Page 166 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 157

Definition 8. The equilibrium point x = 0 is globally attractive if for all x 0 ∈
D(∂ϕ), the solution x(·;t 0 ,x 0 ) of problem (t 0 ,x 0 ,F,0,ϕ) fulfills

lim ||x(t;t 0 ,x 0 )|| = 0.
t→+∞
Let us denote by S(F,ϕ) the set of stationary solutions of (5.37)–(5.40), that
is,

n
S(F,ϕ) ={z ∈ D(∂ϕ) :
F(z),v − z + ϕ(v) − ϕ(z) ≥ 0,∀v ∈ R }. (5.44)
Conditions (5.42) and (5.43) ensure that 0 ∈ S(F,ϕ).
Let us first recall some general abstract theorems of stability in terms of
1
n
generalized Lyapunov functions V ∈ C (R ;R). The following results are par-
ticular cases of those proved in [45].For r> 0, we denote by B r the closed ball
of radius r, that is,
n
B r ={x ∈ R :||x|| ≤ r}.
1
n
Let V ∈ C (R ;R).Weset
E(F,ϕ,V ) ={x ∈ D(∂ϕ) :
F(x),∇V(x)
+ ϕ(x) − ϕ(x −@ V(x)) = 0}. (5.45)
The following results are particular cases of those proved in [45].

Lemma 3. Suppose that the assumptions of Theorem 11 hold together with
conditions (5.36), (5.42), and (5.43). Suppose that there exist R> 0, a> 0, and
1
n
V ∈ C (R ;R) such that
(∀x ∈ D(∂ϕ), ||x|| = R) : V(x) ≥ a
and

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

Then, for any x 0 ∈ D(∂ϕ) with ||x 0 || <R and V(x 0 )<a, the solution
x(·;t 0 ,x 0 ) of problem (t 0 ,x 0 ,F,0,ϕ) satisfies
(∀t ≥ t 0 ) :||x(t;t 0 ,x 0 )|| <R.

Proof. Fix x 0 ∈ D(∂ϕ) with ||x 0 || <R and V(x 0 )<a. It suffices to prove the
result for t> t 0 since ||x(t 0 ;t 0 ,x 0 )|| = ||x 0 || <R. Suppose on the contrary that
∗
there exists t >t 0 such that ||x(t ;t 0 ,x 0 )|| ≥ R. The application x(·;t 0 ,x 0 ) is
∗
continuous, and ||x(t 0 ;t 0 ,x 0 )|| = ||x 0 || <R. We can thus find T> t 0 such that
(∀t ∈[t 0 ,T [) :||x(t;t 0 ,x 0 )|| <R