Page 168 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 159

with a :[0,σ]→ R satisfying a(t) > 0,∀t ∈ (0,σ), and

(∀x ∈ D(∂ϕ) ∩ B σ ) :
F(x),∇V(x) + ϕ(x) − ϕ(x −@ V(x)) ≥ 0.

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

Proof. Without loss of generality, let 0 <ε <σ.Wehave

(∀x ∈ D(∂ϕ), ||x|| = ε) : V(x) ≥ a(ε) > 0.

The function V is continuous, and by assumption V(0) = 0. Therefore, there
exists δ(ε) > 0 such that

||x 0 || ≤ δ(ε) =⇒ |V(x 0 )| < a(ε).

We choose
0 <η(ε)< min{ε,δ(ε)}.

Let us now apply Lemma 3 with R = ε and a = a(ε). We note that if x 0 ∈
D(∂ϕ) satisfies ||x 0 || ≤ η(ε), then V(x 0 )<a(ε) and ||x 0 || <ε. The conclusion
of Lemma 3 leads to

(∀t ≥ t 0 ) :||x(t;t 0 ,x 0 || <ε,

which ensures that the trivial solution of (5.37)–(5.40) is stable.

Example 66. Let us consider problem (5.37)–(5.40) with

2
(∀x ∈ R ) : F(x) = (x 2 ,−sin(x 1 ))
and
2
(∀x ∈ R ) : ϕ(x) = 2 (x).
R
+
We choose
x 2
2 2
(∀x ∈ R ) : V(x) = 1 − cos(x 1 ) + .
2
We have
2
(∀x ∈ R ) :∇V(x) = (sin(x 1 ),x 2 )
and
2
(∀x ∈ R ) :
F(x),∇V(x) = 0.