Page 170 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 161

Thus, for all x 0 ∈ D(∂ϕ) with ||x 0 || ≤ δ, we get

τ
(∀t ≥ t 0 ) : c||x(t;t 0 ,x 0 )|| ≤ a(||x(t;t 0 ,x 0 )||) ≤ V(x 0 )e −λ(t−t 0 ) .
Thus

lim ||x(t;t 0 ,x 0 )|| = 0.
t→+∞
The following result is of particular interest for checking if the trivial sta-
tionary solution is globally attractive.
Theorem 16. Suppose that the assumptions of Theorem 11 hold together with
conditions (5.36), (5.42), and (5.43). Suppose that there exist a symmetric and
positive definite matrix V ∈ R n×n and α> 0 such that
2
(∀x ∈ D(∂ϕ)) :
F(x),V x ≥ α||x||
and

(∀x ∈ D(∂ϕ)) : ϕ(x) − ϕ(x − Vx) ≥ 0.
Then there exist constants c 1 > 0 and c 2 > 0 such that

(∀t ≥ t 0 ) :||x(t;t 0 ,x 0 )|| ≤ c 1 ||x 0 ||e −c 2 (t−t 0 ) . (5.46)

Proof. Let x 0 ∈ D(∂ϕ), and let

(∀t ≥ t 0 ) : x(t) = x(t;t 0 ,x 0 ).

We have
dx n

(t) + F(x(t)),v − x(t) + ϕ(v) − ϕ(x(t)) ≥ 0, ∀v ∈ R , a.e. t ≥ t 0 ,
dt
and thus

dx

(t) + F(x(t)),−Vx(t) + ϕ(x(t) − V x(t)) − ϕ(x(t)) ≥ 0, a.e. t ≥ t 0 ,
dt
which results in
dx

(t) + F(x(t)),V x(t) + ϕ(x(t)) − ϕ(x(t) − V x(t)) ≤ 0, a.e. t ≥ t 0 .
dt
Thus
1 d 2

V x(t),x(t) ≤−α||x(t)|| , a.e. t ≥ t 0 .
2 dt