Page 171 - Complementarity and Variational Inequalities in Electronics

162 Complementarity and Variational Inequalities in Electronics

Let us denote respectively by λ 1 > 0 and λ max > 0 the smallest and greatest
eigenvalues of the symmetric positive definite matrix V .Wehave
1 d α

V x(t),x(t) ≤−
V x(t),x(t) a.e. t ≥ t 0 .
2 dt λ max
Using the Gronwall inequality, we get
2α (t−t 0 )
−
(∀t ≥ t 0 ) :
V x(t),x(t) ≤
Vx 0 ,x 0 e λ max .
Thus

λ max α (t−t 0 )
(∀t ≥ t 0 ) :||x(t)|| ≤ ||x 0 ||e − λ max .
λ 1
The result follows from the last inequality by setting

λ max α
c 1 = ,c 2 = .
λ 1 λ max
Inequality (5.46) entails that the trivial solution of (5.37)–(5.40) is asymp-
totically stable. It is also globally attractive in the sense that

(∀x 0 ∈ D(∂ϕ)) : lim ||x(t;t 0 ,x 0 )|| = 0.
t→+∞
Remark 33. Let
n
(∀x ∈ R ) : F(x) = Mx
with M ∈ R n×n .If M is positive stable, then there exists a symmetric positive
T
definite matrix V ∈ R n×n such that VM + M V is positive definite. Thus
n
(∀x ∈ R ,x = 0) :
Mx,V x > 0.
Therefore, there exists α> 0 such that

2
n
(∀x ∈ R ) :
Mx,Gx ≥ α||x|| .
Indeed, suppose on the contrary that
n 2
(∀α> 0)(∃x ∈ R ) :
Mx,Gx <α||x|| .
n
We can find a sequence {x n }⊂ R such that
1 2
(∀n ∈ N) :
Mx n ,Gx n < ||x n || .
n