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Chapter 5
Elements of Stability
This chapter is entirely dedicated to the study of the stability of solutions to
certain systems of differential equations.
In the first section we introduce and
illustrate the main concepts referring to stability. The second one is concerned
with several necessary and sufficient conditions for various types of stability in
the particular case of firstorder systems of linear differential equations. In the
third section we present some sufficient conditions under which the asymptotic
stability of the null solution of a firstorder differential system is inherited by the
null solution of a certain perturbed system, provided the perturbation is small
enough. In the fourth section we prove several sufficient conditions for stability
expressed by means of some functions decreasing along the trajectories, while
in the fifth section we include several results regarding the stability of solutions
of dissipative systems.
In the sixth section we analyze the stability problem
referring to automatic control systems, while the seventh section is dedicated to
some considerations concerning instability and chaos.
As each chapter of this
book, this one also ends with an Exercises and Problems section.
5.1
Types of Stability
In its usual meaning,
stability
is that property of a particular state of a
given system of preserving the features of its evolution, as long as the
perturbations of the initial data are sufficiently small. This meaning comes
from Mechanics, where it describes that property of the equilibrium state
of a conservative system of being insensitive “
`a la longue
” to any kind of
perturbations of “small intensity”.
Mathematically speaking, this notion
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156
Elements of Stability
has many other senses, all coming from the preceding one, and describing
various kinds of continuity of a given global solution of a system as function
of the initial data, senses which are more or less different from one another.
The rigorous study of stability has its origins in the works of Celestial
Mechanics of both Poincar´
e and Maxwell, and has culminated in 1892 with
the doctoral thesis of the founder of this modern branch of differential
equations, Lyapunov.
As we have already shown in Theorem 2.5.2, under certain regularity
conditions on the function
f
, the map
η
7→
x
(
·
, a, η
) — the unique saturated
solution of the Cauchy problem
‰
x
0
=
f
(
t, x
)
x
(
a
) =
η
CP
(
D
)
— is locally Lipschitz from Ω to
C
([
a, b
];
R
n
), for each
b
∈
(
a, b
ξ
), where
[
a, b
ξ
) is the domain of definition of the saturated solution
x
(
·
, a, ξ
). A much
more delicate problem, and of great practical interest, is that of finding
sufficient conditions on the function
f
such that, on one hand,
x
(
·
, a, ξ
)
be defined on [
a,
+
∞
) and, on the other hand, the map
η
7→
x
(
·
, a, η
) be
continuous from a neighborhood of
ξ
to the space of continuous functions
from [
a,
+
∞
) to
R
n
, endowed with the uniform convergence topology.
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 Spring '08
 WOCK
 Linear Algebra, Algebra, Equations, lim, Stability theory, World Scientiﬁc Book

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