Page 165 - Complementarity and Variational Inequalities in Electronics

156 Complementarity and Variational Inequalities in Electronics

x is right-differentiable on [t 0 ,+∞), (5.38)
x(t) ∈ D(∂ϕ), t ≥ t 0 , (5.39)
dx

(t) + F(x(t)),v − x(t)
dt
n
+ ϕ(v) − ϕ(x(t)) ≥ 0, ∀v ∈ R , a.e. t ≥ t 0 , (5.40)
x(t 0 ) = x 0 . (5.41)
We assume also that

0 ∈ D(∂ϕ) (5.42)
and
F(0) ∈−∂ϕ(0). (5.43)

Then
(∀t ≥ 0) : x(t;t 0 ,0) = 0.
This last relation implies that the trivial stationary solution 0 is the unique so-
lution of problem (t 0 ,x 0 ,F,0,ϕ). The stationary solution 0 is called stable
if small perturbations of the initial condition x(t 0 ) = 0 lead to solutions that
remain in a neighborhood of 0 for all t ≥ t 0 or, precisely:

Definition 5. The equilibrium point x = 0 is said to be stable in the sense of
Lyapunov if for every ε> 0, there exists η = η(ε) > 0 such that for any x 0 ∈
D(∂ϕ) with ||x 0 || ≤ η, the solution x(·;t 0 ,x 0 ) of problem (t 0 ,x 0 ,F,0,ϕ)
satisfies ||x(t;t 0 ,x 0 )|| ≤ ε, ∀t ≥ t 0 .

If in addition the trajectories of the perturbed solutions are attracted by 0,
then we say that the stationary solution is asymptotically stable; precisely:

Definition 6. The equilibrium point x = 0 is asymptotically stable if (1) it is
stable and (2) there exists δ> 0 such that for any x 0 ∈ D(∂ϕ) with ||x 0 || ≤ δ,
the solution x(·;t 0 ,x 0 ) of problem (t 0 ,x 0 ,F,0,ϕ) fulfills

lim ||x(t;t 0 ,x 0 )|| = 0.
t→+∞
Note that the equilibrium point x = 0 is called attractive as soon as part (2)
of Definition 6 is satisfied.

Definition 7. The equilibrium point x = 0 is attractive if there exists δ> 0
such that for any x 0 ∈ D(∂ϕ) with ||x 0 || ≤ δ, the solution x(·;t 0 ,x 0 ) of problem
(t 0 ,x 0 ,F,0,ϕ) fulfills

lim ||x(t;t 0 ,x 0 )|| = 0.
t→+∞