Page 135 - Complementarity and Variational Inequalities in Electronics

126 Complementarity and Variational Inequalities in Electronics

We see that system (4.113) is equivalent to the variational inequality VI(F,q, ),
that is,
2
2
I ∈ R : F(I) + q,v − I + (v) − (I) ≥ 0,∀v ∈ R . (4.115)
Here, D( ) = D( ) ∞ = R − × R + ,

2 5 5
(∀x = (x 1 ,x 2 ) ∈ R ) : F(x),x = R(x 1 + x 2 ) x 1 + R(x 1 + x 2 ) x 2
6
= R(x 1 + x 2 ) ≥ 0,
and
2
(∀(x 1 ,x 2 ) ∈ R ) : F(x 1 ,x 2 ) =∇ϒ(x 1 ,x 2 ),
where
R
2 6
(∀(x 1 ,x 2 ) ∈ R ) : ϒ(x 1 ,x 2 ) = (x 1 + x 2 ) .
6
The function ϒ is convex, and thus
Rt 5 6
r (x 1 ,x 2 ) = ϒ ∞ (x 1 ,x 2 ) = lim (x 1 + x 2 )
F t→+∞ 6

+∞ if x 1 + x 2
= 0
=
0 if x 1 + x 2 = 0.
Let (x 1 ,x 2 ) ∈ R − × R + be such that (x 1 ,x 2 )
= 0. If x 1 + x 2
= 0, then
r (x 1 ,x 2 ) =+∞, and condition (4.83) in Theorem 10 is satisfied. If x 1 +
F
x 2 = 0, then r (x 1 ,x 2 ) = 0, and condition (4.83) holds if and only if
F
(∀x 2 > 0) :−q 1 x 2 + q 2 x 2 = (E 2 − E 1 )x 2 > 0.
Here E 2 >E 1 , and the variational inequality VI(F,q, ) has at least one solu-
¯
¯ ¯
tion. Note that if I = (i 1 ,i 2 ) and I = (i 1 ,i 2 ) are two solutions, then
¯
¯
F(I) − F(I),I − I ≤ 0.
The function F =∇ϒ is monotone, and thus
¯
¯
F(I) − F(I),I − I = 0,
which is equivalent to
5
5
5
5
¯
¯
R((i 1 + i 2 ) − (i 1 + i 2 ) )(i 1 − i 1 ) + R((i 1 + i 2 ) − (i 1 + i 2 ) )(i 2 − i 2 ) = 0
¯
¯
¯
¯
or also
5
5
¯
R((i 1 + i 2 ) − (i 1 + i 2 ) )((i 1 + i 2 ) − (i 1 + i 2 )) = 0.
¯
¯
¯