Page 130 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 121

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 3 (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 ).

Therefore the dynamical behavior of the circuit in Fig. 4.5 is described by the
system

B ⎛ ⎞
i 4

dx −1 1 1 ⎜ ⎟
⎜ −V 3 ⎟
= x + 0 0 ⎜ ⎟, (4.98)
dt RC C C ⎝ i 1 ⎠
i 2
y C N y L F

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞
−V 4 1 0 −10 0 i 4 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
i 3 ⎜ 1 0
⎟U
⎜ ⎟ ⎜ 0 ⎟ 1 −1 ⎟⎜ −V 3 ⎟ ⎜ 0 ⎟
⎜ ⎟ = ⎜ ⎟x + ⎜ ⎟⎜ ⎟+⎜
⎝ 0 −10
⎝ −V 1 ⎠ ⎝ 1 ⎠ 0 ⎠⎝ i 1 ⎠ ⎝ −1 ⎠
0 0 1 0 0 1
−V 2 i 2
(4.99)
and
y ∈−∂ (y L ). (4.100)
Assuming that U remains constant, that is, U(·) ≡ U, the stationary solutions
(or fixed points) of (4.98)–(4.100) satisfy the problem

−ax + By L = 0
(4.101)
4
Ny L + Cx + FU,v − y L + (v) − (y L ) ≥ 0, ∀ v ∈ R ,
1 1
with a = > 0. From the first equation of (4.101) we deduce that x = By L ,
RC a
1
so that y = (N + CB)y L + FU, and our mathematical model reduces to the
a
1
problem VI((N + CB), ,FU), that is,
a
1 4
(N + CB)y L + FU,v − y L + (v) − (y L ) ≥ 0, ∀ v ∈ R . (4.102)
a