Page 175 - Complementarity and Variational Inequalities in Electronics

166 Complementarity and Variational Inequalities in Electronics

We may thus apply Theorem 14 with F(x) = Mx, ϕ = , and V(x) = x to
conclude that the trivial stationary solution of (5.48) is stable in the sense of
Lyapunov.

5.4 INVARIANCE THEORY

Suppose that the assumptions of Theorem 11 hold together with condi-
tions (5.36), (5.42), and (5.43).For x 0 ∈ D(∂ϕ), we denote by γ(x 0 ) the orbit

γ(x 0 ) ={x(τ;t 0 ,x 0 );τ ≥ t 0 },
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where x(·;t 0 ,x 0 ) :[t 0 ,+∞) → R ,t → x(t;t 0 ,x 0 ), denotes the unique solu-
tion of problem (t 0 ,x 0 ,F,0,ϕ). We also denote by (x 0 ) the limit set

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(x 0 ) ={z ∈ R :∃{τ i }⊂[t 0 ,+∞);τ i →+∞ and x(τ i ;t 0 ,x 0 ) → z}.
We say that a set D ⊂ D(∂ϕ) is invariant if

x 0 ∈ D =⇒ γ(x 0 ) ⊂ D,

that is,
x 0 ∈ D =⇒ (∀t ≥ t 0 ) : x(t;t 0 ,x 0 ) ∈ D.
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We denote by d(s,M) the distance from a point s ∈ R to a set M ⊂ R , that
is, d(s,M) = inf m∈M ||s − m||.
Remark 34. Let x 0 ∈ D(∂ϕ).
i) It is clear that
γ(x 0 ) ⊂ D(∂ϕ), (x 0 ) ⊂ D(∂ϕ).

ii) It is easy to check that
(x 0 ) ⊂ γ(x 0 ).
iii) If γ(x 0 ) is bounded, then (x 0 ) = ∅. Indeed, if γ(x 0 ) is bounded, then we
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can find a sequence x(τ i ;t 0 ,x 0 )(τ i ≥ t 0 ) such that x(τ i ;t 0 ,x 0 ) → z ∈ R .
Therefore z ∈ (x 0 ).
iv) If γ(x 0 ) is bounded, then

lim d(x(τ;t 0 ,x 0 ), (x 0 )) = 0.
τ→+∞

Indeed, if we suppose the contrary, then we can find ε> 0 and {τ i }⊂
[t 0 ,+∞) such that τ i →+∞ and d(x(τ i ;t 0 ,x 0 ), (x 0 )) ≥ ε. The sequence
x(τ i ;t 0 ,x 0 ) is bounded, and along a subsequence, we may suppose that