Page 184 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 175

Here
4
E(F,ψ K ,V ) ={x ∈ K : 4x = 0}={(0,x 2 );x 2 ∈ R}.
1
Let z = (0,z 2 ) ∈ E(F,ψ K ,V ). Suppose that γ(z) ⊂ E(F,ψ K ,V ) and set
dx
x(·) = x(·,t 0 ,x 0 ), x (·) = (·,t 0 ,x 0 ). From the dynamics in E(F,ψ K ,V ) we
dt
have

−x 2 (t)v 1 + x (t)(v 2 − x 2 (t)) ≥ 0, ∀v 1 ≥ 0,v 2 ∈ R, a.e. t ≥ t 0 . (5.61)
2
Setting v 2 = x 2 (t) in (5.61), we get
−x 2 (t)v 1 ,∀v 1 ≥ 0, a.e. t ≥ t 0 .

Thus x 2 (t) ≤ 0, a.e. t ≥ t 0 , and by continuity we get z 2 ≤ 0. Setting v 1 = 0
in (5.55), we get

x (t) = 0, a.e. t ≥ t 0 .
2
2
We obtain x 2 (t) = z 2 ,a.e. t ≥ t 0 . Therefore, M ={z ∈ R : z 1 = 0 and z 2 ≤ 0}
is the largest invariant subset of E(F,ψ K ,V ), and for any x 0 ∈ K,wehave
lim d(x(t;t 0 ,x 0 ),M) = 0.
t→+∞
Corollary 13. Suppose that the assumptions of Theorem 11 hold together with
conditions (5.36), (5.42) and (5.43) hold. Suppose also that D(∂ϕ) is closed.
1
n
Let V ∈ C (R ;R) be a function such that the function ϕ(·) − ϕ(·−@ V(·)) is
lower semicontinuous on D(∂ϕ), V(0) = 0,
(∀x ∈ D(∂ϕ)) : V(x) ≥ a(||x||)
with strictly increasing a : R + → R such that a(0) = 0,

(∀x ∈ D(∂ϕ)) :
F(x),∇V(x) + ϕ(x) − ϕ(x −@ V(x)) ≥ 0,

and
E(F,ϕ,V ) ={0}.
Then the trivial solution of (5.37)–(5.40) is (a) the unique stationary solution
of (5.37)–(5.40), (b) asymptotically stable, and (c) globally attractive.
Proof. Assertion (a) is a consequence of Proposition 26. The stability is a di-
rect consequence of Theorem 14. Moreover, we may apply Corollary 12 with
M ={0} (since E(F,ϕ,V ) ={0}) to obtain that, for any x 0 ∈ D(∂ϕ),

lim x(τ;t 0 ,x 0 ) = 0.
τ→+∞
Assertions (b) and (c) follow.