Stability analysis for impulsive stochastic fuzzy p-Laplace dynamic equations under Neumann or Dirichlet boundary condition

Boundary Value Problems20132013:133

DOI: 10.1186/1687-2770-2013-133

Received: 3 March 2013

Accepted: 4 May 2013

Published: 20 May 2013

Abstract

Under Neumann or Dirichlet boundary conditions, the stability of a class of delayed impulsive Markovian jumping stochastic fuzzy p-Laplace partial differential equations (PDEs) is considered. Thanks to some methods different from those of previous literature, the difficulties brought by fuzzy stochastic mathematical model and impulsive model have been overcome. By way of the Lyapunov-Krasovskii functional, Itô formula, Dynkin formula and a differential inequality, new LMI-based global stochastic exponential stability criteria for the above-mentioned PDEs are established. Some applications of the obtained results improve some existing results on neural networks. And some numerical examples are presented to illustrate the effectiveness of the proposed method due to the significant improvement in the allowable upper bounds of time delays.

MSC:34D20, 34D23, 34B45, 34B37, 34K20.

Keywords

differential inequality Laplace diffusion Markovian jumping

1 Introduction

In this paper, we are concerned with the following delayed impulsive Markovian jumping stochastic fuzzy p-Laplace partial differential equations (PDEs):
{ d v i ( t , x ) = [ k = 1 m x k ( D i k ( t , x , v ) | v i ( t , x ) | p 2 v i x k ) d v i ( t , x ) = b i ( v i ( t , x ) ) + j = 1 n c ˆ i j ( r ( t ) ) F j ( v j ( t , x ) ) d v i ( t , x ) = + j = 1 n c ˇ i j ( r ( t ) ) F j ( v j ( t , x ) ) + j = 1 n d ˆ i j ( r ( t ) ) G j ( v j ( t τ j ( t ) , x ) ) d v i ( t , x ) = + j = 1 n d ˇ i j ( r ( t ) ) G j ( v j ( t τ j ( t ) , x ) ) ] d t d v i ( t , x ) = + j = 1 n σ i j ( v j ( t , x ) , v j ( t τ j ( t ) ) , x ) d w j ( t ) , d v i ( t , x ) = for all  t 0 , t t k , x Ω , v ( t + , x ) = M k ( r ( t ) ) v ( t , x ) + N ( r ( t ) ) h ( v ( t τ ( t ) , x ) ) , t = t k , k = 1 , 2 , , v ( θ , x ) = ϕ ( θ , x ) , ( θ , x ) [ τ , 0 ] × Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ1_HTML.gif
(1.1)
equipped with the boundary condition
B [ v i ( t , x ) ] = 0 , ( t , x ) [ τ , + ) × Ω , i = 1 , 2 , , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ2_HTML.gif
(1.1a)
where p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq1_HTML.gif is a positive scalar, Ω R m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq2_HTML.gif is a bounded domain with a smooth boundary Ω of class C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq3_HTML.gif by Ω , v ( t , x ) = ( v 1 ( t , x ) , v 2 ( t , x ) , , v n ( t , x ) ) T R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq4_HTML.gif. The smooth functions D j k ( t , x , v ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq5_HTML.gif. Denote τ ( t ) = ( τ 1 ( t ) , τ 2 ( t ) , , τ n ( t ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq6_HTML.gif, and τ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq7_HTML.gif ( 0 τ j ( t ) τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq8_HTML.gif) corresponds to the transmission delays at time t. v ( t τ ( t ) , x ) = ( v 1 ( t τ 1 ( t ) , x ) , v 2 ( t τ 2 ( t ) , x ) , , v n ( t τ n ( t ) , x ) ) T R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq9_HTML.gif. t k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq10_HTML.gif is called impulsive moment, satisfying 0 < t 1 < t 2 < < t k < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq11_HTML.gif with lim k t k = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq12_HTML.gif. v ( t k + , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq13_HTML.gif and v ( t k , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq14_HTML.gif denote the left-hand and right-hand limits at  t k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq10_HTML.gif, respectively. h ( v ( t k τ ( t ) , x ) ) = ( h 1 ( v 1 ( t k τ 1 ( t k ) ) ) , h 2 ( v 2 ( t k τ 2 ( t k ) ) ) , , h n ( v n ( t k τ n ( t k ) ) ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq15_HTML.gif, and h j ( v j ( t k τ j ( t k ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq16_HTML.gif is the impulsive perturbation at time t k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq10_HTML.gif. We always assume v ( t k + , x ) = v ( t k , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq17_HTML.gif. b j ( v j ( t , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq18_HTML.gif, F j ( v j ( t , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq19_HTML.gif and G j ( v j ( t , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq20_HTML.gif are continuous functions. ⋀ and ⋁ denote the fuzzy AND and OR operation, respectively. Each w j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq21_HTML.gif is scalar standard Brownian motion defined on a complete probability space ( Ω , F , P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq22_HTML.gif with a natural filtration { F t } t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq23_HTML.gif. The noise perturbation σ i j : R × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq24_HTML.gif is a Borel measurable function. { r ( t ) , t 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq25_HTML.gif is a right-continuous Markov process on the probability space which takes values in the finite space S = { 1 , 2 , , N } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq26_HTML.gif with generator Π = { π i j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq27_HTML.gif given by
P ( r ( t + δ ) = j r ( t ) = i ) = { π i j δ + o ( δ ) , j i , 1 + π i j δ + o ( δ ) , j = i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equa_HTML.gif

where π i j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq28_HTML.gif is transition probability rate from i to j ( j i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq29_HTML.gif) and π i i = j = 1 , j i s π i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq30_HTML.gif, δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq31_HTML.gif and lim δ 0 o ( δ ) / δ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq32_HTML.gif. In addition, the transition rates of the Markovian chain are considered to be partially available, namely, some elements in transition rates matrix Π are time-invariant but unknown. For notational clarity, we denote S = S k n i S u n i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq33_HTML.gif with S k n i { j ,  if  π i j  is known } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq34_HTML.gif and S u n i { j ,  if  π i j  is unknown, and  j i } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq35_HTML.gif for a given i S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq36_HTML.gif. ϱ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq37_HTML.gif is a nonnegative scalar, satisfying ϱ i max j S u n i π i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq38_HTML.gif for any given i S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq36_HTML.gif. In mode r ( t ) = r S = { 1 , 2 , , N } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq39_HTML.gif, we denote c ˆ i j ( r ( t ) ) = c ˆ i j ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq40_HTML.gif, d ˆ i j ( r ( t ) ) = d ˆ i j ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq41_HTML.gif, c ˇ i j ( r ( t ) ) = c ˇ i j ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq42_HTML.gif and d ˇ i j ( r ( t ) ) = d ˇ i j ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq43_HTML.gif. Besides, impulse parameters matrices M k ( r ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq44_HTML.gif and N ( r ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq45_HTML.gif are denoted by M k r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq46_HTML.gif and N r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq47_HTML.gif for convenience. The boundary condition (1.1a) is called the Dirichlet boundary condition if B [ v i ( t , x ) ] = v i ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq48_HTML.gif, and the Neumann boundary condition if B [ v i ( t , x ) ] = v i ( t , x ) ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq49_HTML.gif, where v i ( t , x ) ν = ( v i ( t , x ) x 1 , v i ( t , x ) x 2 , , v i ( t , x ) x m ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq50_HTML.gif denotes the outward normal derivative on Ω.

Remark 1.1 PDEs (1.1) own a wide range of physics and engineering backgrounds. They admit the following three Cohen-Grossberg neural networks (CGNNs) as their special cases.
{ d v ( t , x ) = { ( D ( t , x , v ) p v ( t , x ) ) A ( v ( t , x ) ) [ B ( v ( t , x ) ) C f ( v ( t , x ) ) d v ( t , x ) = D g ( v ( t τ ( t ) , x ) ) ] } d t + σ ( v ( t , x ) , v ( t τ ( t ) ) ) d w ( t ) d v ( t , x ) = for all  t 0 , t t k , x Ω , v ( t k + , x ) = M k v ( t k , x ) + N h ( v ( t k τ ( t k ) , x ) ) , k = 1 , 2 , , v ( θ , x ) = ϕ ( θ , x ) , ( θ , x ) [ τ ( t ) , 0 ] × Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ3_HTML.gif
(1.2)
{ v ( t , x ) t = D Δ v ( t , x ) A ( v ( t , x ) ) [ B ( v ( t , x ) ) C f ( v ( t , x ) ) D g ( v ( t τ ( t ) , x ) ) ] , v ( t , x ) t = for all  t t 0 , t t k , x Ω , v ( t k + , x ) = M k v ( t k , x ) + N h ( v ( t k τ ( t k ) , x ) ) , k = 1 , 2 , , v ( θ , x ) = ϕ ( θ , x ) , ( θ , x ) [ τ ( t ) , 0 ] × Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ4_HTML.gif
(1.3)
{ d x i ( t ) = { a i ( x i ( t ) ) [ b i ( x i ( t ) ) j = 1 n c ˆ i j f j ( x j ( t ) ) j = 1 n c ˇ i j f j ( x j ( t ) ) d x i ( t ) = j = 1 n d ˆ i j g j ( x j ( t τ ) ) j = 1 n d ˇ i j g j ( x j ( t τ ) ) ] } d t d x i ( t ) = + j = 1 n σ i j ( x j ( t ) , x j ( t τ ) ) d w j ( t ) , x ( t k + ) = M k x ( t k ) + N h ( v ( t k τ ( t k ) ) ) , k = 1 , 2 , , x i ( t ) = ϕ i ( t ) , τ t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ5_HTML.gif
(1.4)

where D ( t , x , v ) p v ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq51_HTML.gif denotes the Hadamard product of matrix D ( t , x , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq52_HTML.gif and p v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq53_HTML.gif (see, [1] or [2]), and D ( t , x , v ) = ( D j k ( t , x , v ) ) n × m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq54_HTML.gif satisfies D j k ( t , x , v ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq5_HTML.gif for all j, k, (t, x, v). σ ( v ( t , x ) , v ( t τ ( t ) ) ) = ( σ i j ( v j ( t , x ) , v j ( t τ j ( t ) ) ) ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq55_HTML.gif, and w ( t ) = ( w 1 ( t ) , w 2 ( t ) , , w n ( t ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq56_HTML.gif. Throughout this paper, for the mode r ( t ) = r S = { 1 , 2 , , N } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq39_HTML.gif, we denote C ( r ( t ) ) = C r = ( c i j ( r ) ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq57_HTML.gif, and D ( r ( t ) ) = D r = ( d i j ( r ) ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq58_HTML.gif. Δ v ( t , x ) = ( Δ v 1 ( t , x ) , Δ v 2 ( t , x ) , , Δ v n ( t , x ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq59_HTML.gif, and Δ v j ( t , x ) = k = 1 m x k ( v j ( t , x ) x k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq60_HTML.gif. x ( t ) = ( x 1 ( t ) , x 2 ( t ) , , x n ( t ) ) T R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq61_HTML.gif.

The stability of p-Laplace diffusion stochastic CGNNs (1.2) was discussed by Xiongrui Wang, Ruofeng Rao and Shouming Zhong in 2012 [2], and the stability of deterministic system (1.3) was investigated by Xinhua Zhang, Shulin Wu and Kelin Li in 2011 [3]. Impulsive fuzzy CGNNs with nonlinear p-Laplace diffusion has never been studied as far as we know, and such a situation motivates our present study. Both the nonlinear p-Laplace diffusion and fuzzy mathematical model bring a great difficulty in setting up LMI criteria for the stability, and the stochastic functional differential equations model with nonlinear diffusion makes it harder. To study the stability of fuzzy CGNNs with diffusion, we have to construct a Lyapunov-Krasovskii functional in a non-matrix form (see, e.g., [4]). But stochastic mathematical formulae are always described in matrix forms. Furthermore, an impulsive model makes it harder. Recently, some new methods were employed to study the exponential stability for Markovian jumping, fuzzy neural networks in some related literature (see, e.g., [514]). Inspired by some methods and the idea of [3, 4] and the other above-mentioned papers, we overcame the difficulties brought by the Markovian jumping fuzzy impulsive model. By way of the Lyapunov-Krasovskii functional, Itô formula, Dynkin formula, the variational methods in Sobolev space W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq62_HTML.gif (Lemma 2.1), and a differential inequality, new LMI-based global exponential stability criteria for the above-mentioned PDEs are established; we obtain an LMI-based global stochastic exponential stability criterion of PDEs (1.1). Some applications to neural networks improve some existing results, which are illustrated by some numerical examples thanks to the significant improvement in the allowable upper bounds of time delays.

2 Preliminaries

Throughout this paper, we always assume that the following five conditions hold.

(H1) There exists a positive definite diagonal matrix B = diag ( B 1 , B 2 , , B n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq63_HTML.gif such that
b j ( r ) r B j , j = 1 , 2 , , n ,  and  0 r R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equb_HTML.gif
(H2) There exist positive definite diagonal matrices F = diag ( F 1 , F 2 , , F n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq64_HTML.gif, G = diag ( G 1 , G 2 , , G n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq65_HTML.gif and H = diag ( H 1 , H 2 , , H n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq66_HTML.gif such that
| F j ( r 1 ) F j ( r 2 ) | F j | r 1 r 2 | , | G j ( r 1 ) G j ( r 2 ) | G j | r 1 r 2 | , | h j ( r 1 ) h j ( r 2 ) | H j | r 1 r 2 | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equc_HTML.gif

for all r 1 , r 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq67_HTML.gif, j = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq68_HTML.gif.

(H3) There exist nonnegative symmetric matrices U = ( μ i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq69_HTML.gif and V = ( ν i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq70_HTML.gif such that
trace [ σ T ( u , v ) σ ( u , v ) ] u T U u + v T V v , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equd_HTML.gif

where u , v R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq71_HTML.gif, σ ( u , v ) = ( σ i j ( u , v ) ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq72_HTML.gif.

(H4) b j ( 0 ) = F j ( 0 ) = G j ( 0 ) = h j ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq73_HTML.gif, σ i j ( 0 , 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq74_HTML.gif, i , j = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq75_HTML.gif.

It is obvious from (H4) that system (1.1) admits a zero solution v ( t , x ; 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq76_HTML.gif corresponding to the initial data ϕ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq77_HTML.gif. For simplicity, we write v ( t , x ; ϕ ) = v ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq78_HTML.gif. Denote B ( v ( t , x ) ) = ( b 1 ( v 1 ( t , x ) ) , b 2 ( v 2 ( t , x ) ) , , b n ( v n ( t , x ) ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq79_HTML.gif, F ( v ( t , x ) ) = ( F 1 ( v 1 ( t , x ) ) , F 2 ( v 2 ( t , x ) ) , , F n ( v n ( t , x ) ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq80_HTML.gif, G ( v ( t , x ) ) = ( G 1 ( v 1 ( t , x ) ) , , G n ( v n ( t , x ) ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq81_HTML.gif. Denote σ ( t ) = σ ( v ( t , x ) , v ( t τ ( t ) , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq82_HTML.gif for short.

For convenience’s sake, we introduce the following standard notations similar to those of [2].

L 2 ( R × Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq83_HTML.gif, L F 0 2 ( [ τ , 0 ] × Ω ; R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq84_HTML.gif, Q = ( q i j ) n × n > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq85_HTML.gif (<0), Q = ( q i j ) n × n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq86_HTML.gif (⩽0), Q 1 Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq87_HTML.gif ( Q 1 Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq88_HTML.gif), Q 1 > Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq89_HTML.gif ( Q 1 < Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq90_HTML.gif), λ max ( Φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq91_HTML.gif, λ min ( Φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq92_HTML.gif, the identity matrix I and the symmetric terms ∗.

In addition, we denote | C | = ( | c i j | ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq93_HTML.gif for any matrix C = ( c i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq94_HTML.gif; | u ( t , x ) | = ( | u 1 ( t , x ) | , | u 2 ( t , x ) | , , | u n ( t , x ) | ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq95_HTML.gif for any u ( t , x ) = ( u 1 ( t , x ) , u 2 ( t , x ) , , u n ( t , x ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq96_HTML.gif.

Next, we give the following lemma, which is completely similar to [[1], Lemma 2.3]. It can be derived by the Gauss formula (see, e.g., [2]).

Lemma 2.1 ([[1], Lemma 2.3], [[11], Lemma 6])

Let P = diag ( p 1 , p 2 , , p n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq97_HTML.gif be a positive definite matrix, and let v be a solution of system (1.1) with the boundary condition (1.1a). Then we have
Ω v T P ( ( D ( t , x , v ) p v ) ) d x = k = 1 m j = 1 n Ω p j D j k ( t , x , v ) | v j | p 2 ( v j x k ) 2 d x = Ω ( ( D ( t , x , v ) p v ) ) T P v d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Eque_HTML.gif

3 Main results

Theorem 3.1 Assume that p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq1_HTML.gif. If the following three conditions hold:

(C1) there exist a sequence of positive scalars α ¯ r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq98_HTML.gif ( r S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq99_HTML.gif) and positive definite diagonal matrices P r = diag ( p r 1 , p r 2 , , p r n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq100_HTML.gif ( r S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq99_HTML.gif) such that the following LMI conditions hold:
Θ r > 0 , r S , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ6_HTML.gif
(3.1)
P r < α ¯ r I , r S , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ7_HTML.gif
(3.2)
where matrices C ˆ r = ( c ˆ i j ( r ) ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq101_HTML.gif, D ˆ r = ( d ˆ i j ( r ) ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq102_HTML.gif, C ˇ r = ( c ˇ i j ( r ) ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq103_HTML.gif, D ˇ r = ( d ˇ i j ( r ) ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq104_HTML.gif, | d ˆ ( r ) | = max i , j | d ˆ i j ( r ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq105_HTML.gif, G 0 = max j G j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq106_HTML.gif, | d ˇ ( r ) | = max i , j | d ˇ i j ( r ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq107_HTML.gif, and
Θ r = ( j S k n r π r j P j + ϱ r j S u n r P j 2 P r B + P r ( | C ˆ r | + | C ˇ r | ) F + F ( | C ˆ r T | + | C ˇ r T | ) P r + n ( | d ˆ ( r ) | + | d ˇ ( r ) | ) G 0 P r + α ¯ r U ) ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equf_HTML.gif

(C2) a > b 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq108_HTML.gif, where a = min r S ( λ min Θ r λ max P r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq109_HTML.gif, b = max r S ( λ max ( α ¯ r V + n ( | d ˆ ( r ) | + | d ˇ ( r ) | ) G λ max P r ) λ min P r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq110_HTML.gif;

(C3) there exists a constant δ > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq111_HTML.gif such that inf k Z ( t k t k 1 ) > δ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq112_HTML.gif, δ 2 τ > ln ( ρ e λ τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq113_HTML.gif and λ ln ( ρ e λ τ ) δ τ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq114_HTML.gif, where ρ = max { 1 , a j + b j e λ τ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq115_HTML.gif with a j sup r , j ( 2 λ max ( | M j r T | P r | M j r | ) λ min P r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq116_HTML.gif, b j max r S ( 2 λ max ( H | N r T | P r | N r | H ) λ min P r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq117_HTML.gif for all j Z = { 1 , 2 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq118_HTML.gif, and λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq119_HTML.gif is the unique solution of the equation λ = a b e λ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq120_HTML.gif.

Then the null solution of impulsive Markovian jumping stochastic fuzzy system (1.1) is globally stochastically exponentially stable in the mean square with the convergence rate 1 2 ( λ ln ( ρ e λ τ ) δ τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq121_HTML.gif.

Proof Consider the Lyapunov-Krasovskii functional
V ( t , v ( t ) , r ) = Ω i = 1 n p r i v i 2 ( t , x ) d x , r S , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equg_HTML.gif

where v ( t , x ) = ( v 1 ( t , x ) , v 2 ( t , x ) , , v n ( t , x ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq122_HTML.gif is a solution for stochastic fuzzy system (1.1). Sometimes we may denote v ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq123_HTML.gif by v, v i ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq124_HTML.gif by v i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq125_HTML.gif, and σ ( v ( t , x ) , v ( t τ ( t ) , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq126_HTML.gif by σ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq127_HTML.gif for simplicity.

Let ℒ be the weak infinitesimal operator. Then it follows by Lemma 2.1 that
L V ( t , v ( t ) , r ) = 2 k = 1 m i = 1 n Ω p r i D i k ( t , x , v ) | v i | p 2 ( v i x k ) 2 d x 2 i = 1 n Ω p r i v i [ b i ( v i ) j = 1 n c ˆ i j ( r ) F j ( v j ) j = 1 n c ˇ i j ( r ) F j ( v j ) j = 1 n d ˆ i j ( r ) G j ( v j ( t τ ( t ) , x ) ) j = 1 n d ˇ i j ( r ) G j ( v j ( t τ ( t ) , x ) ) ] d x + Ω v T j S π r j P j v d x + Ω trace ( σ T ( t ) P r σ ( t ) ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ8_HTML.gif
(3.3)
On the other hand, we have
2 i = 1 n p r i | v i | j = 1 n | d ˆ i j ( r ) | G j | v j ( t τ ( t ) , x ) | v T ( n | d ˆ ( r ) | G 0 P r ) v + v T ( t τ ( t ) , x ) ( n | d ˆ ( r ) | α ¯ r G ) v ( t τ ( t ) , x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ9_HTML.gif
(3.4)
2 i = 1 n p r i | v i | j = 1 n | d ˇ i j ( r ) | G j | v j ( t τ ( t ) , x ) | v T ( n | d ˇ ( r ) | G 0 P r ) v + v T ( t τ ( t ) , x ) ( n | d ˇ ( r ) | α ¯ r G ) v ( t τ ( t ) , x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ10_HTML.gif
(3.5)

From π r r < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq128_HTML.gif and the definition of ϱ r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq129_HTML.gif, it is clear that j S π r j P j j S k n r π r j P j + ϱ r j S u n r P j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq130_HTML.gif.

So, we can conclude by (H1)-(H4)
L V ( t , v ( t ) , r ) Ω | v T | ( j S k n r π r j P j + ϱ r j S u n r P j 2 P r B + 2 P r ( | C ˆ r | + | C ˇ r | ) F + n ( | d ˆ ( r ) | + | d ˇ ( r ) | ) G 0 P r + α ¯ r U ) | v | d x + Ω α ¯ r [ v T ( t τ ( t ) , x ) ( V + n ( | d ˆ ( r ) | + | d ˇ ( r ) | ) G ) v ( t τ ( t ) , x ) ] d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ11_HTML.gif
(3.6)
Completely similar to (2.7)-(2.9) in [2], we can get by the Itô formula
D + V ( t , v ( t ) , r ) Ω | v T | Θ r | v | d x + Ω α ¯ r [ v T ( t τ ( t ) , x ) ( V + n ( | d ˆ ( r ) | + | d ˇ ( r ) | ) G ) v ( t τ ( t ) , x ) ] d x min r S ( λ min Θ r λ max P r ) V ( t , v ( t ) , r ) + max r S ( λ max ( α ¯ r V + n ( | d ˆ ( r ) | + | d ˇ ( r ) | ) G λ max P r ) λ min P r ) [ V ( t , v ( t ) , r ) ] τ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ12_HTML.gif
(3.7)
Owing to V ( t , v ( t ) , r ) = Ω | v T | P r | v | d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq131_HTML.gif, we can get
V ( t k , v ( t k ) , r ) = Ω v T ( t k , x ) P r v ( t k , x ) d x = Ω ( M k r v ( t k , x ) + N r h ( v ( t k τ ( t k ) , x ) ) ) T P r ( M k r v ( t k , x ) + N r h ( v ( t k τ ( t k ) , x ) ) ) d x 2 λ max ( | M k r T | P r | M k r | ) λ min P r V ( t k , v ( t k ) , r ) + 2 λ max ( H | N r T | P r | N r | H ) λ min P r Ω | v T ( t k τ ( t k ) , x ) | P r | v ( t k τ ( t k ) , x ) | d x a k V ( t k , v ( t k ) , r ) + b k [ V ( t k , v ( t k ) , r ) ] τ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ13_HTML.gif
(3.8)
From (C3), it is not difficult to conclude that ρ k + 1 e k λ τ e ( δ 2 λ ) τ e δ t k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq132_HTML.gif, where λ, a k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq133_HTML.gif, b k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq134_HTML.gif, ρ are defined in (C3), and so ρ = max k Z { 1 , a k + b k e λ τ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq135_HTML.gif. Then, by (C2), the differential inequality lemma ([[2], Lemma 1.6]) yields E V ( t ) ρ [ E V ( 0 ) ] τ e ( λ ln ( ρ e λ τ ) δ τ ) t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq136_HTML.gif, or
( λ min P r ) E v ( t ) 2 2 ρ ( λ max P r sup τ s 0 E ϕ ( s ) 2 2 ) e ( λ ln ( ρ e λ τ ) δ τ ) t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ14_HTML.gif
(3.9)
i.e.,
E v ( t ) 2 2 ρ [ max r S ( λ max P r λ min P r ) sup τ s 0 E ϕ ( s ) 2 2 ] e ( λ ln ( ρ e λ τ ) δ τ ) t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ15_HTML.gif
(3.10)

Therefore, we can see by the definition of global stochastic exponential stability (see, e.g., [15]) that the null solution of impulsive Markovian jumping stochastic fuzzy system (1.1) is globally stochastically exponentially stable in the mean square with the convergence rate 1 2 ( λ ln ( ρ e λ τ ) δ τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq121_HTML.gif. □

Particularly for the case of p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq137_HTML.gif, we get from the Poincaré inequality (see, e.g., [[16], Lemma 2.4]) that λ 1 Ω | v i ( t , x ) | 2 d x Ω | v i ( t , x ) | 2 d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq138_HTML.gif, where λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq139_HTML.gif is the lowest positive eigenvalue of the boundary value problem
{ Δ φ ( t , x ) = λ φ ( t , x ) , x Ω , B [ φ ( t , x ) ] = 0 , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equh_HTML.gif

Theorem 3.2 Let p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq137_HTML.gif, and D 0 = inf i , k , ( t , x v ) D i k ( t , x , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq140_HTML.gif. Then all the conclusions of Theorem  3.1 are true if its conditions are satisfied except that the Θ r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq141_HTML.gif is replaced by Θ r = ( 2 λ 1 D 0 P r + j S k n r π r j P j + ϱ r j S u n r P j 2 P r B + P r ( | C ˆ r | + | C ˇ r | ) F + F ( | C ˆ r T | + | C ˇ r T | ) P r + n ( | d ˆ ( r ) | + | d ˇ ( r ) | ) G 0 P r + α ¯ r U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq142_HTML.gif.

Proof Indeed, if p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq137_HTML.gif, we can get by the Poincaré inequality
2 k = 1 m i = 1 n Ω p r i D i k ( t , x , v ) | v i | p 2 ( v i x k ) 2 d x = 2 k = 1 m i = 1 n Ω p r i D i k ( t , x , v ) ( v i x k ) 2 d x 2 D 0 Ω i = 1 n p r i k = 1 m ( v i x k ) 2 d x 2 λ 1 D 0 Ω v T P r v d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equi_HTML.gif

Then, by (3.3), we can similarly complete the rest of the proof by way of the methods in (3.4)-(3.10). □

4 Applications of main results in neural networks

Let B ( v ( t , x ) ) = A ( v ( t , x ) ) B ( v ( t , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq143_HTML.gif with A ( v ( t , x ) ) = diag ( a 1 ( v 1 ( t , x ) ) , a 2 ( v 2 ( t , x ) ) , , a n ( v n ( t , x ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq144_HTML.gif and B ( v ( t , x ) ) = ( b 1 ( v 1 ( t , x ) ) , b 2 ( v 2 ( t , x ) ) , , b n ( v n ( t , x ) ) ) T R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq145_HTML.gif, F ( v ( t , x ) ) = A ( v ( t , x ) ) f ( v ( t , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq146_HTML.gif, and G ( v ( t , x ) ) = A ( v ( t , x ) ) g ( v ( t , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq147_HTML.gif satisfy the following.

(H1) There exist positive definite diagonal matrices A ̲ = diag ( a ̲ 1 , a ̲ 2 , , a ̲ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq148_HTML.gif and A ¯ = diag ( a ¯ 1 , a ¯ 2 , , a ¯ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq149_HTML.gif such that
0 < a ̲ i a i ( r ) a ¯ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equj_HTML.gif

for all r R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq150_HTML.gif, i = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq151_HTML.gif.

(H2) There exists a positive definite diagonal matrix B = diag ( B 1 , B 2 , , B n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq152_HTML.gif such that
b j ( r ) r B j , j = 1 , 2 , , n ,  and  0 r R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equk_HTML.gif
(H3) There exist positive definite diagonal matrices F = diag ( F 1 , F 2 , , F n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq64_HTML.gif, G = diag ( G 1 , G 2 , , G n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq153_HTML.gif and H = diag ( H 1 , H 2 , , H n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq66_HTML.gif such that
| f j ( r 1 ) f j ( r 2 ) | F j | r 1 r 2 | , | g j ( r 1 ) g j ( r 2 ) | G j | r 1 r 2 | , | h j ( r 1 ) h j ( r 2 ) | H j | r 1 r 2 | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equl_HTML.gif

for all r 1 , r 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq67_HTML.gif, j = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq68_HTML.gif.

(H4) There exist nonnegative symmetric matrices U = ( μ i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq154_HTML.gif and V = ( ν i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq70_HTML.gif such that
trace [ σ T ( u , v ) σ ( u , v ) ] u T U u + v T V v , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equm_HTML.gif

where u , v R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq71_HTML.gif, σ ( u , v ) = ( σ i j ( u , v ) ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq72_HTML.gif.

(H5) b j ( 0 ) = f j ( 0 ) = g j ( 0 ) = h j ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq155_HTML.gif, σ i j ( 0 , 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq156_HTML.gif, i , j = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq75_HTML.gif.

Applying our main results to Cohen-Grossberg neural networks (CGNNs), we can conclude the following corollary from Theorem 3.1 directly.

Corollary 4.1 If the following three conditions hold:

(D1) there exist a positive scalar α ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq157_HTML.gif and a positive definite diagonal matrix P such that the following LMI conditions hold:
Θ = ( 2 P A ̲ B + P A ¯ ( | C ˆ | + | C ˇ | ) F + F ( | C ˆ T | + | C ˇ T | ) A ¯ P + n ( | d ˆ | + | d ˇ | ) G 0 P A ¯ + α ¯ U ) > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ16_HTML.gif
(4.1)
P < α ¯ I , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ17_HTML.gif
(4.2)

where matrices C ˆ = ( c ˆ i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq158_HTML.gif, D ˆ = ( d ˆ i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq159_HTML.gif, C ˇ = ( c ˇ i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq160_HTML.gif, D ˇ = ( d ˇ i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq161_HTML.gif, | d ˆ | = max i , j | d ˆ i j | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq162_HTML.gif, A ¯ 0 = max i a ¯ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq163_HTML.gif, G 0 = max j G j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq106_HTML.gif, | d ˇ | = max i , j | d ˇ i j | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq164_HTML.gif;

(D2) a > b 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq108_HTML.gif, where a = λ min Θ λ max P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq165_HTML.gif, b = λ max ( α ¯ V + n ( | d ˆ | + | d ˇ | ) A ¯ 0 G λ max P ) λ min P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq166_HTML.gif;

(D3) there exists a constant δ > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq111_HTML.gif such that inf k Z ( t k t k 1 ) > δ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq112_HTML.gif, δ 2 τ > ln ( ρ e λ τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq113_HTML.gif and λ ln ( ρ e λ τ ) δ τ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq114_HTML.gif, where ρ = max { 1 , a j + b j e λ τ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq115_HTML.gif with a j sup j Z ( 2 λ max ( | M j T | P | M j | ) λ min P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq167_HTML.gif, b j 2 λ max ( H | N T | P | N | H ) λ min P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq168_HTML.gif for all j Z = { 1 , 2 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq118_HTML.gif, and λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq119_HTML.gif is the unique solution of the equation λ = a b e λ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq120_HTML.gif.

Then the null solution of impulsive stochastic fuzzy system (1.4) is globally stochastically exponentially stable in the mean square with the convergence rate 1 2 ( λ ln ( ρ e λ τ ) δ τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq121_HTML.gif.

Remark 4.1 Corollary 4.1 not only extends [[4], Theorem 3.1] from non-impulsive stochastic fuzzy CGNNs to impulsive stochastic fuzzy CGNNs, but also improves the criterion of [[4], Theorem 3.1] from the non-matrix form to the more condensed matrix form, which can be efficiently tested and verified by computer Matlab LMI toolbox.

If Markovian jumping and fuzzy factors are ignored, we can conclude the following corollary.

Corollary 4.2 Assume that p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq1_HTML.gif. If the following three conditions hold:

(E1) there exist a sequence of positive scalars α ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq157_HTML.gif and positive definite diagonal matrices P such that the following LMI conditions hold:
Θ ˜ > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ18_HTML.gif
(4.3)
P < α ¯ I , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ19_HTML.gif
(4.4)

where matrices C = ( c i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq94_HTML.gif, D = ( d i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq169_HTML.gif, | d | = max i , j | d i j | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq170_HTML.gif, A ¯ 0 = max i a ¯ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq163_HTML.gif, G 0 = max j G j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq106_HTML.gif, and Θ ˜ = ( 2 P A ̲ B + P A ¯ | C | F + F | C T | A ¯ P + n | d | G 0 P A ¯ + α ¯ U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq171_HTML.gif;

(E2) a > b 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq108_HTML.gif, where a = λ min Θ ˜ λ max P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq172_HTML.gif, b = λ max ( α ¯ V + n | d | A ¯ 0 G λ max P ) λ min P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq173_HTML.gif;

(E3) there exists a constant δ > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq111_HTML.gif such that inf k Z ( t k t k 1 ) > δ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq112_HTML.gif, δ 2 τ > ln ( ρ e λ τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq113_HTML.gif and λ ln ( ρ e λ τ ) δ τ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq114_HTML.gif, where ρ = max { 1 , a j + b j e λ τ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq115_HTML.gif with a j sup j Z ( 2 λ max ( | M j T | P | M j | ) λ min P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq167_HTML.gif, b j max r S ( 2 λ max ( H | N T | P | N | H ) λ min P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq174_HTML.gif for all j Z = { 1 , 2 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq118_HTML.gif, and λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq119_HTML.gif is the unique solution of the equation λ = a b e λ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq120_HTML.gif,

then the null solution of impulsive stochastic system (1.2) is globally stochastically exponentially stable in the mean square with the convergence rate 1 2 ( λ ln ( ρ e λ τ ) δ τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq121_HTML.gif.

Remark 4.2 If letting N = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq175_HTML.gif, system (1.2) was investigated by [2]. However, LMIs criterion of Corollary 4.2 is more feasible and effective than that of [[2], Theorem 2.1]. In fact, we know from the Schur complement theorem that the LMI condition of [[2], Theorem 2.1] is equivalent to the inequality ϒ = 2 P 1 A ̲ B P 1 A ¯ | C | | C | T A ¯ P 1 P 1 A ¯ | D | | D | T A ¯ P 1 F 2 P 1 U P 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq176_HTML.gif, where the term P A ¯ | C | | C T | A ¯ P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq177_HTML.gif actually makes parameters amplify against ϒ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq178_HTML.gif if λ min A ¯ > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq179_HTML.gif or λ min | C | > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq180_HTML.gif. In other words, Corollary 4.2 can judge what [[2], Theorem 2.1] cannot do, which may be illustrated by Example 5.2 (below).

Corollary 4.3 If the following three conditions hold:

(F1) there exist a positive scalar α ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq157_HTML.gif and a positive definite diagonal matrix P such that the following LMI conditions hold:
Θ ¯ > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ20_HTML.gif
(4.5)

where matrices C = ( c i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq94_HTML.gif, D = ( d i j ) n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq169_HTML.gif, | d | = max i , j | d i j | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq170_HTML.gif, A ¯ 0 = max i a ¯ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq163_HTML.gif, G 0 = max j G j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq106_HTML.gif, and Θ ¯ = ( 2 λ 1 P D 2 P A ̲ B + P A ¯ | C | F + F | C T | A ¯ P + n | d | G 0 P A ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq181_HTML.gif;

(F2) a > b 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq108_HTML.gif, where a = λ min Θ ¯ λ max P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq182_HTML.gif, b = λ max ( n | d | A ¯ 0 G λ max P ) λ min P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq183_HTML.gif;

(F3) there exists a constant δ > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq111_HTML.gif such that inf k Z ( t k t k 1 ) > δ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq112_HTML.gif, δ 2 τ > ln ( ρ e λ τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq113_HTML.gif and λ ln ( ρ e λ τ ) δ τ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq114_HTML.gif, where ρ = max { 1 , a j + b j e λ τ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq115_HTML.gif with a j sup j Z ( 2 λ max ( | M j T | P | M j | ) λ min P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq167_HTML.gif, b j max r S ( 2 λ max ( H | N T | P | N | H ) λ min P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq174_HTML.gif for all j Z = { 1 , 2 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq118_HTML.gif, and λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq119_HTML.gif is the unique solution of the equation λ = a b e λ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq120_HTML.gif,

then the null solution of impulsive deterministic system (1.3) is globally stochastically exponentially stable in the mean square with the convergence rate 1 2 ( λ ln ( ρ e λ τ ) δ τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq121_HTML.gif.

Remark 4.3 For the same reason as in Remark 4.2, the LMI (4.5) of Corollary 4.3 is more feasible and effective than that of [[3], Theorem 3.1], which may be illustrated by a numerical example below (Example 5.1).

5 Numerical examples

In this section, two examples are given to illustrate that the criteria of Corollary 4.2 and Corollary 4.3 can judge what some existing criteria cannot do. The third numerical example is presented to illustrate the effectiveness of our main results (Theorems 3.1-3.2).

Example 5.1 Consider impulsive system (1.3) with the following parameters:
A ̲ = diad ( 0.1 , 0.1 ) , A ¯ = diag ( 0.2 , 0.2 ) , B = diag ( 2 , 2 ) , F = diag ( 0.1 , 0.2 ) , M = diag ( 1.2 , 1.2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equ21_HTML.gif
(5.1)
In this section, we denote
[ a , b ; c , d ] = ( a b c d ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equn_HTML.gif

Assume, in addition, G = diag ( 1 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq184_HTML.gif, D = diag ( 0.005 , 0.005 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq185_HTML.gif, N = diag ( 0.0003 , 0.0003 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq186_HTML.gif, C = [ 3.2 , 0.003 ; 0.003 , 3.2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq187_HTML.gif, D = [ 0.11 , 0.003 ; 0.003 , 0.12 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq188_HTML.gif, H = F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq189_HTML.gif, M k M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq190_HTML.gif, k = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq191_HTML.gif . v = ( v 1 ( t , x ) , v 2 ( t , x ) ) T R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq192_HTML.gif, x = ( x 1 , x 2 ) T Ω = { ( x 1 , x 2 ) T R 2 : | x j | < 2 , j = 1 , 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq193_HTML.gif. Assume that the boundary condition is the Dirichlet boundary one, and then λ 1 = π 2 = 9.8696 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq194_HTML.gif (see, e.g., [16]). Assume that the lower limit of the time interval between impulses inf k Z ( t k t k 1 ) = 19 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq195_HTML.gif. From the differential inequality lemma [[2], Lemma 1.6] we know that δ τ < 19 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq196_HTML.gif and δ > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq111_HTML.gif, and hence the upper limit of time delay τ < 19 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq197_HTML.gif.

With the above data, one can use computer Matlab LMI toolbox to solve the LMI (C1) of [[3], Theorem 3.1], and obtain t min = 0.0516 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq198_HTML.gif, P = diag ( 0.5169 , 0.5147 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq199_HTML.gif. Next, we need verify (C2) in [[3], Theorem 3.1]. However, a direct computation derives a ˜ = λ min Θ λ max P = 0.0624 0.5169 = 0.1207 < 0.1749 = 0.09 0.5147 = λ max ( G 2 ) λ min P = b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq200_HTML.gif, which implies a ˜ < b ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq201_HTML.gif. Hence, [[3], Theorem 3.1] cannot judge the stability of impulsive system (1.3) with the above data.

However, we can solve LMI (4.5) by Matlab LMI toolbox, and obtain t min = 0.5298 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq202_HTML.gif, P = diag ( 2.4462 , 2.3729 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq203_HTML.gif. Further computation yields a = λ min Θ ¯ λ max P = 0.5417 2.4462 = 0.2214 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq204_HTML.gif, b = λ max ( n | d | A ¯ 0 G λ max P ) λ min P = 0.0352 2.3729 = 0.0148 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq205_HTML.gif, and hence a > b 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq108_HTML.gif. In addition, a direct calculation derives ρ = max j { 1 , a j + b j e λ τ } = 2.9689 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq206_HTML.gif. Let δ = 1.8093 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq207_HTML.gif, τ = 10.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq208_HTML.gif, and then δ τ = 18.9977 < 19 = inf k Z ( t k t k 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq209_HTML.gif, λ = 0.1499 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq210_HTML.gif, δ 2 τ ln ( ρ e λ τ ) = 31.7105 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq211_HTML.gif and λ ln ( ρ e λ τ ) δ τ = 0.0098 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq212_HTML.gif.

All the conditions (F1)-(F3) of Corollary 4.3 are satisfied, then by Corollary 4.3 the null solution of impulsive deterministic system (1.3) is globally stochastically exponentially stable in the mean square with the convergence rate 0.0049 and the allowable upper bound of time delays τ = 10.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq208_HTML.gif.

Example 5.2 Under the Neumann boundary condition, we consider stochastic system (1.2) with the data (4.1) and the following parameters:
G = diag ( 0.3 , 0.3 ) , D ( t , v , x ) = [ 0.005 , 0.003 ; 0.003 , 0.005 ] , C = [ 2.2 , 0.003 ; 0.003 , 2.2 ] , D = [ 0.11 , 0.003 ; 0.003 , 0.12 ] , U = diag ( 0.0003 , 0.0003 ) , V = [ 0 , 0 ; 0 , 0 ] = N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equo_HTML.gif

Let τ ( t ) τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq213_HTML.gif, and then μ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq214_HTML.gif. Now, one can use Matlab LMI toolbox to solve the LMI condition of [[2], Theorem 2.1] and obtain t min = 0.0628 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq215_HTML.gif, P 1 = diag ( 0.9084 , 0.9237 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq216_HTML.gif, P 2 = diag ( 0.0920 , 0.0778 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq217_HTML.gif. Further computation yields that min { λ min ϒ ˜ λ max P 1 , ( 1 μ ) } = 0.0832 < 1.1568 = λ max G 2 λ min P 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq218_HTML.gif, which implies that the condition (C2) of [[2], Theorem 2.1] is not satisfied. Hence, the stability of system (1.2) with the above data cannot be judged by [[2], Theorem 2.1].

However, we solve LMIs (4.3)-(4.4), and obtain t min = 0.9343 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq219_HTML.gif, α ¯ = 42.5544 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq220_HTML.gif, P = diag ( 24.7366 , 23.9523 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq221_HTML.gif. Moreover, we can get by direct computation that a = 0.1978 > 0.0149 = b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq222_HTML.gif, and ρ = 2.9743 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq223_HTML.gif. Let τ = 10.3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq224_HTML.gif, δ = 1.8445 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq225_HTML.gif so that δ τ = 18.9979 < 19 = inf k Z ( t k t k 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq226_HTML.gif, δ 2 τ ln ( ρ e λ τ ) = 32.5410 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq227_HTML.gif and λ ln ( ρ e λ τ ) δ τ = 0.0053 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq228_HTML.gif. All (E1)-(E3) of Corollary 4.2 are satisfied, then by Corollary 4.2 the null solution of impulsive stochastic system (1.2) is globally stochastically exponentially stable in the mean square with the convergence rate 0.0027 and the allowable upper bound of time delays τ = 10.3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq224_HTML.gif.

Example 5.3 Consider impulsive stochastic Markovian jumping fuzzy system (1.1) with the following parameters:
D ( t , x , v ) = [ 0.003 , 0.005 ; 0.004 , 0.006 ] , B = diag ( 2 , 2 ) , F = diag ( 0.1 , 0.2 ) , H = G = F , U = diag ( 0.0003 , 0.0003 ) = V = N r , M k r diag ( 1.2 , 1.2 ) , r S = { 1 , 2 , 3 } , k = 1 , 2 , . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equp_HTML.gif

In addition, C ˆ 1 = [ 0.11 , 0.003 ; 0.003 , 0.12 ] = D ˆ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq229_HTML.gif, C ˇ 1 = [ 0.16 , 0.003 ; 0.003 , 0.18 ] = D ˇ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq230_HTML.gif, C ˆ 2 = [ 0.13 , 0.003 ; 0.003 , 0.15 ] = D ˆ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq231_HTML.gif, C ˇ 2 = [ 0.17 , 0.003 ; 0.003 , 0.19 ] = D ˇ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq232_HTML.gif, C ˆ 3 = [ 0.12 , 0.003 ; 0.003 , 0.13 ] = D ˆ 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq233_HTML.gif, C ˇ 3 = [ 0.175 , 0.003 ; 0.003 , 0.196 ] = D ˇ 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq234_HTML.gif.

The two cases of the transition rates matrices are considered as follows:
Case (1): Π = ( 0.2 0.1 0.1 0.2 0.3 0.1 0.3 0.1 0.4 ) , Case (2): Π = ( 0.2 ? ? 0.2 0.3 0.1 0.3 0.1 0.4 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_Equq_HTML.gif

v = ( v 1 ( t , x ) , v 2 ( t , x ) ) T R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq192_HTML.gif, x = ( x 1 , x 2 ) T Ω = { ( x 1 , x 2 ) T R 2 : | x j | < 2 , j = 1 , 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq193_HTML.gif. Assume that the boundary condition is the Dirichlet boundary one, and then λ 1 = π 2 = 9.8696 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq194_HTML.gif. Assume that the lower limit of the time interval between pulses inf k Z ( t k t k 1 ) = 19 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq195_HTML.gif. From the differential inequality lemma (see, [17] or [[2], Lemma 1.3]), we know that δ τ < 19 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq196_HTML.gif and δ > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq111_HTML.gif, and hence the upper limit of time delay τ < 19 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq197_HTML.gif (see, Remark 5.1).

Now one can use Matlab LMI toolbox to solve the LMI conditions (3.1)-(3.2) of Theorem 3.1 for Case (1) and p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq1_HTML.gif, and obtain t min = 0.5607 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq235_HTML.gif, and α 1 = 25.4531 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq236_HTML.gif, α 2 = 25.4421 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq237_HTML.gif, α 3 = 25.4273 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq238_HTML.gif, P 1 = diag ( 6.6617 , 6.7783 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq239_HTML.gif, P 2 = diag ( 6.6320 , 6.7643 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq240_HTML.gif, P 3 = diag ( 6.6066 , 6.7304 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq241_HTML.gif.

Moreover, a direct computation derives a = 3.7256 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq242_HTML.gif, b = 0.1399 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq243_HTML.gif. So, the condition (C2) a > b 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq108_HTML.gif holds in this case.

By the definition of a j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq244_HTML.gif, b j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq245_HTML.gif, further computation derives ρ = max j { 1 , a j + b j e λ τ } = 2.9375 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq246_HTML.gif. Let τ = 14 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq247_HTML.gif and δ = 1.3569 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq248_HTML.gif. Then solving the equation λ = a b e λ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq120_HTML.gif yields λ = 0.2299 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq249_HTML.gif. By these data, one can calculate that δ τ = 18.9972 < 19 = inf k Z ( t k t k 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq250_HTML.gif, δ 2 τ ln ( ρ e λ τ ) = 21.4822 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq251_HTML.gif and λ ln ( ρ e λ τ ) δ τ = 0.0037 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq252_HTML.gif, which implies the condition (C3) of Theorem 3.1 holds. By Theorem 3.1, the null solution of impulsive Markovian jumping stochastic fuzzy system (1.1) is globally stochastically exponentially stable in the mean square with the convergence rate 0.00185 and the allowable upper bounds of time delays τ = 14 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq247_HTML.gif.

Assume that the boundary condition is the Dirichlet boundary one, and then λ 1 = π 2 = 9.8696 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq194_HTML.gif (see, e.g., [16]). Similarly, we can solve the corresponding conditions of Theorem 3.2 for Case (1) and p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq137_HTML.gif, and obtain t min = 0.5580 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq253_HTML.gif, α ¯ 1 = 25.3516 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq254_HTML.gif, α ¯ 2 = 25.3408 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq255_HTML.gif, α ¯ 3 = 25.3262 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq256_HTML.gif, and P 1 = diag ( 6.5896 , 6.7043 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq257_HTML.gif, P 2 = diag ( 6.5604 , 6.6905 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq258_HTML.gif, P 3 = diag ( 6.5354 , 6.6572 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq259_HTML.gif.

Similarly, we can calculate and obtain a = 3.7294 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq260_HTML.gif, b = 0.1399 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq261_HTML.gif, ρ = 2.9371 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq262_HTML.gif.

Let δ = 1.3473 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq263_HTML.gif and τ = 14.1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq264_HTML.gif, and then λ = 0.2284 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq265_HTML.gif, δ τ = 18.9972 < 19 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq266_HTML.gif, δ 2 τ ln ( ρ e λ τ ) = 21.2557 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq267_HTML.gif and λ ln ( ρ e λ τ ) δ τ = 0.0029 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq268_HTML.gif. By Theorem 3.2, the null solution of impulsive Markovian jumping stochastic fuzzy system (1.1) with p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq137_HTML.gif is stochastically exponentially stable in the mean square with the convergence rate 0.00145 and the allowable upper bounds of time delays τ = 14.1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq264_HTML.gif.

Obviously in Case (2) we can assume ϱ 1 = 0.2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq269_HTML.gif. Next, we employ Matlab LMI toolbox to solve LMI conditions (3.1)-(3.2) of Theorem 3.1 for Case (2) and p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq1_HTML.gif, and obtain t min = 0.3196 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq270_HTML.gif, α ¯ 1 = 25.7897 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq271_HTML.gif, α ¯ 2 = 25.6385 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq272_HTML.gif, α ¯ 3 = 25.6264 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq273_HTML.gif, P 1 = diag ( 7.0171 , 7.1446 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq274_HTML.gif, P 2 = diag ( 6.7114 , 6.8458 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq275_HTML.gif, P 3 = diag ( 6.6913 , 6.8174 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq276_HTML.gif. So, a = 3.5752 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq277_HTML.gif, b = 0.1399 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq243_HTML.gif, ρ = 2.9377 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq278_HTML.gif.

Let δ = 1.4072 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq279_HTML.gif and τ = 13.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq280_HTML.gif, and then λ = 0.2350 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq281_HTML.gif, δ τ = 18.9973 < 19 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq282_HTML.gif, δ 2 τ ln ( ρ e λ τ ) = 22.4827 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq283_HTML.gif and λ ln ( ρ e λ τ ) δ τ = 0.0113 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq284_HTML.gif. By Theorem 3.1, the null solution of impulsive Markovian jumping stochastic fuzzy system (1.1) is stochastically exponentially stable in the mean square with the convergence rate 0.00565 and the allowable upper bounds of time delays τ = 13.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq285_HTML.gif.

Similarly, we can solve the corresponding conditions of Theorem 3.2 for Case (1) and p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq137_HTML.gif, and obtain t min = 0.5614 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq286_HTML.gif, α ¯ 1 = 25.5625 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq287_HTML.gif, α ¯ 2 = 25.4289 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq288_HTML.gif, α ¯ 3 = 25.4165 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq289_HTML.gif, and P 1 = diag ( 6.8412 , 6.9638 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq290_HTML.gif, P 2 = ( 6.5701 , 6.7003 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq291_HTML.gif, P 3 = ( 6.5494 , 6.6714 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq292_HTML.gif, and then a = 3.5777 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq293_HTML.gif, b = 0.1399 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq294_HTML.gif, ρ = 2.9371 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq295_HTML.gif. Let δ = 1.3969 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq296_HTML.gif and τ = 13.6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq297_HTML.gif, and then λ = 0.0845 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq298_HTML.gif.

Further computation yields δ τ = 18.9973 < 19 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq282_HTML.gif, δ 2 τ ln ( ρ e λ τ ) = 22.2850 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq299_HTML.gif and λ ln ( ρ e λ τ ) δ τ = 0.0096 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq300_HTML.gif. By Theorem 3.2, the null solution of impulsive Markovian jumping stochastic fuzzy system (1.1) with p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq137_HTML.gif is stochastically exponentially stable in the mean square with the convergence rate 0.0048 and the allowable upper bounds of time delays τ = 13.6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq297_HTML.gif.

Table 1 shows that the upper bounds of time delay decrease when there exist unknown elements of a transition rates matrix. This means that unknown elements of transition rates bring a great difficulty in judging the stability.
Table 1

Allowable upper bounds of time delays and the convergence rate

 

Theorem 3.1 ( p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq301_HTML.gif)

Theorem 3.2 ( p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq302_HTML.gif)

 

Case (1)

Case (2)

Case (1)

Case (2)

Upper bound τ

14

13.5

14.1

13.6

Convergence rate

0.00185

0.00565

0.00145

0.0048

In some related literature [18, 19], their impulsive assumption is M k T P M k < P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq303_HTML.gif. However, our impulse matrix M k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq304_HTML.gif may not satisfy the assumption of decreasing impulse. In all the above numerical examples, impulsive parameters matrices M k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq304_HTML.gif satisfy λ min M k = 1.2 > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq305_HTML.gif so that M k T P M k > P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq306_HTML.gif. Thereby, the increasing impulse not only brings some unstable factors to CGNNs, but also limits the time-delays’ upper limit τ < inf k Z ( t k t k 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq307_HTML.gif (see [17] or [[2], Lemma 1.6]).

Remark 5.1 The parameters of impulsive deterministic system (1.3) do not satisfy the conditions of [[3], Theorem 3.1] so that we are not sure whether system (1.3) is stable, for the conditions of [[3], Theorem 3.1] are only sufficient ones, not necessary for the stability of system (1.3). However, we can conclude the stability by our Corollary 4.3, which implies that Corollary 4.3 allows for more effectiveness and less conservatism than [[3], Theorem 3.1]. By the same token as in Remark 5.2, Corollary 4.2 is better than [[2], Theorem 2.1].

Remark 5.2 Table 1 shows that the diffusion plays a positive role in the criterion of Theorem 3.2, which admits a wider range of time delays. Table 1 also illustrates the effectiveness and less conservatism of Theorems 3.1-3.2 due to the significant improvement in the allowable upper bounds of time delays.

Remark 5.3 Finding a solution x to the LMI system A ( x ) < B ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq308_HTML.gif is called the feasibility problem. So, in Examples 5.1-5.3, the system is feasible if t min < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq309_HTML.gif, and infeasible if t min > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-133/MediaObjects/13661_2013_Article_387_IEq310_HTML.gif (see [[11], Remark 29(3)] for detail).

Declarations

Acknowledgements

The authors would like to thank the referees for their valuable suggestions. This work was supported by the Scientific Research Fund of Science Technology Department of Sichuan Province (2012JY010), and the Scientific Research Fund of Sichuan Provincial Education Department (12ZB349).

Authors’ Affiliations

(1)
Institution of Mathematics, Yibin University
(2)
College of Mathematics and Software Science, Sichuan Normal University

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© Rao and Pu; licensee Springer. 2013

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