Here a sufficient condition is given for stability of a first order system. First of all, here is a
fundamental estimate for the entries of a fundamental matrix.
Lemma D.5.1Let the functions, r_{k}be given in the statement of Theorem D.4.8and supposethat A is an n×n matrix whose eigenvalues are
{λ1,⋅⋅⋅,λn}
. Suppose that these eigenvalues areordered such that
Re (λ1) ≤ Re (λ2) ≤ ⋅⋅⋅ ≤ Re(λn) < 0.
Then if 0 > −δ >Re
(λn)
is given, there exists a constant, C such that for each k = 0,1,
⋅⋅⋅
,n,
−δt
|rk(t)| ≤ Ce (4.28)
(4.28)
for all t > 0.
Proof: This is obvious for r_{0}
(t)
because it is identically equal to 0. From the definition of the
r_{k},r_{1}^{′} = λ_{1}r_{1},r_{1}
(0)
= 1 and so r_{1}
(t)
= e^{λ1t} which implies
|r1(t)| ≤ eRe(λ1)t.
Suppose for some m ≥ 1 there exists a constant, C_{m} such that
|rk(t)| ≤ CmtmeRe(λm)t
for all k ≤ m for all t > 0. Then
r′m+1 (t) = λm+1rm+1 (t)+ rm(t),rm+1(0) = 0
and so
∫
λm+1t t −λm+1s
rm+1 (t) = e 0 e rm (s)ds.
Then by the induction hypothesis,
∫
Re(λm+1)t t|| −λm+1s|| m Re(λm)s
|rm+1 (t)| ≤ e 0 e Cms e ds
∫ t
≤ eRe(λm+1)t smCme − Re(λm+1)seRe(λm)sds
∫0t
≤ eRe(λm+1)t smCmds = -Cm--tm+1eRe(λm+1)t
0 m + 1
It follows by induction there exists a constant, C such that for all k ≤ n,
n Re(λ )t
|rk(t)| ≤ Ct e n
and this obviously implies the conclusion of the lemma.
The proof of the above lemma yields the following corollary.
Corollary D.5.2Let the functions, r_{k}be given in the statement of Theorem D.4.8and supposethat A is an n×n matrix whose eigenvalues are
{λ1,⋅⋅⋅,λn}
. Suppose that these eigenvalues areordered such that
Re (λ ) ≤ Re(λ ) ≤ ⋅⋅⋅ ≤ Re (λ ).
1 2 n
Then there exists a constant C such that for all k ≤ m
|rk(t)| ≤ CtmeRe (λm)t.
With the lemma, the following sloppy estimate is available for a fundamental matrix.
Theorem D.5.3Let A be an n×n matrix and let Φ
(t)
be the fundamental matrix for A. Thatis,
Φ′(t) = AΦ (t),Φ (0) = I.
Suppose also the eigenvalues of A are
{λ1,⋅⋅⋅,λn}
where these eigenvalues are ordered suchthat
Re (λ1) ≤ Re (λ2) ≤ ⋅⋅⋅ ≤ Re(λn) < 0.
Then if 0 > −δ >Re
(λ )
n
, is given, there exists a constant, C such that
| |
||Φ (t) ||
ij
≤ Ce^{−δt}for allt > 0. Also
|Φ(t)x| ≤ Cn3 ∕2e−δt|x|. (4.29)
(4.29)
Proof: Let
{| | }
M ≡ max ||Pk(A)ij|| for all i,j,k .
Then from Putzer’s formula for Φ
(t)
and Lemma D.5.1, there exists a constant, C such
that
Definition D.5.4Let f : U → ℝ^{n}where U is an open subset of ℝ^{n}such that a ∈ U andf
(a)
= 0.A point, a where f
(a)
= 0 is called an equilibrium point.Then a is asymptotically stableif for any ε > 0 there exists r > 0 such that whenever
|x0 − a |
< r and x
(t)
the solution to theinitial value problem,
′
x = f (x),x(0) = x0,
it follows
lt→im∞ x(t) = a, |x(t)− a| < ε
A differential equation of the form x^{′} = f
(x)
is called autonomousas opposed to anonautonomous equation of the form x^{′} = f
(t,x)
. The equilibrium point a is stableiffor every ε > 0 there exists δ > 0 such that if
|x0 − a|
< δ, then if x is the solutionof
x ′ = f (x),x(0) = x0, (4.30)
(4.30)
then
|x (t)− a|
< ε for all t > 0.
Obviously asymptotic stability implies stability.
An ordinary differential equation is called almost linear if it is of the form
x′ = Ax + g(x)
where A is an n × n matrix and
lim g-(x)= 0.
x→0 |x|
Now the stability of an equilibrium point of an autonomous system, x^{′} = f
(x)
can always be
reduced to the consideration of the stability of 0 for an almost linear system. Here is why. If you
are considering the equilibrium point, a for x^{′} = f
(x)
, you could define a new variable, y by
a + y = x. Then asymptotic stability would involve
|y(t)|
< ε and lim_{t→∞}y
(t)
= 0 while
stability would only require
|y(t)|
< ε. Then since a is an equilibrium point, y solves the following
initial value problem.
y′ = f (a+ y )− f (a),y(0) = y0,
where y_{0} = x_{0}− a.
Let A = Df
(a)
. Then from the definition of the derivative of a function,
y′ = Ay + g (y),y(0) = y0 (4.31)
(4.31)
where
lim g-(y-)= 0.
y→0 |y|
Thus there is never any loss of generality in considering only the equilibrium point 0 for an almost linear
system.^{1}
Therefore, from now on I will only consider the case of almost linear systems and the equilibrium
point 0.
Theorem D.5.5Consider the almost linear system of equations,
′
x = Ax + g (x) (4.32)
(4.32)
where
g(x)-
lxim→0 |x| = 0
and g is a C^{1}function. Suppose that for all λ an eigenvalue of A,Reλ < 0. Then 0 isasymptotically stable.
Proof: By Theorem D.5.3 there exist constants δ > 0 and K such that for Φ
(t)
the
fundamental matrix for A,
− δt
|Φ (t)x| ≤ Ke |x|.
Let ε > 0 be given and let r be small enough that Kr < ε and for
|x|
<
(K + 1)
r,
|g (x )|
< η
|x|
where η is so small that Kη < δ, and let
|y0|
< r. Then by the variation of constants formula, the
solution to 4.32, at least for small t satisfies
∫
t
y (t) = Φ(t)y0 + 0 Φ (t − s)g (y(s)) ds.
The following estimate holds.
∫ t ∫ t
|y(t)| ≤ Ke −δt|y |+ Ke −δ(t−s)η|y (s)|ds < Ke−δtr+ Ke −δ(t−s)η |y (s)|ds.
0 0 0
Therefore,
∫ t
eδt|y (t)| < Kr + K ηeδs|y(s)|ds.
0
By Gronwall’s inequality,
δt Kηt
e |y(t)| < Kre
and so
(K η−δ)t (Kη−δ)t
|y(t)| < Kre < εe
Therefore,
|y (t)|
< Kr < ε for all t and so from Corollary D.3.4, the solution to 4.32 exists for all
t ≥ 0 and since Kη − δ < 0,