PDF - Universiteit Twente

7.1. Exponentially stable semigroups on a Banach space
The powers of Ad satisfy the norm estimates
�A n d � = �(I − A −1
1 )−n� � ∞
1
≤
(n − 1)!
1
≤
(n − 1)!
= ˜ MA1
(n − 1)!
= ˜ MA1
(n − 1)!
0
� ∞
e −t �e A−1
1 t �t n−1 dt
e −t ˜ MA1 g(t)tn−1 dt
0
� ∞
e
0
−t g(t)t n−1 dt
� t1
e
0
−t g(t)t n−1 dt + ˜ � ∞
MA1
e
(n − 1)! t1
−t g(t)t n−1 dt. (7.8)
The estimate for the time interval from 0 to t1 is simple using the fact that
g is non-decreasing
�
˜MA1
t1
e
(n − 1)! 0
−t g(t)t n−1 dt ≤ ˜ MA1g(t1) (n − 1)!
� ∞
e
0
−t t n−1 dt = ˜ MA1g(t1). (7.9)
For the estimate for the time interval from t1 to ∞ we use two observations.
First Stirling’s approximation for (n − 1)!
(n − 1)! ≥ � � �n−1 n − 1
2π(n − 1)
.
e
Secondly, by the choice of t1, see (7.7), we have that
e −(1−ε)t t n−1 ≤ e −(1−ε)α(n−1) (α(n − 1)) n−1 , for t ≥ t1.
89