Page 49 - Complementarity and Variational Inequalities in Electronics

The Variational Inequality Problem Chapter | 3 39

We set

(x) = ϕ ∗ (−x)
(∀x ∈ R) : θ D 3
D 3
is finite and continuous.
and assume the existence of a point x 0 ∈ R at which θ D 3
Then

(x) =−∂ϕ ∗ (−x).
(∀x ∈ R) : ∂θ D 3
D 3
Therefore
(−V 3 ).
V 3 ∈ ∂ϕ D 1 (i 3 ) ⇔ i 3 ∈−∂θ D 3
We set

4
(∀x ∈ R ) : (x) = ϕ D 4 (x 1 ) + θ D 3 (x 2 ) + ϕ D 1 (x 3 ) + ϕ D 2 (x 4 ).

This results in that the dynamical behavior of the circuit in Fig. 3.3 is described
by the system
⎛ ⎞
B
i 4

dV −1 1 1 ⎜ ⎟
⎜ −V 3 ⎟
= V + 0 0 ⎜ ⎟, (3.9)
dt RC 1 C 1 C 1 ⎝ i 1 ⎠
i 2
y C N y L F

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞
1 0 −10 0 0
−V 4 i 4
⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
i 3 ⎜ 1 0
⎟u,
⎜ ⎟ ⎜ 0 ⎟ 1 −1 ⎟⎜ −V 3 ⎟ ⎜ 0 ⎟
⎜ ⎟ = ⎜ ⎟V + ⎜ ⎟⎜ ⎟+⎜
⎝ 0 −10
⎝ −V 1 ⎠ ⎝ 1 ⎠ 0 ⎠⎝ i 1 ⎠ ⎝ −1 ⎠
−V 2 0 0 1 0 0 i 2 1
(3.10)
and
y ∈−∂ (y L ). (3.11)
Remark 10. The relation y ∈−∂ (y L ) is equivalent to

⎛ ⎞
⎛ ⎞
y 1 ∂ϕ D 4 (y L,1 )
⎜ ⎟
y 2 ⎜ ∂θ D 3 (y L,2 ) ⎟
⎜ ⎟
⎜ ⎟
∈−⎜ ⎟
y 3 ⎝ ∂ϕ D 1 (y L,3 ) ⎠
⎝ ⎠
y 4 (y L,4 )
∂ϕ D 2