Linear Algebra, Theory And Applications, 2012a

C.5. GEOMETRIC THEORY OF AUTONOMOUS SYSTEMS 429
Suppose for some m ≥ 1 there exists a constant, C m such that
for all k ≤ m for all t>0. Then
and so
Then by the induction hypothesis,
|r k (t)| ≤C m t m e Re(λ m)t
r ′ m+1 (t) =λ m+1 r m+1 (t)+r m (t) ,r m+1 (0) = 0
∫ t
r m+1 (t) =e λm+1t e −λm+1s r m (s) ds.
|r m+1 (t)| ≤ e Re(λ m+1)t
≤
≤
∫ t
0
∫ t
e Re(λ m+1)t
0
∫ t
e Re(λ m+1)t
0
0
∣
∣e ∣ −λ m+1s C m s m e Re(λm)s ds
s m C m e − Re(λ m+1)s e Re(λ m)s ds
s m C m ds =
C m
m +1 tm+1 e Re(λ m+1)t
It follows by induction there exists a constant, C such that for all k ≤ n,
|r k (t)| ≤Ct n e Re(λ n)t
and this obviously implies the conclusion of the lemma.
The proof of the above lemma yields the following corollary.
Corollary C.5.2 Let the functions, r k be given in the statement of Theorem C.4.8 and
suppose that A is an n × n matrix whose eigenvalues are {λ 1 , ··· ,λ n } . Suppose that these
eigenvalues are ordered such that
Re (λ 1 ) ≤ Re (λ 2 ) ≤···≤Re (λ n ) .
Then there exists a constant C such that for all k ≤ m
|r k (t)| ≤Ct m e Re(λm)t .
With the lemma, the following sloppy estimate is available for a fundamental matrix.
Theorem C.5.3 Let A be an n × n matrix and let Φ(t) be the fundamental matrix for A.
That is,
Φ ′ (t) =AΦ(t) , Φ (0) = I.
Suppose also the eigenvalues of A are {λ 1 , ··· ,λ n } where these eigenvalues are ordered such
that
Re (λ 1 ) ≤ Re (λ 2 ) ≤···≤Re (λ n ) < 0.
∣
∣ ∣∣
Then if 0 > −δ >Re (λ n ) , is given, there exists a constant, C such that ∣Φ(t) ij ≤ Ce
−δt
for all t>0. Also
|Φ(t) x| ≤Cn 3/2 e −δt |x| . (3.30)