Chemistry Reference
In-Depth Information
By using the definition of the Riemann integral we have
t
i 1
/
f
.
i
/
X
N
X
N
t
i 1
/
f
.
i
/.t
i
.t
i
i
D
1
i
D
1
where a
D
t
0
<t
1
<
<t
N
D
b and
i
2
Œt
i 1
;t
i
for all i. By letting N
!1
we conclude (A.4).
In practical applications three particular vector norms are usually used, namely
k
v
k
1
D
max
fj
v
1
j
;
j
v
2
j
;:::;
j
v
n
jg
;
k
v
k
1
Dj
v
1
jCj
v
2
jCCj
v
n
j
and
D
p
j
v
1
j
k
v
k
2
2
Cj
v
2
j
2
CCj
v
n
j
2
;
where
v
i
is the ith element of
v
for i
D
1;2;:::;n. The norm
k
:
k
2
is usually called
the
Euclidean
norm.
All three norms satisfy conditions (1)-(3). Notice that
k
:
k
2
is the n-dimensional
generalization of the well known definition of the lengths of 2 and 3 dimensional
real vectors.
Let
T
be an invertible n
n matrix, and
k
:
k
a given vector norm. Then a new
vector norm can be defined as
k
u
k
T
Dk
Tu
k
:
Clearly this norm also satisfies conditions (1)-(3).
Assume now that
x
.0/ is the initial state of a system. Then the following stability
types can be considered.
Definition A.1.
An equilibrium
x
is called stable (or marginally stable) if for all
"
1
>0 there exists an ">0 such that
kx
.0/
xk
<"implies that for all t>0,
kx
.t/
xk
<"
1
.
Definition A.2.
An equilibrium
x
is asymptotically stable (or locally asymptoti-
cally stable) if it is stable and there is an ">0such that
kx
.0/
xk
<"implies
that
x
.t/ converges to
x
as t
!1
.
Definition A.3.
An equilibrium
x
is globally asymptotically stable in
D
if it is
stable and for arbitrary
x
.0/
2 D
,
x
.t/ converges to
x
as t
!1
.
The (marginal) stability of an equilibrium means that the entire state trajectory
remains close to the equilibrium if the initial state is selected close enough to the
equilibrium. If in addition the state trajectory converges to the equilibrium as t
!1
,
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