ABSTRACT
In order to maximize the righthand side of inequality within the con
straint we have to choose then p
has to verify
#
$
p
#
$
Any domain X
p
is convenient as an initial set and then domain X
dened by the
union of the X
p
such than p
veries is a possible set for practical stability
We also set X
A
X
see Fig
b second study let
s
s X
A
X
and we want to minimize X
F
then p
F
We have now to verify and
%
B
B
B
&
e
e
e
e
'
C
C
C
A
p
exp
p
F
e exp
p
#
$
Minimization of p
F
leads to the same result as in Example that is
p
F
p
p
T
The admissible initial set is then characterized by
#
$
p
#
$
This leads to the set X
Fig obtained by union of X
p
such that p
Figure Practical stability of with the settling time s with regard to f
X
X
A
X
F
S
d
g second study
Conclusions
This chapter was devoted to the vectornorms approach It appears that such a
tool addresses the dierent aspects of stability questions the stability properties
themselves including practical stability with settling time but also the related
question of the domains estimation and what was not presented here the ques
tion of constrained control that appears as a direct application of the preceding
invariance properties
Vector norms constitute a simple case of vector Lyapunov functions and the
main presented results can be enlarged to this more general class provided the con
sidered vector Lyapunov functions verify the following connectivity
property
the considered VLF p is regular radially unbounded and such that the
related following set
X
c
fx
n
px cg
is connected for any positive c
which property is veried for vector norms because of the triangular inequality
Therefore the presented overvaluing comparison systems appear to be a general
and practical approach to complex systems
# On the one hand the actual remaining drawback of such an approach is the
loss of information that may appear if the overvaluing procedure is too strong
The solution of this point is linked to the improvement of the state#space
model conditioning However the loss of information is encountered in
disturbing
parameters to be taken into account
Lastly it is important to remark that this chapter was presented in the frame
work of nonlinear timeinvariant systems but the method of vector norms address
es more general classes of models for example nonlinear timevarying systems or
nonlinear timedelay systems with possible timevarying delays
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Mathematical Society of Japan
VI Zubov Methods of AM Liapunov and Their Applications in Russian
Leningrad Leningrad Gos University English translation Groningen
P Noordho Ltd
BLANK PAGE
Author index
Barbashin E A
Bellman R
Bhatia N P #
Bitsoris G
Borne P
Brayton R K
Chetaev N G ix
Chiang H D
Dini U #
Genesio R
Gentina JC
Gruji'c Lj T Gruyitch L T
Hahn W #
Halanay A
Hocking J G
Infante E F
Kalman R E
Kappel F
Knobloch H W
Koteliansky D M #
Krasovskii N N
Lagrange JL xix #
LaSalle J P ix
#
Laurent F
Lefschetz S ix
Lurie A I xiii #
Lyapunov A M ix xv xix xx
#
#
# #
# #
# #
Malkin I G
Martynyuk A A
Matrosov V M
Mawhin J
McClamroch H H
McShane E J
Michel A N
Nemytskii V V
Poincar'e H
Popov V M xix
Postnykov V N
Richard J P
Rouche N
-
Siljak D D
Spivak M
Stepanov V V
Szeg)o G P #
Thorp J S
Tong C H
Vanelli A
Vinograd R E
Walker J A
Wa-zewski T
Weiss L
Weissenberger S
Yoshizawa T
Young G S
Zubov V I
Subject index
X is unstable
limit point
neighbourhood xiii
neighbourhood xiii
limit point
Ounique boundedness
Ouniquely bounded # #
Ouniquely bounded neighbourhood
Ouniquely bounded neighbourhood of
a set
Ouniquely bounded set
# #
absolute stability #
absolute stability with nite attrac
tion time
absolutely stable
absolutely stable set
absolutely stable set on N
L with
nite attraction time
absolutely stable state
absolutely stable state onN
L with
nite attraction time
aggregation function
analytic function
approach viaOuniquely bounded sets
asymptotic stability xi #
asymptotic stability domain
#
asymptotic stability domain estimate
asymptotic stability domain of a set
asymptotic stability domain of a state
asymptotic stability in the whole
asymptotic stability of a set
asymptotically stable
asymptotically stable equilibrium state
asymptotically stable set
asymptotically stable set with respect
to motions
asymptotically stable state
attraction xi xii
attraction domain #
attraction domain of a set
attraction domain of the origin
attraction time xvi
attraction with nite attraction time
attractive
attractive X
attractive point
attractive set
attractive set with nite attraction time
attractive state
attractive state with nite attraction
time
backwardtime Eulerian derivative
backwardtime lower right lower left
Dini derivative of
backwardtime right left Dini deriva
tive of
backwardtime unique motion
backwardtime upper right upper left
Dini derivative of
Barbashin#Krasovskii criterion
Borne and Gentina criterion
Borne and Richard criterion
boundary of xiv
boundedness
boundedness of motions
centre w of unique boundedness
centre of unique boundedness
centre of uniquely bounded neighbour
hood
centre of uniquely bounded neighbour
hoodness of a set
closeness xix
closure of xiv
commutative group
compact set
comparison function
comparison function of the class
comparison functions
comparison scalar system
comparison system
complete global asymptotic stability
complete global asymptotic stability
of sets
completely asymptotically stable in
the large
completely globally asymptotically
stable
completely globally asymptotically
stable set
completely globally asymptotically
stable state
completely globally stable set with
state with
nite attraction time
constructive algorithm
constructive computer Lyapunov func
tion generation
continuity of motions
continuous dependence
controllable
decomposition
decoupled
decoupled overvaluing system
desired output of the system
desired output vector xvi
dierential inequalities
Dini derivative
direct method of Lyapunov
disjoint
distance xix
distance function xvi
disturbance
domain xii
domain of asymptotic stability
#
#
domain of asymptotic stability of the
asymptotically stable set
domain of asymptotic stability with
nite attraction time
domain of attraction
domain of attraction of X
domain of attraction of a set
domain of exponential stability of a
set with respect to
domain of exponential stability of a
state with respect to
domain of practical contractive sta
bility with settling time of a
set with respect to
domain of practical contractive sta
bility with settling time of a
system with respect to
domain of practical contraction of a
state
settling time of a set with re
spect to
domain of practical contraction with
settling time of a system with
respect to
domain of practical stability of a set
with respect to
domain of practical stability of a sys
tem with respect to
domain of stability
domain of stability of a set
domain of system practical stability
with settling time
domain of the corresponding stability
property xx
domains of Lyapunov stability prop
erties
domains of practical contraction with
settling time
domains of practical stability
domains of practical stability with
settling time
domains of stability properties
dynamic behaviour
dynamical system #
dynamical systems
ellipsoid closed set xii
empty set xvi
equilibrium
equilibrium point #
equilibrium regime
equilibrium state
#
estimate E
p
of D
p
estimate domain xii
estimate of D
p
estimate of a domain
estimate of practical stability domain
estimate of the strict domain of ex
ponential stability of a set
estimate of the asymptotic stability
#
estimate of the asymptotic stability
domain of a set
estimate of the asymptotic stability
domains
estimate of the domain of asymptotic
stability of X
estimate of the domain of asymp
totic stability with nite at
traction time
estimate of the domain of practical
contraction with settling
time of the system motions
with respect to
estimate of the domain of practical
stability of the system with
respect to
estimate of the domain of practical
stability with settling time
estimate of the exponential stability
domain
estimate of the exponential stability
domain of X
estimate of the exponential stability
domain of a set
estimate of the practical stability do
main
estimate of the stability domain
estimate of the stability domain of a
set
estimate of the strict asymptotic sta
bility domain #
estimate of the strict attraction do
main
estimate of the strict domain of
attraction of a set
estimate of the strict domain of sta
bility #
estimate of the strict domain of sta
bility of a set
estimate of the system practical sta
bility
bility with settling time with
respect to
Euclidean norm xvii
Eulerian derivative
existence
existence and uniqueness of motions
existence of generalised motions
existence of motion
exponential estimate
exponential stability
exponential stability domain estimate
exponential stability domain with re
spect to
exponential stability domains
exponential stability of X
exponentially stable
exponentially stable X
exponentially stable set
exponentially stable state
family of all Lurie functions xiii
nal time xvi
nite reachability time
nite time
nite time interval
forced regime
forwardtime
forwardtime Eulerian derivative
forwardtime lower right lower left
Dini derivative of
forwardtime right left Dini deriva
tive of
forwardtime unique generalised mo
tion
forwardtime unique motion
forwardtime upper right upper left
Dini derivative of
free regime
frequency matrix function
functional family #
general oneshot approach
generalised dynamical system
# #
#
# #
#
generalised solution
generating function xiv #
#
# #
generating function of a uniquely
bounded set
generating function of an Ouniquely
bounded set
generation of a function
generation of a system Lyapunov func
tion
global asymptotic stability
global positive deniteness
global stability in the whole in the
large
globally asymptotically stable
globally asymptotically stable stable
in the whole state
globally asymptotically stable set
globally attractive set attractive in
the whole in the large
globally attractive set with nite at
traction time
globally attractive state attractive in
the whole in the large
globally attractive state with nite at
traction time
globally exponentially stable
globally exponentially stable set
exponentially stable in the
whole in the large
globally exponentially stable state
exponentially stable in the
whole in the large
globally Lipschitzian
globally negative denite
globally negative denite derivative
globally positive denite function
globally positive denite function with
respect to
globally positive denite radially un
bounded function
globally positive semidenite with re
spect to a set
globally stable stable in the whole
stable in the large
globally stable set with nite attrac
tion time
globally stable state with nite attrac
tion time
greatest smallest partial limit
group property
hermitian part
Holders norm
Holders norm
homogenous function
hyperbolic
hyperstability concept xix
ideal relay function xiv
importance eigenvalue xvi
importance value
importance vector xiv
increasing with respect to the diago
nal elements
independent scalar xiv
innite time interval
initial moment xiv
integral curve xv
interior of xiv
invariance
invariance features
invariance principle
invariance properties of limit sets
invariance properties of sets
invariance with respect to system mo
tions
invariant
points
invariant set relative to the system
inverse function
irreducible
Jacobian
Jacobian matrix
Knobloch#Kappel theorem
Koteliansky conditions #
Krasovskii criterion
Kronecker delta xvi
Lstability concept xix
Lagrange stability xix
Lagrange stable point relative to
Lagrange stable set relative to
LaSalle principle
leading principal minors
left lower semicontinuous
limit cycle
limit points
limit sets
linear part of the system
Lipschitz condition
Lipschitz constant
Lipschitz continuous
Lipschitz function
Lipschitzian
local overvaluingmatrix
local overvaluing pair
local overvaluing system
Lurie form
Lurie matrix xiii
Lurie matrix equations
Lurie nonlinearities
Lurie system #
Lyapunov closeness
Lyapunov function
Lyapunov function construction
Lyapunov functional family #
Lyapunov method
Lyapunov sense
Lyapunov stability
Lyapunov stability concept xix xx
Lyapunov stability criteria
Lyapunov stability domains
Lyapunov stability theory
Lyapunovs conditions
Lyapunovs denition of stability
Lyapunovs methodology for nonlin
ear systems
Lyapunovs methodology for timeinva
riant linear systems
Lyapunovs original denition
Lyapunovs original methodology
Mmatrix
matrix Popov criterion
matrix transfer function
max norm
maximal eigenvalue xvi
maximal Lyapunov function
measure of matrix
minimal eigenvalue xvi
motion xv # #
#
multiplicative semigroup
natural form
negative denite # #
negative denite function with respect
to a set on a set
negative denite matrix
negative deniteness #
negative limit point
negative limit set xiii
negative semidenite matrix
negatively invariant
negatively invariant set
negatively Lagrange stable
negatively precompact
negatively stable point
new methodology
nominal desired pair with respect
to
nominal desired regime with respect
to
nominal motion with respect to
nonattractive
nondenite
nonLyapunov stability concept
nonradially unbounded function
nonsemidenite
nonsingular
norm of matrix
novel development of the Lyapunov
method
observable
oneshot approach
oneshot construction
open neighbourhood xiii
open set
output vector xvi
overvaluation lemma
overvaluing comparison systems
overvaluing input vector
overvaluing matrix
overvaluing pair
overvaluing system
Pmatrix
parallelepipedic closed set xii
partial dierential equation
period
periodic regime
persistency ix xix
perturbed motions
polyhedral neighbourhood
Popov criterion
Popov frequency approach
positive denite
# #
#
#
positive denite function with respect
to
positive denite matrix
positive denite on
positive denite with respect to
#
positive denite with respect to a set
positive deniteness
#
positive deniteness criterion
positive deniteness with respect
to
positive function
positive importance vector
positive invariance
positive invariant
positive limit
positive limit point
positive limit set xiii
positive lower limit
positive semidenite matrix
positive semidenite with respect
to
positive semidenite with respect to
a set
positive semidenite with respect to
a set in the whole
positive semidenite with respect to
a set on a set
positively negatively Lagrange
stable
positively negatively precompact
positively invariant #
positively invariant relative to
positively invariant set
positively Lagrange stable
positively precompact
positively stable point
positively stable set
practical contraction with settling
time
practical Lyapunov sense
practical stability
practical stability concept xix
practical stability criteria
practical stability domains
practical stability with settling time
practical stability with settling time
with regard to
practical system stability with settling
time
practically contractively stable with
settling time with respect
to
practically stable with respect to
practically stable with settling time
with respect to
precompact
precompact generalised motion
precompact relative to
precompact set relative to
precompactness
principal minors
quadratic form
qualitative features of stability do
mains properties
quasiincreasing
quasiincreasing function
RVN # regular vector norm
radial unboundedness
radially increasing
radially increasing on a neighbour
hood
radially increasing on the boundary of
a set
radially increasing on the boundary of
a set to
radially increasing positive denite
function on a set
radially increasing with respect to
radially increasing with respect to set
on its neighbourhood
radially unbounded
radially unbounded globally positive
denite
radially unbounded with respect to
rational function
recursive equations
recursive relation
region of attraction
regular
relative to the system
rhomboid closed set xii
right upper right lower limit
saturation function xiv
scalar Lyapunov function xv
second method of Lyapunov
semideniteness property
set S
e
of the equilibrium states
set of admitted system states
set of all the equilibrium points xiv
set of permitted inputs
set of the equilibrium states
settling time xvi
sign denite function
sign function xiv
sign semidenite function
simplicial cone
smoothness property
solution motion of the system
solution function
solutions
specic smoothness property
stability xii xix
stability domain # #
stability domain D
s
A of the set A
stability domain estimate
stability domain of X
stability domain of a set
stability in the Lyapunov sense
stability of X
stability of a set
stability of equilibrium points
stability of motion
stability problem
stability properties of sets
stability theorems
stability with nite attraction time
stable # #
stable closed set
stable equilibrium state
stable limit cycle
stable point
stable set
stable set with nite attraction time
stable state with nite attraction time
stable unperturbed motion
stable with respect to
state portrait
state variable xv
state vector xv
stationary point
stationary regime
stationary state
steady state
strict asymptotic stability domain
strict asymptotic stability domain of
a set
strict asymptotic stability domain of
a state
strict attraction domain
strict domain of attraction of a set
strict domain of exponential stability
of a set with respect to
strict domain of exponential stability
of a respect
strict domain of stability
strict exponential stability domain
strict Lyapunov sense
strict stability domain
strong smoothness property #
#
#
surjective
symmetric matrix
symmetric part
system
system aggregation function
system is practically contractive with
settling time with respect
to
system is practically stable with re
spect to
system Lyapunov function
systemmotions
system regimes
system solutions
systems mathematical model
systems with continuous motions
time interval xix
timeinvariant continuoustime sys
tems
timevarying continuoustime sys
tems
timevarying systems
trajectory xv
twostage approach
uniformly continuous
uniformly nonsingular monotone
unique generalised motion
unique boundedness
unique equilibrium state
unique generalised motion
unique motion #
unique solution
uniquely bounded xiv #
uniquely bounded neighbourhood #
uniquely bounded neighbourhood of a
set
uniquely bounded set
uniqueness
uniqueness of motions
unperturbed motion
unstable
unstable equilibrium in the asymp
totic stability domain of a set
unstable unperturbed motion
Van der Pol equation
Vanelli#Vidyasagar approach
Vanelli#Vidyasagar recursive algorithm
Vanelli#Vidyasagar results
Vanelli#Vidyasagar theorem
vector Lyapunov functions
vector norm
Wa-zewski conditions
weak smoothness property #
#
#
weakly invariant #
weakly invariant set relative to the
system
with respect to a set
Yakubovich#Kalman lemma
Yoshizawa criterion
Zmatrix
Zubov theorem