Page 189 - Complementarity and Variational Inequalities in Electronics

180 Complementarity and Variational Inequalities in Electronics

and

λ 1 y 1
0 ≤ ⊥ ≥ 0
λ 2 y 2
with
C

⎛ ⎞

x 1
y 1 01 −1
= ⎝ x 2 ⎠ .
y 2 01 0
x 3
We have seen that we may set
z(t) = Rx(t)

with
⎛ ⎞
1
√ 0 0
C 4
√
⎜ ⎟
R = ⎝ 0 L 3 0 ⎟
⎜
√ ⎠
0 0 L 2
and
3
(∀z ∈ R ) : (z) = ψ 2 (CR −1 z)
R
+
so as to reduce the study of the circuit to the variational inequality
dz −1

(t) − RAR z(t),v − z(t)
dt
n
+ (v) − (z(t)) ≥ 0,∀v ∈ R , a.e. t ≥ 0. (5.64)
√ √
1
Note that z is easily calculated: z 1 = √ x 1 ,z 2 = L 3 x 2 ,z 3 = L 2 x 3 .Wealso
C 4
have
3
(∀z ∈ R ) : (z) = K (z)
with
3
K ={z ∈ R : CR −1 z ≥ 0}.
Here
⎛ ⎞
0 √ 1 − √ 1
CR −1 = ⎝ L 3 L 2 ⎠ ,
0 √ 1 0
L 3