In this paper we investigate the exponential stability of impulsive control for neural networks with time-varying delay by using a Lyapunov-Krasovskii functional. One numerical example is given to demonstrate the effectiveness of the obtained results

Li, Jianli

Tu, Xinni

English

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MathLAB Journal Vol 2 No 1 (2019) http://purkh.com/index.php/mathlab
Impulsive Control for Exponential Stability of Neural
Networks with Time-varying Delay
Xinni Tu, Jianli Li
Department of Mathematics, Hunan Normal University Changsha, Hunan 410081, China,
Maidan Svobody, 4, office 6-32, Kharkiv 61022, Ukraine.
e-mail: ljianli@sina.com (J. Li) 2574314427@qq.com (X. Tu)
Abstract
In this paper we investigate the exponential stability of impul-sive control for neural networks with time-varying delay
by using a Lyapunov-Krasovskii functional. One numerical example is given to demonstrate theeffectiveness of the
obtained results.
Keywords: Impulsive control; Exponential stability; Neural networks; Lyapunov-Krasovskii functional.
Mathematics Subject Classification: 34A37; 34K10.
Introduction
Neural networks has been used in various fields, such as pattern recognition, signal processing and other fields [1-2]. In
recent years, time delays has been occurred frequently. Therefore, some authors pay more attention to neural networks
with time-varying delay [3-6] , such as
x′(t) = −Cx(t) +Af(x(t)) +Bf(x(t− d(t))), (1)
where x(.) = [x1(.), x2(.), ..., xn(.)]
T ∈ Rn is the neuron state vector; f(x(.)) = [f1(x(.)),
vspace5mmf2(x(.)), ..., fn(x(.))]
T ∈ Rn denotes the neuron activation function; C = diag(c1, c2, ..., cn) is a diagonal
matrix with ci > 0; and A and B are the connection weight matrix and the delayed connection weight matrix.
respectively, the time delay d(t) is a time-varying differentiable function .
Impulsive control which reduces the control cost is an effective and ideal control technique [7-9]. Impulsive effect
exists widely in many evolutionary processes. The paper investigates a delayed neural networks with impulses, which
is neither purely continuous-time nor purely discrete-time ones.
In this paper, we consider the system (1) subjected to certain impulsive state displacements at fixed moments of
time: {
x
′
(t) = −Cx(t) +Af(x(t)) +Bf(x(t− d(t))), t 6= tk,
x(t+) = Rx(t), t = tk,
(2)
where R is positive definite block-diagonal matrix. xj(t
−
k ) = xj(tk), which mean xj(t) is left continuous at each tk.
The moments of impulsive satisfy t1 < t2 < · · · < tk < tk+1 < ... and lim
k→+∞
tk =∞.
The paper constructs a new Lyapunov-Krasovskii functional (LKF) firstly. Then, we can obtain the LKF which is
non-increasing . And a stability condition is established by integral inequality. What’s different from the exists is that
we are going to consider the impulsive differential equations. Finally, a numerical example is given to demonstrate
the effectiveness of the obtained results.
2. Preliminaries
In this paper, we assume that the time delay d(t) is a time-varying differentiable function that satisfies
0 ≤ d(t) ≤ h, d′(t) ≤ µ, (3),
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MathLAB Journal Vol 2 No 1 (2019) http://purkh.com/index.php/mathlab
where h and µ are constants.
In addition, it is assumed that each neuron activation function fj(.) satisfies the following condition
0 ≤ fj(xj)
xj
≤ Lj , fj(0) = 0,∀xj 6= 0, j = 1, 2, .., n. (4)
Definition 1The system (2) is said to be exponentially stable if there exist constants k > 0 and M ≥ 1 such that
||x(t)|| ≤Mφe−kt, (5)
where φ = sup
−h≤θ≤0
||x(θ)||, k is called the exponential convergence rate.
Lemma 1[3]For any vector a, b ∈ Rn, the inequality
2aT b ≤ aTXa+ bTX−1b (6)
holds, in which X is any positive matrix (i. e, X > 0).
Lemma 2[3] Suppose that (4) holds, then∫ u
v
[fj(s)− fj(v)]ds ≤ [u− v][fj(u)− fj(v)], j = 1, 2, ..., n. (7)
Lemma 3[4] Assume that (4) holds, then we have∫ u
v
[fj(s)− fj(v)]ds ≥ 1
2Lj
[fj(u)− fj(v)]2, j = 1, 2, ..., n. (8)
3. Main results
Theorem 1.The system (2) with (4) and a time-varying delay satisfying condition (3) is exponential stable and have
the exponential convergence rate k, if there exist P = PT > 0,M = MT > 0,W = WT > 0, N = NT > 0, D =
diag(d1, d2, ..., dn) ≥ 0, such that the following matrix is feasible
Γ1 =

Θ1
Θ2
Θ3
Θ4
 < 0,
Γ2 =
[
Θ5
Θ6
]
< 0,
where
Θ1 = 2kP − 2PC + 2PAL+ P 2 + 4kLD + e2khM + hN ,
Θ2 = −2DCL−1 + 2DA+D2 + e2khW ,
Θ3 = (µ− 1)M ,Θ4 = 2BTB + (µ− 1)W ,
Θ5 = R
TPR− P + 2RTLDR,Θ6 = −DL−1,
L = diag(L1, L2, ..., Ln).
Proof.Construct the following Lyapunov-Krasovskii functional
V (x(t)) = V1(x(t)) + V2(x(t)) + V3(x(t)),
V1(x(t)) = e
2ktxT (t)Px(t) + 2
n∑
j=1
dje
2kt
∫ xj
0
fj(s)ds,
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MathLAB Journal Vol 2 No 1 (2019) http://purkh.com/index.php/mathlab
V2(x(t)) = e
2kh
∫ t
t−d(t) e
2ks[xT (s)Mx(s) + fT (x(s))Wf(x(s))]ds,
V3(x(t)) =
∫ 0
−h
∫ t
t+θ
e2ksxT (s)Nx(s)dsdθ,
where P = PT > 0,M = MT > 0,W = WT > 0, N = NT > 0, D = diag(d1, d2, ..., dn) ≥ 0 are to be determined.
Firstly, we define the following vector
η1(t) = [x
T (t) fT (x(t)) xT (t− d(t)) fT (x(t− d(t)))],
η2(t) = [x
T (t) fT (x(t))].
For t 6= tk, by using Lemma 1 and 2, calculating the derivative of Vi(x(t))(i = 1, 2, 3) along the trajectories of the
system (2) yields.
V
′
1 (x(t)) = 2ke
2ktxT (t)Px(t) + 2e2ktxT (t)Px
′
(t) + 4
n∑
j=1
kdje
2kt
∫ xj
0
fj(s)ds
+2
n∑
j=1
dje
2ktfj(xj(t))x
′
j(t)
≤ 2ke2ktxT (t)Px(t) + 2e2ktxT (t)Px′(t) + 4ke2ktfT (x(t))Dx(t)
+2e2ktfT (x(t))Dx
′
(t)
≤ 2ke2ktxT (t)Px(t)− 2e2ktxT (t)PCx(t) + 2e2ktxT (t)PALx(t)
+e2ktxT (t)P 2x(t) + e2ktfT (x(t− d(t)))BTBf(x(t− d(t)))
+4ke2ktxT (t)LDx(t)− 2e2ktfT (x(t)DCL−1f(x(t))
+2e2ktfT (x(t))DAf(x(t)) + e2ktfT (x(t))D2f(x(t))
+e2ktfT (x(t− d(t)))BTBf(x(t− d(t)))
= e2kt[xT (t)(2kP − 2PC + 2PAL+ P 2 + 4kLD)x(t)
+fT (x(t))(−2DCL−1 + 2DA+D2)f(x(t))
+fT (x(t− d(t))(2BTB)f(x(t− d(t)))].
V
′
2 (x(t)) = e
2khe2kt(xT (t)Mx(t) + fT (x(t)Wf(x(t)))− e2khe2k(t−d(t))
(1−d′(t))[xT (t−d(t))Mx(t−d(t))+fT (x(t−d(t))Wf(x(t−d(t)))]
≤ e2kt[xT (t)e2khMx(t) + fT (x(t))e2khWf(x(t)) + xT (t− d(t))
(µ− 1)Mx(t− d(t)) + fT (x(t− d(t))(µ− 1)Wf(x(t− d(t)))].
V
′
3 (x(t)) = he
2ktxT (t)Nx(t)−
∫ t
t−h
e2ksxT (s)Nx(s)ds ≤ e2ktxT (t)hNx(t).
From the previous expression, we can get
V
′
(x(t)) ≤ e2kt[xT (t)(2kP − 2PC + 2PAL+ P 2 + 4kLD + e2khM + hN)x(t)
+fT (x(t))(−2DCL−1+2DA+D2+e2khW )f(x(t))+xT (t−d(t))(µ−1)
Mx(t− d(t)) + fT (x(t− d(t))(2BTB + (µ− 1)W )f(x(t− d(t)))]
= e2ktη1(t)Γ1η
T (t).
From Γ1 < 0 , we have V
′
(x(t)) < 0.
When t = tk(k = 1, 2, ...), by using Lemmas 2 and 3 , we can obtain
V1(x(t
+
k ))− V1(x(tk)) = e2kt
+
k xT (t+k )Px(t
+
k ) + 2
n∑
j=1
dje
2kt+
k
∫ x+
j
0
fj(s)ds
−e2ktkxT (tk)Px(tk)− 2
n∑
j=1
dje
2ktk
∫ xj
0
fj(s)ds
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MathLAB Journal Vol 2 No 1 (2019) http://purkh.com/index.php/mathlab
≤ e2ktkxT (tk)(RTPR− P )x(tk) + 2e2ktkfT (x(t+k ))Dx(t+k )
−e2ktkfT (x(tk))DL−1f(x(tk))
= e2ktk[xT(tk)(R
TPR−P+2RTLDR)x(tk)−fT(x(tk))DL−1f(x(tk))],
V2(x(t
+
k ))− V2(x(tk)) = e2kh
∫ t+
k
t+
k
−d(t+
k
)
e2ks(xT (s)Mx(s) + fT (x(s))Wf(x(s)))ds
−e2kh
∫ tk
tk−d(tk)
e2ks(xT (s)Mx(s) + fT (x(s))Wf(x(s)))ds
= 0,
V3(x(t
+
k ))− V3(x(tk)) =
∫ 0
−h
∫ t+
k
t+
k
+θ
e2ksxT (s)Nx(s)dsdθ −
∫ 0
−h
∫ tk
tk+θ
e2ksxT (s)Nx(s)dsdθ
= 0,
then, we can obtain
V (x(t+k ))− V (x(tk)) ≤ e2ktk [xT (tk)(RTPR−P+2RTLDR)x(tk)−fT(x(tk))DL−1f(x(tk))]
= e2ktkη2(tk)Γ2η
T
2 (tk),
Γ2 < 0 implies V (x(t
+
k )) ≤ V (x(tk)).
It follows from V
′
(x(t)) < 0 and V (x(t+k )) ≤ V (x(tk)) that V (x(t)) ≤ V (x(0)) for any t ≥ 0.
However, from Lemma 2 we have
V (x(0)) = xT (0)Px(0) + 2
n∑
j=1
dj
∫ xj
0
fj(s)ds+ e
2kh
∫ 0
−d(0)
e2ks[xT (t)Mx(t)
+fT (x(t))Wf(x(t))]ds+
∫ 0
−h
∫ 0
θ
e2ksxT (t)Nx(t)dsdθ
≤ λmax(P )||φ||2 + 2
n∑
j=1
djxj(0)fj(xj(0)) + e
2khλmax(M)
∫ 0
−d(0)
xT (s)x(s)ds
+e2khλmax(W )
∫ 0
−d(0)
fT (x(s))f(x(s))ds+ λmax(N)
∫ 0
−h
∫ 0
θ
xT (s)x(s)dsdθ
≤ λmax(P )||φ||2 + 2λmax(DL)||φ||2 + he2khλmax(M)||φ||2
+he2khλmax(W )λmax(L
2)||φ||2 + h2λmax(N)||φ||2
= Λ||φ||2,
where
Λ = λmax(P ) + 2λmax(DL) + he
2khλmax(M) + he
2khλmax(W )λmax(L
2) + h2λmax(N),
φ = sup
−h≤θ≤0
||x(θ)||.
On the other hand, we get
V (x(t)) ≥ e2ktxT (t)Px(t) ≥ e2ktλmin(P )||x(t)||2.
Therefore
||x(t)|| ≤
√
Λ
λmin(P )
||φ||e−kt,
which shows that system (2) is exponentially stable and has the exponential convergence rate k. The proof is com-
pleted.
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MathLAB Journal Vol 2 No 1 (2019) http://purkh.com/index.php/mathlab
4. Examples
Consider the following neural networks system{
x
′
(t) = −Cx(t) +Af(x(t)) +Bf(x(t− d(t))), t 6= tk
x(t+) = Rx(t), t = tk,
where
A =
[ −1 12
1
2 −1
]
, C = diag( 2, 3 ),
B =
[
1
2
1
2
1
2 − 12
]
, L1 = L2 = 1,
R =
[
1
2
1
2
]
.
.
Then we can construct a LKF. Let
M =
[
1 12
1
2
1
2
]
, P = D = diag( 1, 1 ),
N =
[
1
2
1
4
1
4
1
4
]
, k =
1
8
,
W =
[
8
3 1
1 83
]
, µ =
1
4
.
It follows from Γ1 < 0 and Γ2 < 0 that the all conditions of Theorem 1 are satisfied, so the neural networks system is
exponential stability.
Conclusions
We investigate the impulsive neural networks with time-varying delay. By constructing a new Lyapunov-Krasovskii
functional (LKF) , a exponential stability condition is established, which extend some previous results. One numerical
example is given to demonstrate the effectiveness of the obtained results.
Conflicts of Interest
Authors declare that there is no conflict of interest.
Funding Statement
This work is supported by the NSFC(11571088, 11471109), the Zhejiang Provincial Natural Science Foundation of
China ( LY14A010024), and Scientific Research Fund of Hunan Provincial Education Department ( 14A098).
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Impulsive Control for Exponential Stability of Neural Networks with Time-varying Delay

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Impulsive Control for Exponential Stability of Neural Networks with Time-varying Delay

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