Page 169 - Complementarity and Variational Inequalities in Electronics

160 Complementarity and Variational Inequalities in Electronics

There exists σ> 0 sufficiently small to ensure that

x 2 1
||x|| ≤ σ =⇒ 1 − cos(x 1 ) ≥ .
4
Thus
x 2 x 2
||x|| ≤ σ =⇒ V(x) ≥ 1 + 2 .
4 4
We also have
2
x ∈ R 2 =⇒ x −@ V(x) = (x 1 − sin(x 1 ),0) ∈ R .
+ +
Thus

2
(x ∈ R ,||x|| ≤ σ) =⇒
F(x),∇V(x) + ϕ(x) − ϕ(x −@ V(x)) = 0,
+
and using Theorem 14, we obtain the stability of the trivial solution of prob-
lem (5.37)–(5.40).

Theorem 15 (Asymptotic stability). Suppose that the assumptions of Theo-
rem 11 hold together with conditions (5.36), (5.42), and (5.43). Suppose that
n
1
there exist σ> 0, λ> 0, and V ∈ C (R ;R) with V(0) = 0 such that
(∀x ∈ D(∂ϕ) ∩ B σ ) : V(x) ≥ a(||x||),
τ
with a :[0,σ]→ R satisfying a(t) ≥ ct , ∀t ∈[0,σ], for some constants c> 0,
τ> 0 and

(∀x ∈ D(∂ϕ) ∩ B σ ) :
F(x),∇V(x) + ϕ(x) − ϕ(x −@ V(x)) ≥ λV (x).

Then the trivial solution of (5.37)–(5.40) is asymptotically stable.

Proof. The stability of the trivial solution follows from Theorem 14. In partic-
ular, there exists δ> 0 such that, for every x 0 ∈ D(∂ϕ) with ||x 0 || ≤ δ,

(∀t ≥ t 0 ) :||x(t;t 0 ,x 0 )|| ≤ σ.

Using the same notation and approach as in Lemma 3, we see that

dV
(t) ≤−λV(t), a.e. t ≥ t 0 .
dt
Using Lemma 2, we obtain
(∀t ≥ t 0 ) : V(t) ≤ V(t 0 )e −λ(t−t 0 ) .