Page 176 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 167

x(τ i ;t 0 ,x 0 ) → x . Thus x ∈ (x 0 ). On the other hand, we get the con-
∗
∗
tradiction d(x∗, (x 0 )) ≥ ε.
v) The set of stationary solutions S(F,ϕ) is invariant. Indeed, if x 0 ∈ S(F,ϕ),
then (∀t ≥ t 0 ) : x(τ;t 0 ,x 0 ) = x 0 , and thus γ(x 0 ) ={x 0 }⊂ S(F,ϕ).

Thanks to Theorem 12, we can prove that the set (x 0 ) is invariant by using
standard topological arguments (see e.g. [80]).
Theorem 17. Suppose that the assumptions of Theorem 11 hold together with
conditions (5.36), (5.42), and (5.43). Suppose also that D(∂ϕ) is closed. Let
x 0 ∈ D(∂ϕ). The set (x 0 ) is invariant.

Proof. Let z ∈ (x 0 ). There exists {τ i }⊂[t 0 ,+∞) such that τ i →+∞ and
x(τ i ;t 0 ,x 0 ) → z.Let τ ≥ t 0 .Wehave z ∈ (x 0 ) ⊂ D(∂ϕ) = D(∂ϕ), and we
may use Theorem 12 to obtain

x(τ;t 0 ,z) = lim x(τ;t 0 ,x(τ i ;t 0 ,x 0 )).
i→∞
Then noting from uniqueness property of the solution that

x(τ;t 0 ,x(τ i ;t 0 ,x 0 )) = x(τ − t 0 + τ i ;t 0 ,x 0 ),

we get
x(τ;t 0 ,z) = lim x(τ − t 0 + τ i ;t 0 ,x 0 ).
i→∞
Thus setting w i = τ − t 0 + τ i , we see that w i ≥ t 0 , w i →+∞ and
x(w i ;t 0 ,x 0 ) → x(τ;t 0 ,z), so that x(τ;t 0 ,z) ∈ (x 0 ).
Our goal is now to prove an extension of the LaSalle invariance principle
applicable to the first-order evolution variational inequality
dx

(t;t 0 ,x 0 ) + F(x(t;t 0 ,x 0 )),v − x(t;t 0 ,x 0 )
dt
n
+ ϕ(v) − ϕ(x(t;t 0 ,x 0 )) ≥ 0, ∀v ∈ R , a.e. t ≥ t 0 .
Lemma 4. Suppose that the assumptions of Theorem 11 hold together with
n
conditions (5.36), (5.42), and (5.43). Let be a compact subset of R .We
n
1
assume that there exists V ∈ C (R ;R) such that
(∀x ∈ D(∂ϕ) ∩ ) :
F(x),∇V(x) + ϕ(x) − ϕ(x −@ V(x)) ≥ 0. (5.49)

Let x 0 ∈ D(∂ϕ).If γ(x 0 ) ⊂ , then there exists a constant k ∈ R such that
(∀x ∈ (x 0 )) : V(x) = k.