12 J. Chen et al. / Neural Networks 100 (2018) 10–24
for i, j = 1, 2, . . . , n and t ≥ 0, where switching jumps T
c
j
> 0,
T
a
j
> 0 and T
b
j
> 0 are constants,
ˆ
a
ij
,
ˇ
a
ij
,
ˆ
b
ij
,
ˇ
b
ij
,
ˆ
c
i
and
ˇ
c
i
are all
constants and
ˆ
c
i
,
ˇ
c
i
> 0.
Let τ = max{sup
t≥0
τ
ij
(t); i, j = 1, . . ., n} and let C
τ
:=
C[−τ , 0] denote a Banach space of all continuous functions ϕ :
[−τ , 0] → R. Sometime, for x ∈ R
n
, we write x ∈ C
τ
means
x(s) ≡ x in [−τ , 0]. For ϕ ∈ C
τ
, let ∥ϕ∥
c
= sup
s∈[−τ , 0]
∥ϕ(s)∥.
The initial states associated with DFMNN (2.1) are of the form
x
i
(t
0
+ s) = ϕ
i
(s), s ∈ [−τ , 0], i = 1, . . . , n. (2.5)
Let x
t
∈ C ([−τ , 0], R
n
) be defined by x
t
(s) = x(t + s), −τ ≤ s ≤
0, and the initial states (2.5) can be rewritten as
x
t
0
= ϕ ∈ C
τ
. (2.6)
Now, we introduce the following definitions about set-valued
map and differential inclusion. Suppose E ⊂ R
n
, then x → F (x)
is called a set-valued map from E to R
n
, if for each point x ∈ E,
there exists a nonempty set F (x) ⊂ R
n
. A set-valued map F with
nonempty values is said to be upper semicontinuous at x
0
∈ E, if
for any open set N containing F (x
0
), there exists a neighborhood
M of x
0
such that F (M) ⊂ N. The map F (x) is said to have a closed
(convex, compact)image if for each x ∈ E, F (x) is closed (convex,
compact).
Suppose E ⊂ R
n
, co[E] denote the closure of the convex hull of
set E. Let f : R × R
n
→ R
n
be measurable and essentially locally
bounded. Then the Filippov set-valued map K (f ) : R × R
n
→ R
n
is
defined as follows:
K (f )(t, x) ==
δ>0
µ(N)=0
co[f (t, B(x, δ)/N)]
where B(x, δ) is the ball of center x and radius δ, intersection is
taken over all sets N of measure zero and over all δ > 0, µ(N)
is Lebesgue measure of set N.
Calculus for the map K , the following properties can be ob-
tained (see Wang et al., 2014).
Lemma 2.1. The map K : {f |f : R
m
→ R
n
} → {g|g : R
m
→ 2
R
n
}
has the following properties:
(1) Assume that f
j
: R
m
R
n
j
, j ∈ {1, 2, . . . , N} are locally bounded,
then
K [
N
k=1
f
j
]⊆
N
k=1
K [f
j
]
where
is Kronecker product.
(2) Let g : R
m
→ R
p×n
be continuous and f : R
m
→ R
n
be locally
bounded, then
K [gf ](x) = g(x)K [f ](x).
Definition 2.1. A vector-value function x(t) defined on a non-
degenerate interval I ⊂ R is called a Filippov solution of system
D
α
t
0
x(t) = f (t, x
t
), if it is absolutely continuous on any subinterval
[t
1
, t
2
] of I and for a.a. t ∈ I, x(t) satisfies the differential inclusion
D
α
t
0
x(t) ∈ K (f )(t, x
t
). (2.7)
The global existence results of fractional-order functional dif-
ferential inclusions are given in Wang et al. (2009).
According to Definition 2.1, we can define the Filippov solution
of DFMNN (2.1) as follows. A function x(t) is said to be a (Filippov)
solution of DFMNN (2.1) on [0, +∞) with initial condition (2.5) or
(2.6), if x(t) is absolutely continuous on any compact interval of
[0, +∞) and satisfies differential inclusions
D
α
t
0
x
i
(t) ∈ −K [c
i
](x
i
(t))x
i
(t) +
n
j=1
K [a
ij
f
j
](x
j
(t))
+
n
j=1
K [b
ij
g
j
](x
j
(t − τ
ij
(t))) + I
i
.
(2.8)
Or equivalently, there exist ϑ
i
(·) ∈ K [c
i
](·), γ
ij
(·) ∈ K [a
ij
f
j
](·)
and η
ij
(·) ∈ K [b
ij
g
j
](·) satisfy
D
α
t
0
x
i
(t) = −ϑ
i
(x
i
(t))x
i
(t) +
n
j=1
γ
ij
(x
j
(t))
+
n
j=1
η
ij
(x
j
(t − τ
ij
(t))) + I
i
,
a.a. t ≥ t
0
,
y
i
(t
0
+ θ ) = ϕ
i
(θ), θ ∈ [−τ , 0],
¯η
ij
(ϕ
j
(θ)) ∈ K [b
ij
g
j
](ϕ
j
(θ)),
a.a. θ ∈ [−τ , 0].
(2.9)
Consider the DFMNN (2.1) as the driver system, the controlled
response system can be described as follows:
D
α
t
0
y
i
(t) = −c
i
(y
i
(t))y
i
(t) +
n
j=1
a
ij
(y
j
(t))f
j
(y
j
(t))
+
n
j=1
b
ij
(y
j
(t − τ
ij
(t)))g
j
(y
j
(t − τ
ij
(t)))
+ I
i
+ u
i
(t),
(2.10)
where i = 1, . . . , n, t ≥ t
0
0 < α < 1. Similarly
D
α
t
0
y
i
(t) ∈ −K [c
i
]y
i
(t) +
n
j=1
K [a
ij
f
j
](y
j
(t))
+
n
j=1
K [b
ij
g
j
](y
j
(t − τ
ij
(t)))
+ I
i
+ u
i
(t).
(2.11)
Or equivalently, there exist
¯
ϑ
i
(·) ∈ K [c
i
](·), ¯γ
ij
(·) ∈ K [a
ij
f
j
](·) and
¯η
ij
(·) ∈ K [b
ij
g
j
](·) satisfy
D
α
t
0
y
i
(t) =
¯
ϑ
i
(y
i
(t))y
i
(t) +
n
j=1
¯γ
ij
(y
j
(t))
+
n
j=1
¯η
ij
(y
j
(t − τ
ij
(t))) + I
i
+ u(t),
a.a. t ≥ t
0
,
y
i
(t
0
+ θ ) = ψ
i
(θ), θ ∈ [−τ , 0],
¯η
ij
(ψ
j
(θ)) ∈ K [b
ij
g
j
](ψ
j
(θ)),
a.a. θ ∈ [−τ , 0].
(2.12)
From Lemma 2.1, it is obvious that
K [a
ij
f
j
] ⊆ K [a
ij
]
K [f
j
],
K [b
ij
g
j
] ⊆ K [b
ij
]
K [g
j
],
K [c
i
(x
i
)x
i
] = K [c
i
](x
i
)x
i
(2.13)
for i, j = 1, 2, . . . , n. According to the definition of the map K we
can get
K [f
j
](x
j
) =
co{f
+
j
, f
−
j
}
K [g
j
](x
j
) = co{g
+
j
, g
−
j
}
(2.14)
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