
Theorem If a closed set A A
n
is stable then it is positively invariant
rrr
Proof a We essentially repeat the proof of Theorem by Bhatia and Szeg)o
p Let a closed set A A
n
be stable Since it is closed then
N A
A
Since it is stable then X
A implies that every generalised motionXtX
of the
system obeys
M
it follows
that every generalised motion through X
A satises
XtX
N A
A for all t
which proves the theorem
b The theorem can be proved also in the following way Let a closed set A
A
n
be stable Since it is closed then
XA X A
Since it is stable then X
A guarantees that every generalised motion XtX
of the system fulls
XtX
A for all t
and every
because X
A
M
Let
which with and yield
XtX
A for all t
Theorem by Bhatia and Szeg)o p is broadened to hold also for non
dynamical systems
Theorem Let A A
n
be a compact set If A A
i
i I where I is
a nite index set and subsets A
i
of A A
i
A are connected and pairwise disjoint
then for the set A to be stable it is necessary and sucient that every A
i
is stable
rrr
Proof Let A A
n
be compact set and A A
i
i I with all A
i
s
connected pairwise disjoint and with nite index set I Then every A
i
is compact
and
N A N A
i
i I
Necessity Let A be stable Let j I be arbitrarily chosen Let
be such
that
N
A
j
N
A
k
k j I
Such
exists because allA
i
are compact and I is nite Let
be arbitrarily
chosen Then X
N
M
A A implies Denition Section
XtX
N A t
Let X
N
M
A A
j
be arbitrarily chosen Since
M
A
then
# imply
j
M
there is A
j
obeying
X
A
j
A
j
XtX
A
j
t
This means stability of A
j
because XtX
is also arbitrary generalised motion of
the system
Suciency Let A
i
be stable for any i I Let
be arbitrarily chosen and
A min
M
A
i
i I
Since
M
A
i
for every i I and I is nite then A
Let
X
N A A Since A
i
sets are pairwise disjoint and A A
i
i I
then there is j I such that X
N A A
j
The denition of A yields
N A A
j
N
M
A
j
A
j
Hence X
N
M
A
j
A
j
which guaran
tees XtX
N A
j
t
due to stability of A
j
This result and
imply further XtX
N A t
which means stability of A because
XtX
is arbitrary generalised motion of the system
Invariance features of stability domains
properties
Invariance properties of stability domains of sets hold also for stability domains of
equilibrium states which is obvious if we dene a set A fX
e
g With this in mind
it su!ces to consider only invariance properties of stability domains of sets
Theorem Let the system be a generalised dynamical system Let a
set A A
n
be stable Then the sets D
s
A and D
sc
A for any
the domain of stability D
s
A and the strict domain of stability D
sc
A of the
set A are all positively invariant rrr
Proof Let the system be a generalised dynamical system Then it has
unique motions obeying the group property P
Denition and Denition
Let
be arbitrary Let X
D
s
A that implies XtX
N A
for all t
Let and
be arbitrary and y XX
Evidently
yields X X
N A Since the Property P
X
X
X XX
X y then X y N A for all
This
yields y D
s
A ie XX
D
s
A for all
which proves the
positive invariance of D
s
A Let now
(
X
D
sc
A so that Xt
(
X
N A
for all t
From XtX
C
t
D
s
A which holds for a generalised
dynamical system with stable set A it follows that (y X
(
X
for any
is connected with
(
X
Using this and X
(
X
X (y N A for
all
we conclude that (y X
(
X
D
sc
A for any
which
proves positive invariance of D
sc
A Let
*
X
D
s
A be arbitrary There is
Denition Section *
such that Xt
*
X
N *A for all t
Hence
*
X
D
s
*A implies Xt
*
X
D
s
*A D
s
A for all t
and
X
be
Hence
X
D
sc
A yields XtX
D
sc
A D
sc
A for all t
and proves
positive invariance of D
sc
A
The next results broaden the validity of the classical theorems on properties of
the domain of attraction from the framework of dynamical systems to generalised
dynamical systems
Theorem Let the system be a generalised dynamical system Let a
set A A
n
be attractive Then its domain of attraction D
a
A is open and
connected D
a
A D
ac
A Besides D
a
A its closure D
a
A and its boundary
D
a
A if it is nonempty are all invariant rrr
Proof Let the system be a generalised dynamical system Let a set A
A
n
be attractive Then it has the domain of attraction D
a
A Denition
Section and Denition Section Let X
D
a
A be arbitrarily cho
sen and kX
k be also arbitrarily chosen There is X
Deni
tion Section such that XtX
A for all t Let
be such that N A D
a
A Such exists because both D
a
A and N A are
neighbourhoods of A Let m
for any m f g Since the system
is a generalised dynamical system then Denition Property P
there is
for X
such that X
N X
guarantees XX
N y for y XX
Now N A D
a
A m
m f g
and y XX
N A imply N A N A and N A D
a
A
Hence XX
N A and XX
D
a
A Let z XX
z D
a
A
so that for any
there is z
obeying XtX
A for all
t z Since Xt z X tXX
Xt X
Denition
and Denition Property P
then X
N X
ensures for any
the
existence of X
such that XtX
A for all t X
One possible choice for X
is X
z
Hence any
X
N X
is in D
a
A D
a
A is open Its connectedness is due to a of Def
inition Section and the properties P
and P
of generalised dynamical
systems Denition
The system has unique generalised motions Denition and
Property
and Theorem which possess Property Xt X
X tXX
t X
n
If D
a
A were not invariant then there
would exist w D
a
A such that Xw D
a
A However X w
X Xw for every Let v Xw From w D
a
A it follows
Xtw N A for all t w so that X w X v N A
for all w and for any
Therefore v Xw D
a
A
that disproves Xw D
a
A and proves invariance of D
a
A
Since D
a
A is open and invariant then D
a
A and D
a
A if it is nonempty
are invariant Theorem
Comment Since D
a
A of D
a
A is invariant when the system is a
generalised dynamical system then obviously
X
Theorem Let a set A A
n
be asymptotically stable for the system
with generalised motions XtX
C
t
X
C
X
where D
s
A
D
a
A X
Then
D
a
A D
ac
A D
sc
A D
s
A
and
DA D
c
A D
a
A D
ac
A rrr
Proof Let a set A A
n
be asymptotically stable for the system whose
generalised motions XtX
C
t
X
C
X
and X
D
s
A
D
a
A Then XtX
is unique X
X
Let X
D
a
A be arbitrarily
chosen which is possible because the set A has D
a
A D
s
A and DA due to its
asymptotic stability Hence lim fX tX
A t g This XtX
C
t
X
and D
a
A X
imply max fXtX
A t
g
Let
Hence XtX
N A t
which implies X
D
s
A
andX
D
s
Denition Section Hence D
a
A D
s
A so that DA
D
a
A due to DA D
s
A D
a
A FromXtX
C
t
X
and D
a
A
X
it follows that for any X
D
a
A and any t
the point y XtX
is
connected with X
This and existence of
such that X X
D
ac
A due
to lim fX tX
A t g imply that X
is connected with X X
Hence X
D
ac
A and D
a
A D
ac
A By denition D
ac
A D
a
A
Therefore D
a
A D
ac
A Let X
D
ac
A D
a
A and
be
dened as above so that for all t
XtX
N A which together with
lim fXtX
A t g and XtX
C
t
D
a
A imply existence
of
such that X X
D
sc
A connectedness of XtX
at any t
with X X
and XtX
D
sc
A D
sc
A for all t
Hence
X
D
a
A D
ac
A implies X
D
sc
A and D
a
A D
ac
A D
sc
By the
denitions D
sc
A D
s
A D
c
A D
sc
A D
ac
A and DA D
s
A D
a
A
Therefore DA D
a
A D
ac
A D
c
A D
sc
A D
s
A
Comment This theorem explains why many authors use the term region of
attraction for the domain of asymptotic stability It also justies the use of the
asymptotic stability domain for the strict asymptotic stability domain
Comment Various other qualitative features of domains of stability proper
ties were discovered and described by Bhatia and Szego Genesio et al and
Chiang and Thorp
We shall particularly consider stability of equilibrium states on the boundaries of
domains of stability properties
Features of equilibrium states on boundaries
of domains of stability properties
At rst we establish a theorem discovering that the boundary of attraction domain
a
A of its
domain of attraction contains at least one equilibrium state of a given system then
every such an equilibrium state is unstable and nonattractive rrr
Proof Let a set A
n
be attractive and letX
e
D
a
A be an equilibrium state
X
e
is unstable If X
e
were stable then for every
there would be
X
e
such that X
X
e
guarantees XtX
X
e
for all
t
Let X
e
A d and
d
Let X
N
X
e
D
a
A
X
e
Hence XtX
X
e
d
for all t
or
XtX
A
d
for all t
that contradicts X
D
a
A There
fore X
e
cannot be stable
X
e
cannot be attractive If X
e
were attractive then there would exist % such
that X
X
e
% implies lim fX tX
X
e
t g Let X
N % X
e
D
a
A Then lim fXtX
A t g that contradicts
X
D
a
A Hence X
e
D
a
A cannot be attractive
This theorem yields the following
Theorem If a set A
n
is asymptotically stable and its domains of stability
D
s
A of attraction D
a
A and of asymptotic stability DA are interrelated by
DA D
a
A D
s
A
then every equilibrium state X
e
of a given system which is in the boundary DA
X
e
DA is unstable and nonattractive rrr
Proof Since DA D
a
A and they exist as requested then the statement of the
theorem results from Theorem
BLANK PAGE
Chapter
Foundations of the Lyapunov
method
Introductory comment
Denitions of stability properties and of their domains are stated via generalised
motions of systems A test of a stability feature via its denition demands knowledge
of system generalised motions for innitely many dierent initial pointsX
from a
neighbourhood of an equilibrium state or a set Determination of the generalised
motions further requires solving the systems mathematical model which is rarely
possible analytically in the closed form for nonlinear systems described by nonlinear
rstorder vector dierential equation
dX
dt
fX
Papers by Poincar'e inspired Lyapunov to pose the problem of test
ing stability properties by using the equation directly rather than by solving
it If fX AX the system is linear and stability properties are tested via real
parts of the eigenvalues of the matrix A and eventually also via their multiplicities
However when f is nonlinear then it was necessary to discover another essentially
dierent method It was discovered and established in by Lyapunov and
has been wellknown as the direct method of Lyapunov or the second method of Lya
punov or simply as the Lyapunov method It is based on the concept of sign denite
functions v
n
and their behaviour vXtX
along generalised motions
XtX
which is to be tested without solving the equation for XtX
In
order to achieve this goal Lyapunov used the total time derivative of a function v
assumed to be positive denite that will be dened in the next section along an
arbitrary generalised motion of the system Lyapunov supposed that the function
v is dierentiable on a hyperball B
vX C
B
Then its total time
derivative dvX dt along system motion at a point X XtX
is its Eulerian
derivative that can be expressed via fX
vX
d
vX
d
vXtX
grad vX
T
dX
grad vX
T
fX
vX at
every X B
This immediately poses questions of the sense origin and construction of a func
tion v for a given system These questions will be mathematically claried in
the next chapter Their excellent physical clarication in the framework of conser
vative systems can be found in the book by Rouche and Mawhin pp #
In what follows the classic Lyapunov method and the related results
will be briey synthesised For extended studies see # # #
# # # # #
# # # # #
# # # # #
# # #
Sign denite functions
Sign semi denite functions
Denition A function v
n
is
a positive semidenite with respect to a set A A
n
if and only if there is
a neighbourhood N A of the set A such that
the function v is continuous on N A
vX CN A
the function v is nonpositive on the interior
A of the set A
vX X
A
the function v vanishes on the boundary A of the set A
vX X A
the function v is nonnegative on N A
vX X N A
A
and
there exists y N A at which the function v has a positive real value
y N A vy
b positive semidenite with respect to a set A A
n
on a set S if and only
S
or in the
large or globally positive semidenite with respect to a set A if and only if
the conditions under a are satised for N A
n
d The expression with respect to a set A A
n
is omitted if and only if
A O f g
Example Let a function v be dened by
vX a
T
X It is continuous on
n
and vanishes at the origin It is not
positive semidenite because there is not a neighbourhood ofX on which
it is nonnegative
vx sign jxj It is nonnegative on and vanishes at x However it is
not positive semidenite because it is not continuous at x
vx cos x It is nonnegative and continuous on
h
i
However it does
not vanish at x cos Thus it is not positive semidenite
vX
n
X
i
i
x
i
is positive semidenite in the whole
vx sin jxj is positive semidenite on
vx
jxj
jxj jxj
is positive semidenite with respect to A
fx x jxj g in the whole
vx
jxj
jxj jxj jxj
is positive semidenite with respect
to A B
on the set S fx x jxj g but not in the large
vx is not positive semidenite because there are not a neighbourhood
N of x and y N such that vy
Denition A function v
n
is globally negative semidenite with
respect to a set A A
n
fon a set Sg if and only if the function v is globally
positive semidenite with respect to the set A fon the set Sg respectively
Denition a A matrix H
nn
is positive semidenite if and only if its
quadratic form X
T
HX is positive semidenite
b A matrix H
nn
is negative semidenite if and only if its quadratic form
X
T
HX is negative semidenite
for
semidenite properties Vol p Let H h
ij
nn
Its principal
minors H
j
j
j
r
i
i
i
r
are dened by
H
j
j
j
r
i
i
i
r
h
i
j
h
i
j
h
i
j
r
h
i
j
h
i
j
h
i
j
r
+ + +
h
i
r
j
h
i
r
j
h
i
r
j
r
r n
i
k
f ng
j
k
f ng
k r
Theorem In order for a matrix H H
T
nn
to be positive semidenite
it is necessary and sucient that
all its principal minors H
j
j
j
r
i
i
i
r
are nonnegative
H
j
j
j
r
i
i
i
r
r n
i
k
f ng
j
k
f ng
k r
and
at least one of its principal minors is positive
r f n g i
k
f ng j
k
f ng k r
H
j
j
j
r
i
i
i
r
rrr
Note The necessary and su!cient conditions for negative semidenitness of
a matrix H
nn
reduce to the necessary and su!cient conditions for positive
semideniteness of the matrix H Denition b
Sign denite functions
Denitions of sign denite functions
Denition A function v
n
is
a positive denite with respect to a set A A
n
if and only if there is a
neighbourhood N A of the set A such that
the function v is continuous on N A
vX CN A
vX X
A
the function v vanishes on the boundary A of the set A
vX X A
the function v has positive values on N A out of the closure A of the
set A
vX X N A A
b positive denite with respect to a set A A
n
on a set S if and only if the
conditions under a are satised for N A S
c positive denite with respect to a set A A
n
in the whole or in the
large or globally positive denite with respect to a set A if and only if the
conditions under a are satised for N A
n
d The expression with respect to a set A A
n
is omitted if and only if
A O f g
Note A necessary condition for global positive deniteness of a function v
with respect to a set A on a set S is its global positive semideniteness with
respect to the set A on the set S respectively
Example Let a function v be dened by
vX
n
X
i
i
x
i
It is not positive denite because it vanishes on the
hyperplane
x
x
n
x
n
which means that there is not
a neighbourhood N of X such that vX for all X N
vx sin jxj is positive denite on It is not positive denite
on because vx for jxj
vx
jxj
jxj jxj
is globally positive denite with respect to the
set A fx x jxj g However it is not positive denite because
vx on A that means there is not a neighbourhood N of x on which
vx out of the origin
vx
jxj
jxj jxj jxj
is positive denite with respect
to the set A fx x
jxj g on the set S fx x
jxj g It is not positive denite on S because vx
vX
i
i
jx
i
j is positive denite i k n and
i
for every
i k n Under these conditions it is globally positive denite
too
Denition A function v
n
is globally negative denite with respect
to a set A A
n
fon a set Sg if and only if the function v is globally positive
denite with respect to the set A fon the set Sg respectively
Denition a A matrix H
nn
is positive denite if and only if its
quadratic form X
T
HX is positive denite
b A matrix H
nn
is negative denite if and only if its quadratic form is
negative denite
The criterion for positive deniteness of a square symmetric matrix has the following
simple form Vol p
Theorem Positive deniteness criterion In order for a matrix H
h
ij
H
T
nn
to be positive denite it is necessary and sucient that all
its leading principal minors H
k
k
are positive
H
k
k
h
h
h
k
h
h
h
k
h
k
h
k
h
kk
k n
rrr
Note The necessary and su!cient conditions for negative deniteness of a
matrix H H
T
reduce to the necessary and su!cient conditions for positive de
niteness of the matrix H Denition More precisely
Theorem In order for a matrix H h
ij
H
T
nn
to be negative
denite it is necessary and sucient that
k
h
h
h
k
h
h
h
k
h
k
h
k
h
kk
k n
rrr
Note If a matrix H
nn
is not symmetric H H
T
then it is positive
negative denite i its symmetric part H
s
H H
T
is positive negative
denite respectively This follows from the fact that
X
T
HX X
T
H
H
T
H
T
X X
T
H
s
X X
T
H
as
X X
T
H
s
X
X
T
H
as
X
X
T
H H
T
!
