Theorem 1 If there exist a symmetric and positive definite matrix , positive scalar constants , , , and scalar constant such that the following hold:
-
(1)
,
-
(2)
,
-
(3)
,
then the origin of the system (3) is exponentially stable, and
where , for any .
Proof Let us construct the following Lyapunov function:
(5)
from which we obtain
(6)
If , then by (3), (4), and (5) we have
which implies that
(7)
Similarly, if , then we have
which implies that
(8)
It follows from (7) and (8) that:
(1) If , then we have
So
(2) If , then we have
So
(3) If , then we have
So
(4) If , then we have
So
(5) If , then we have
So
(6) If , then we have
So
By induction, we have:
(7) If , i.e., , then we have
(9)
So
(8) If , i.e., , then we have that
(10)
From (9) we know that
(11)
where .
From (10) we know that
Case 1. If , then
Case 2. If , then
So, for any , we have
(12)
where .
It follows from (11) and (12) that, for any ,
(13)
By (5), (6), and (13), we conclude that
where , for any .
So we finish the proof. □
From Lemma 2, we know that the two conditions of Theorem 1 are equivalent to the following two LMIs, respectively:
(14)
(15)