9.3 Convergence Results
359
Computing the time derivative of V along the solutions of (9.19) we obtain
˙
V (y) = [∇V (y)]
T dy
dt
= [∇V (y)]
T diag
dy i
dq 1
, . . . ,
dy n
dq n
dq
dt
= −[∇V (y)]
T diag( ˆ
ϕ
(q 1 ), . . . , ˆ
ϕ
(q n ))∇V (y)
= −
n
i=1
ˆ
ϕ
(q i )
∂V (y)
∂y i
2
= −
n
i=1
ˆ
ϕ
(q i )
dq i
dt
2
.
Since ˆ
ϕ (q i ) > 0 for any q i ∈ R, we have ˙
V (·) ≤ 0 for any q and ˙
V (·) < 0
when q is not an EP, i.e., V (·) is strictly decreasing along nonstationary solutions
of (9.13). This implies by LaSalle’s invariance principle that any solution of (9.13)
converges to the set of EPs of (9.13) as t → +∞ [16].
Now, recall that ˆ
ϕ(·) is analytic in R (Sect. 9.2.1). Then, we can use an argument
based on Łojasiewicz inequality, and the principle of trajectories with finite length,
as in the proof of Theorem 1 in [19], to show that any solution of (9.13) converges
to a singleton, which is necessarily an EP of (9.13), as t → +∞. The argument
holds independently of the geometric structure of the set of EPs, hence also in the
case when (9.13) has infinitely many nonisolated EPs.
Remark 9.2 The convergence result in Theorem 9.1 holds even when (9.13) has
infinitely many nonisolated EPs. However, it is worth to remark that vector fields as
those defining (9.13) enjoy the generic property of possessing isolated EPs. Indeed,
given the interconnection matrix G and the nonlinearity ˆ
ϕ(·), it can be shown via a
technique based on Sard’s lemma as that used in the proof of Property 2 in [19] that,
for almost all Q 0 ∈ R n , in the sense of the Lebesgue measure, there exists a finite
number of isolated EPs of (9.13).
9.3.1 Voltage-Current Domain
The equations describing the dynamics of a M-SCNN in the standard (v, i)-domain
can be obtained according to FCAM (Chap. 5) by differentiating with respect to time
the Eqs. (9.13) in the (ϕ, q)-domain. We have
dq i (t)
dt
=
1
k q
dq M i (t; t 0 )
dt
=
1
k q
i M i (t; t 0 ) =
1
k q R
v C i (t)
(9.20)
359
Computing the time derivative of V along the solutions of (9.19) we obtain
˙
V (y) = [∇V (y)]
T dy
dt
= [∇V (y)]
T diag
dy i
dq 1
, . . . ,
dy n
dq n
dq
dt
= −[∇V (y)]
T diag( ˆ
ϕ
(q 1 ), . . . , ˆ
ϕ
(q n ))∇V (y)
= −
n
i=1
ˆ
ϕ
(q i )
∂V (y)
∂y i
2
= −
n
i=1
ˆ
ϕ
(q i )
dq i
dt
2
.
Since ˆ
ϕ (q i ) > 0 for any q i ∈ R, we have ˙
V (·) ≤ 0 for any q and ˙
V (·) < 0
when q is not an EP, i.e., V (·) is strictly decreasing along nonstationary solutions
of (9.13). This implies by LaSalle’s invariance principle that any solution of (9.13)
converges to the set of EPs of (9.13) as t → +∞ [16].
Now, recall that ˆ
ϕ(·) is analytic in R (Sect. 9.2.1). Then, we can use an argument
based on Łojasiewicz inequality, and the principle of trajectories with finite length,
as in the proof of Theorem 1 in [19], to show that any solution of (9.13) converges
to a singleton, which is necessarily an EP of (9.13), as t → +∞. The argument
holds independently of the geometric structure of the set of EPs, hence also in the
case when (9.13) has infinitely many nonisolated EPs.
Remark 9.2 The convergence result in Theorem 9.1 holds even when (9.13) has
infinitely many nonisolated EPs. However, it is worth to remark that vector fields as
those defining (9.13) enjoy the generic property of possessing isolated EPs. Indeed,
given the interconnection matrix G and the nonlinearity ˆ
ϕ(·), it can be shown via a
technique based on Sard’s lemma as that used in the proof of Property 2 in [19] that,
for almost all Q 0 ∈ R n , in the sense of the Lebesgue measure, there exists a finite
number of isolated EPs of (9.13).
9.3.1 Voltage-Current Domain
The equations describing the dynamics of a M-SCNN in the standard (v, i)-domain
can be obtained according to FCAM (Chap. 5) by differentiating with respect to time
the Eqs. (9.13) in the (ϕ, q)-domain. We have
dq i (t)
dt
=
1
k q
dq M i (t; t 0 )
dt
=
1
k q
i M i (t; t 0 ) =
1
k q R
v C i (t)
(9.20)