Page 178 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 169

By continuity

lim V(x(τ i ;t 0 ,x 0 )) = V(y).
i→+∞
Therefore V(y) = k. Since y was chosen arbitrary in (x 0 ),wehave
∀y ∈ (x 0 ) : V(y) = k.

Lemma 5. Suppose that the assumptions of Theorem 11 hold together with
n
1
conditions (5.36), (5.42), and (5.43). We assume that there exists V ∈ C (R ;R)
such that
(∀x ∈ D(∂ϕ)) :
F(x),∇V(x) + ϕ(x) − ϕ(x −@ V(x)) ≥ 0.
For a ∈ R,set
n
a (V ) ={x ∈ R : V(x) ≤ a}.
The set D(∂ϕ) ∩ a (V ) is invariant.

Proof. Let x 0 ∈ D(∂ϕ) ∩ a (V ). Then x 0 ∈ D(∂ϕ) and V(x 0 ) ≤ a.If τ ≥ t 0 ,
then x(τ;t 0 ,x 0 ) ∈ D(∂ϕ), and as in the proof of Lemma 4, we check that
V(x(·;t 0 ,x 0 )) is decreasing on [t 0 ,+∞). Thus

V(x(τ;t 0 ,x 0 )) ≤ V(x(t 0 ;t 0 ,x 0 )) = V(x 0 ) ≤ a.

Therefore
γ(x 0 ) ⊂ D(∂ϕ) ∩ a (V ).

Theorem 18 (Invariance theorem). Suppose that the assumptions of Theo-
rem 11 hold together with conditions (5.36), (5.42), and (5.43). Suppose also
n
1
n
that D(∂ϕ) is closed. Let ⊂ R be a compact set, and V ∈ C (R ;R) be a
function such that the function ϕ(·) − ϕ(·−@ V(·)) is lower semicontinuous on
D(∂ϕ) ∩ and
(∀x ∈ D(∂ϕ) ∩ ) :
F(x),∇V(x) + ϕ(x) − ϕ(x −@ V(x)) ≥ 0.
We set

E (F,ϕ,V ) = E(F,ϕ,V ) ∩
and denote by M the largest invariant subset of E (F,ϕ,V ). Then, for each
x 0 ∈ D(∂ϕ) such that γ(x 0 ) ⊂ , we have

lim d(x(τ;t 0 ,x 0 ),M) = 0.
τ→+∞