Page 164 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 155

− [F + ωI](x (t)) −[F + ωI](x(t)),x (t) − x(t)

2
≤ ω||x (t) − x(t)|| , a.e. t ≥ t 0 .
n
0
n
Recalling that x ∈ C ([t 0 ,+∞);R ) and dx ∈ L (t 0 ,+∞;R ), we may write
∞
dt loc
d 2 2
||x (t) − x(t)|| ≤ 2ω||x (t) − x(t)|| , a.e. t ≥ t 0 . (5.29)
dt
We may apply Lemma 2 with T> τ − t 0 , α = 0, b(·) = 0, a(·) = 2ω, and
2

w(·) =||x (·) − x(·)|| to get
2 2 2ω(t−t 0 )
||x (t) − x(t)|| ≤||x − x 0 || e ,∀t ∈[t 0 ,t 0 + T ]. (5.30)
0
It follows that

||x (τ) − x(τ)|| ≤ δ e 2ω(τ−t 0 ) = ε.
A direct application of Theorem 11 gives a result that can be used to
study Q(Rx 0 ,RAR −1 ,RDu, ) = (0,Rx 0 ,RAR −1 ,Du, ) (and therefore
also problem P(x 0 ,A,B,C,D,u, )).
Theorem 13. Let A ∈ R n×n and be as defined in (5.11). Suppose that
0 n 1 n

u ∈ C ([0,+∞);R ) with u ∈ L (0,+∞;R ). Let t 0 ∈ R and z 0 = Rx 0 ∈
loc
0
n
D(∂ ). Then there exists a unique z ∈ C ([0,+∞);R ) such that
dz n
∞
∈ L (t 0 ,+∞;R ), (5.31)
loc
dt
z is right-differentiable on [t 0 ,+∞), (5.32)
z(t 0 ) = z 0 , (5.33)
z(t) ∈ D(∂ ), t ≥ t 0 , (5.34)
dz −1

(t) − RAR z(t) − DRu(t),v − z(t)
dt
n
+ (v) − (z(t)) ≥ 0, ∀v ∈ R , a.e. t ≥ t 0 . (5.35)
5.3 LYAPUNOV STABILITY OF A STATIONARY SOLUTION

Suppose in addition to the assumptions of Theorem 11 that

(∀t ≥ t 0 ) : f(t) = 0. (5.36)
So, we consider problem (t 0 ,x 0 ,F,0,ϕ):

dx n
∞
∈ L (t 0 ,+∞;R ), (5.37)
loc
dt