9.3 Convergence Results
357
where Q 0 ∞ = max i {|Q 0i |}. Then, for any ρ ≥ ρ min the hypercube
K ρ = {q ∈ R
n
: :q ∞ ≤ ρ}
is a positively invariant set for the dynamics of (9.13), i.e., if q(0) ∈ K ρ , we have
q(t) ∈ K ρ for any t ≥ 0, where q(·) is the solution of (9.13) with initial condition
q(0).
Proof As it is intuitively clear, it is enough to show that the vector field at the righthand side of (9.13) points inward on the boundary of the hypercube K ρ . This ensures
that any solution starting in K ρ cannot leave the same hypercube. In mathematical
terms we should have
− ρ +
n
i=1
G ik ˆ
ϕ(q i ) + Q 0k ≤ 0
ρ +
n
i=1
G ik ˆ
ϕ(q i ) + Q 0k ≥ 0
(9.18)
for any k = 1, 2, . . . , n and ρ ≥ ρ min . These are satisfied if −ρ+|
n
i=1 G ik ˆ
ϕ(q i )|+
|Q 0k | ≤ −ρ +
n
i=1 |G ik | | ˆ
ϕ(q i )| + |Q 0k | ≤ −ρ +
n
i=1 |G ik |
| ˆ
ϕ(ρ)| + |Q 0k | ≤ 0.
Since ρ ≥ ¯
q M , and ˆ
ϕ(·) is odd, we have 0 < ˆ
ϕ(ρ) ≤ 1 +
2R on
R off
(1 + ΔR) (ρ − ¯
q M ) so that inequalities (9.18) hold if
−ρ +
n
i=1
|G ik |
1 +
2R on
R off
(1 + ΔR) (ρ − ¯
q M )
+ |Q 0k | ≤ 0.
Note that
n
i=1 |G ik | ≤ ≤G 1 , and |Q 0k | ≤ ≤Q 0 ∞ , for any k = 1, 2, . . . , n, so
that by (9.16) we conclude that (9.18) are satisfied if
ρ ≥
G 1
1 − ¯
q M
2R on
R off
(1 + ΔR)
+ +Q 0 ∞
1 − −G 1
2R on
R off
(1 + ΔR)
.
Property 9.2 Suppose that (9.16) is satisfied. Then, the following hold.
(i) Any solution q(·) of (9.13) is bounded, i.e., q(t) ≤ max{{q(0) ∞ , ρ min } for
t ≥ 0.
(ii) There exists at least an EP of (9.13).