Page 194 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 185

∗
∗
∗
∗
z ⊥ Mz ⇔ z = z = 0.
3
2
∗
Setting w = Mz , we note that
∗
1
∗ ∗
w =−√ z ,
2
1
L 3 C 4
1 R 1
∗ ∗ R 1 + R 3 ∗ ∗
w = √ z + z − √ z ,
2 1 2 3
L 3 C 4 L 3 L 2 L 3
R 1 R 1 + R 2 ∗
∗ ∗
w =−√ z + z .
3 2 3
L 2 L 3 L 2
Thus
3
S(M, ) ={z ∈ R : z 1 ≥ 0,z 2 = 0,z 3 = 0}.
The set S(M, ) is an invariant subset of E(M, ,V ). We claim that it is
∗
the largest one. Indeed, let us study the dynamics in E(M, ,V ).Let z ∈
E(M, ,V ), that is, z = 0 and z = 0. We have
∗
∗
2 3
∗
z 1 (0) = z ,z 2 (0) = 0,z 3 (0) = 0,
1
(∀t ≥ 0) : z 2 (t) = 0 and z 3 (t) = 0,
and
1

z (t)(v 1 − z 1 (t)) + √ z 1 (t)v 2 ≥ 0,∀v 1 ∈ R,v 2 ≥ 0, a.e. t ≥ 0.
1
L 3 C 4
Setting v 1 = z 1 (t), we get
z 1 (t)v 2 ,∀v 2 ≥ 0, a.e. t ≥ 0.
It follows by continuity that (∀t ≥ 0) : z 1 (t) ≥ 0. It follows in particular that
z ≥ 0. Setting v 2 = 0, we get
∗
1
z (t) = 0, a.e. t ≥ 0.

1
Thus
∗
(∀t ≥ 0) : z 1 (t) = z ≥ 0.
1
Thus any invariant subset of E(M, ,V ) is a subset of S(M, ). Therefore, for
any z 0 ∈ K,wehave

lim d(z(t;0,z 0 ),S(M, )) = 0.
t→+∞