Page 140 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 131

D
u

⎛ ⎞
10
V i
= 0,
⎜ ⎟
+ ⎝ 01 ⎠
2V c
00
and we suppose that the electrical superpotentials of the four diodes D 1 , D 2 ,
D 3 , D 4 are respectively given by ϕ 1 ,ϕ 2 ,ϕ 3 ,ϕ 4 ∈ 0 (R;R ∪{∞}):
V 1 ∈ ∂ϕ 1 (x 1 ),
V 2 ∈ ∂ϕ 2 (x 2 ) = ∂ϕ 2 (x 1 − x 7 ),
V 3 ∈ ∂ϕ 3 (x 3 ) = ∂ϕ 3 (x 6 − x 1 ),
V 4 ∈ ∂ϕ 4 (x 4 ) = ∂ϕ 4 (x 7 + x 6 − x 1 ).

Setting
C

⎛ ⎞
0 0 1 ⎛ ⎞
x 7
⎜ −10 1 ⎟
⎜
⎟
⎝ x 6 ⎠
0
y = ⎜ ⎟
⎝ 1 −1 ⎠
x 1
1 1 −1
and defining the function
4
(∀x ∈ R ) : (x) = ϕ 1 (x 1 ) + ϕ 2 (x 2 ) + ϕ 3 (x 3 ) + ϕ 4 (x 4 ),
we may write
V ∈ ∂ (y)
and then consider problem NRM(A,B,C,D,u, ). It is easy to see that in prac-
tice,
T T
2112 = C 132 ∈]0,+∞[ 4
is a point at which is finite and continuous since electrical superpotentials ϕ i
(1 ≤ i ≤ 4) of any type of diode are finite and continuous on ]0,+∞[. There-
T
fore, Assumption (H1) holds. We note also that C = B, and thus Assump-
tion (H2) holds with P = I. As a consequence of Proposition 21, problem
NRM(A,B,C,D,u, ) can be studied via problem VI(−A,−Du, ) where
= ◦ C.Here −A is symmetric and positive semidefinite.

Example 63. Let us here assume that all diodes are ideal, that is,
.
ϕ 1 ≡ ϕ 2 ≡ ϕ 3 ≡ ϕ 4 ≡ R +