Page 174 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 165

with

01 −1
C =
01 0
and
2
(∀X ∈ R ) : (X) = ϕ D (X 1 ) + ϕ Z (X 2 ).
We have seen that we may set

z(t) = Rx(t)

and
3 −1
(∀z ∈ R ) : (z) = (CR z)
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. (5.48)
Setting M =−RAR −1 ,wehave

⎛ 1 ⎞
0 − √ 0
L 3 C 4
⎜ ⎟
⎜ 1 R 1 +R 3 R 1 ⎟
M = ⎜ √ − √ ⎟.
L 3 C 4 L 3
⎝ L 2 L 3 ⎠
0 − √ R 1 R 1 +R 2
L 2 L 3 L 2
The matrix M is positive semidefinite. Indeed, we have
⎛ ⎞
0 0 0
T ⎜ 2(R 1 +R 3 ) 2R 1 ⎟
M + M = ⎜ 0 − √ ⎟
L 3
⎝ L 2 L 3 ⎠
0 − √ 2R 1 2(R 1 +R 2 )
L 2 L 3 L 2
with
2(R 1 + R 3 )
T T
1 (M + M ) = 0, 2 (M + M ) = > 0,
L 3
2(R 1 + R 2 )
T T T
3 (M + M ) = > 0, 12 (M + M ) = 0, 13 (M + M ) = 0,
L 2
and
4(R 1 R 2 + R 1 R 3 + R 2 R 3 )
T T
23 (M + M ) = > 0, 123 (M + M ) = 0.
L 2 L 3