Page 153 - Complementarity and Variational Inequalities in Electronics

144 Complementarity and Variational Inequalities in Electronics

Here we have
⎛ ⎞
R i R 1 0
⎜ ⎟
−PA = ⎝ −γR i R 1 (R 2 + R 0 ) ⎠ .
1 −1 1
The matrix −PA is a P-matrix since

1 (−PA) = R i > 0, 2 (−PA) = R 1 > 0, 3 (−PA) = 1 > 0,
12 (−PA) = (1 + γ)R 1 R i > 0, 13 (−PA) = R i > 0,
23 (−PA) = R 1 + R 2 + R 0 > 0,

and

123 (−PA) = R i (R 1 + R 2 + R 0 ) + R 1 (γ R i + R 2 + R 0 )> 0.
3
Moreover, ∈ D (R ;R ∪{+∞}) (with 1 ≡ ϕ ZD , 2 ≡ 3 ≡ 0). We may
thus apply Theorem 7 to ensure that the system in NRM(A,B,C,D,U i ,ϕ ZD )
has a unique solution.
Thus, for a driven time-dependent input t à U i (t), the output time-
dependent voltage t à U o (t) defined by (see Fig. 4.15)

U o (t) = γR i I i (t) − R o I 2 (t)

is uniquely defined with the current functions t à I i (t) and t à I 2 (t) that are
uniquely determined in solving VI(−PA,−PDU i (t), ).