Page 161 - Complementarity and Variational Inequalities in Electronics

152 Complementarity and Variational Inequalities in Electronics

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There exists z 0 ∈ R such that is finite and continuous at y 0 = CR −1 z 0 .In-
deed, we have
⎛ √ ⎞
C 4 0 0 ⎛ ⎞
1 1
0 1 −1 ⎜ 1 ⎟ 0 √ − √
−1
CR = ⎜ 0 √ 0 ⎟ = ⎝ L 3 L 2 ⎠ ,
0 1 0 ⎝ L 3 ⎠ 0 √ 1 0
0 0 √ 1 L 3
L 2
and
z 0
y 0
⎛ ⎞

⎛ ⎞
1 1 1
1 0 √ − √ ⎜ √
= ⎝ L 3 L 2 ⎠ ⎟
1 0 √ 1 0 ⎝ L 3 ⎠
L 3 0
is a point at which is finite and continuous. We may thus set
z(t) = Rx(t)

and
3
(∀z ∈ R ) : (z) = (CR −1 z)
so as to reduce the study of the circuit to the variational inequality

dz −1

(t) − RAR z(t) − RDu(t),v − z(t)
dt
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+ (v) − (z(t)) ≥ 0,∀v ∈ R , a.e. t ≥ 0. (5.19)
Note that
⎛ 1 ⎞
0 − √ 0
L 3 C 4
⎜ ⎟
−RAR −1 = ⎜ √ 1 R 1 +R 3 − √ R 1 ⎟.
⎜
⎟
L 3
L 3 C 4
⎝ L 2 L 3 ⎠
0 − √ R 1 R 1 +R 2
L 2 L 3 L 2
The Kalman–Yakubovich–Popov lemma ensures that the matrix −RAR −1 is
positive semidefinite.
5.2 EXISTENCE AND UNIQUENESS THEOREM
The following existence and uniqueness result is essentially a consequence of
Kato’s theorem [59] (see also [45] for the details).

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1
n
0

Theorem 11. Suppose that f ∈ C ([t 0 ,+∞);R ) with f ∈ L (t 0 ,+∞;R ).
loc
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Let ϕ ∈ 0 (R ;R∪{+∞}), and let F : R → R be a continuous operator such