Page 173 - Complementarity and Variational Inequalities in Electronics

164 Complementarity and Variational Inequalities in Electronics

Proof. We claim that B σ ∩ S(F,ϕ) ={0}. Indeed, setting = B σ and using
Proposition 24, we obtain

B σ ∩ S(F,ϕ) ⊂ B σ ∩ E(F,ϕ,V ) ={0}.

The following result can be proved by following the same arguments as those
used in the proof of Proposition 25.

Proposition 26. Suppose that the assumptions of Theorem 11 hold together with
1
n
conditions (5.36), (5.42), and (5.43). Suppose that there exists V ∈ C (R ;R)
such that
(∀x ∈ D(∂ϕ)) :
F(x),∇V(x) + ϕ(x) − ϕ(x −@ V(x)) ≥ 0

and
E(F,ϕ,V ) ={0}.

Then S(F,ϕ) ={0}, that is, the trivial stationary solution of (5.37)–(5.40) is the
unique stationary solution of (5.37)–(5.40).

5.3.1 A Nonregular Circuit (Continuation)

Let us again consider the mathematical model that corresponds to the circuit
depicted in Fig. 5.2 with

(∀t ≥ 0) : u(t) = 0.

We have
A

⎛ ⎞ ⎛ ⎞
dx 1 0 1 0 ⎛ ⎞
dt x 1
⎜ ⎟ ⎜ 1 (R 1 +R 3 ) R 1 ⎟
⎜ dx 2 ⎟ = ⎜ − − ⎟⎝ x 2 ⎠
L 3 C 4 L 3 L 3
⎝ ⎝ ⎠
dt ⎠
dx 3 0 R 1 − (R 1 +R 2 ) x 3
dt L 2 L 2
B

⎛ ⎞
0 0

1 1 y L,1
⎜ ⎟
− ⎜ ⎟
L 3
⎝ L 3 ⎠ y L,2
1
− 0
L 2
and
y L ∈ ∂ (Cx)