Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.243j
(Fall
2003):
DYNAMICS
OF
NONLINEAR
SYSTEMS
by
A.
Megretski
Lecture
6:
Storage
Functions
And
Stability
Analysis
1
This
lecture
presents
results
describing
the
relation
between
existence
of
Lyapunov
or
storage
functions
and
stability
of
dynamical
systems.
6.1
Stability
of
an
equilibria
In
this
section
we
consider
ODE
models
x
˙(
t
)
=
a
(
x
(
t
))
,
(6.1)
where
a
:
X
�
R
n
is
a
continuous
function
defined
on
an
open
subset
X
of
R
n
.
Remem
ber
that
a
point
¯
x
0
)
=
0,
i.e.
if
x
(
t
)
≥
¯
x
0
≤
X
is
an
equilibrium
of
(6.1)
if
a
(¯
x
0
is
a
solution
of
(6.1).
Depending
on
the
behavior
of
other
solutions
of
(6.1)
(they
may
stay
close
to
x
0
,
or
converge
to
¯
¯
x
0
as
t
�
→
,
or
satisfy
some
other
specifications)
the
equilibrium
may
be
called
stable,
asymptotically
stable,
etc.
Various
types
of
stability
of
equilibria
can
be
derived
using
storage
functions.
On
the
other
hand,
in
many
cases
existence
of
storage
functions
with
certain
properties
is
impled
by
stability
of
equilibria.
6.1.1
Locally
stable
equilibria
Remember
that
a
point
¯
x
0
≤
X
is
called
a
(locally)
stable
equilibrium
of
ODE
(6.1)
if
for
every
� >
0
there
exists
� >
0
such
that
all
maximal
solutions
x
=
x
(
t
)
of
(6.1)
with
x
0
 �
�
are
deinfed
for
all
t
∀
0,
and
satisfy

x
(
t
)
−
¯
0

< �
for
all
t
∀
0.

x
(0)
−
¯
x
The
statement
below
uses
the
notion
of
a
lower
semicontinuity
:
a
function
f
:
Y
�
R
,
defined
on
a
subset
Y
of
R
n
,
is
called
lower
semicontinuous
if
lim
inf
f
(¯)
∀
f
(¯
x
x
x
�
)
�
¯
�
≤
Y.
r
�
0
,r>
0
x
→
Y
:

¯
x
�

<r
¯
x
−
¯
1
Version
of
September
24,
2003
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�
2
Theorem
6.1
¯
x
0
≤
X
is
a
locally
stable
equilibrium
of
(6.1)
if
and
only
if
there
exist
c
>
0
and
a
lower
semicontinuous
function
V
:
B
c
(¯
x
0
)
�
R
,
defined
on
x
0
)
=
{
¯
x
0
∞
<
c
}
B
c
(¯
x
:
∞
x
−
¯
and
continuous
at
x
0
,
such
that
V
(
x
(
t
))
is
monotonically
nonincreasing
along
the
solu
¯
tions
of
(6.1),
and
V
(¯
x
)
�
¯
x
0
)
/
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 Spring '09
 AlexandreMegretski
 Dynamics, Continuous function, Stability theory, Extreme Value Theorem, Rn �� Rn

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