Page 187 - Complementarity and Variational Inequalities in Electronics

178 Complementarity and Variational Inequalities in Electronics

and
T
(∀x ∈ K) : ψ K (x) − ψ(x −[G + G ]x) = 0.
There exists α> 0 such that
2
T
(∀x ∈ K :
Ax,[G + G ]x ≥ α||x|| .
Indeed, suppose on the contrary that
2
T
(∀α> 0)(∃x ∈ K) :
Ax,[G + G ]x <α||x|| .
We can find a sequence {x n }⊂ K such that
1
T 2
(∀n ∈ N) :
Ax n ,[G + G ]x n < ||x n || .
n
−1
Setting z n = x n ||x n || , we obtain
1
T
(∀n ∈ N) :
Az n ,[G + G ]z n < .
n
We have (∀n ∈: N) : z n ∈ K since K is assumed to be a closed convex cone. We
⊂ K
also have (∀n ∈: N) :||z n || = 1, and thus there exists a subsequence z n k
→ z with z ∈ K and ||z|| = 1. Taking the limit as k →∞, we get
such that z n k
T

Az,[G + G ]z ≤ 0, a contradiction since z ∈ K and z = 0.
Moreover, using (5.62), we get a constant σ> 0 such that
α
||x|| ≤ σ ⇒||F 1 (x)|| ≤ ||x||.
T
||G + G ||
Thus, if ||x|| ≤ σ, then
1 T

F 1 (x),∇V(x) =
F 1 (x),[G + G ]x
2
1 T 1 2
≥− ||G + G ||||F 1 (x)||||x|| ≥ − α||x|| ,
2 2
which results in
α 2
(∀x ∈ D(∂ϕ),||x|| ≤ σ) :
Ax + F 1 (x),∇V(x) ≥ ||x|| ,
2
and the conclusion follows from Corollary 14.

5.5 A NONREGULAR CIRCUIT WITH IDEAL DIODES
Let us consider the circuit of Section 5.1.1 but with ideal diodes (see Fig. 5.3).
We have
⊥ x 2 ≥ 0
0 ≤−u D 4