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Together with Lyapunov-Krasovskii functional theory and reciprocal convex technique, a new sufficient condition is derived to guarantee the global stability for recurrent neural networks with both time-varying and continuously distributed delays, in which one improved delay-partitioning technique is employed. The LMI-based criterion heavily depends on both the upper and lower bounds on state delay and its derivative, which is different from the existent ones and has more application areas as the lower bound of delay derivative is available. Finally, some numerical examples can illustrate the reduced conservatism of the derived results by thinning the delay interval.

Recently, various classes of neural networks have been increasingly studied in the past few decades, due to their practical importance and successful applications in many areas such as optimization, image processing, and associative memory design. In those applications, the key feature of the designed neural network is to be convergent. Meanwhile, since there inevitably exist communication delay which is the main source of oscillation and instability in various dynamical systems, great efforts have been made to analyze the dynamical behaviors of time-delay systems including delayed neural networks (DNNs), and many elegant results have been reported; see [

Presently, during tackling the effect of time delay, the delay-partitioning idea has been verified to be more effective in reducing the conservatism and widely employed [

In the paper, we make some great efforts to investigate asymptotical stability for recurrent neural networks with both time-varying and continuously distributed delay, in which both the upper and lower bounds of time delay and its derivative are treated. Through applying an improved delay-partitioning idea, one LMI-based condition is derived based on combination of reciprocal convex technique and convex one, which can present the pretty delay dependence and computational efficiency. Finally, we give three numerical examples to illustrate the reduced conservatism.

For symmetric matrices

Consider the DNNs with continuously distributed delay of the following form:

The following assumptions on system (

The delay

For the constants

As pointed out in [

Suppose

In order to establish the stability criterion, firstly, the following lemmas are introduced.

For any constant matrix

Let the functions

Then the problem to be addressed in next section can be formulated as developing a condition ensuring that the DNNs (

In the section, through utilizing the reciprocal convex technique idea in [

Given a positive integer

Based on (

Presently, the convex combination technique has been widely employed to tackle time-varying delay owing to the truth that it could reduce the conservatism more effectively than the previous ones, see [

One can easily check that the theorem in this work achieves some great improvements over the one in [

Secondly, owing to the introduction of reciprocal convex approach, Theorem

When

Owing to the introduction of delay-partitioning idea in this work, the difficulty and complexity in checking the theorem will become more and more evident when the integer

In the section, three numerical examples will be presented to illustrate that our results are superior over the ones by convex combination technique.

We revisit the system considered in [

Calculated MAUBs

Methods | 0.6 | 0.8 | 0.9 | 1.2 | |

Li et al. [ | 3.4878 | 2.8458 | 1.9150 | 1.1167 | |

3.7458 | 3.1150 | 2.1153 | 1.3189 | ||

Theorem | 3.5664 | 2.9316 | 2.0552 | 1.2107 | |

3.8543 | 3.2311 | 2.2115 | 1.4445 |

Calculated MAUBs

Methods | 0.6 | 0.8 | 0.9 | 1.2 | |

Zhang et al. [ | 3.5209 | 2.8654 | 1.9508 | — | |

Li et al. [ | 3.5872 | 2.8815 | 1.9657 | 1.2055 | |

Theorem | 3.6435 | 2.9443 | 2.0112 | 1.2899 |

Based on Tables

Calculated MAUBs

Methods | 0.8 | 0.9 | Unknown | |

Li et al. [ | 2.8815 | 1.9657 | 1.2055 | |

3.1488 | 2.1968 | 1.4078 | ||

Theorem | 2.9668 | 2.0113 | 1.3115 | |

3.2350 | 2.2778 | 1.4890 |

Consider the delayed neural networks (

Calculated MAUBs

Methods | 0.1 | 0.5 | 0.9 | Unknown | |

Hu et al. [ | 3.33 | 3.16 | 3.10 | 3.09 | |

3.65 | 3.32 | 3.26 | 3.24 | ||

Li et al. [ | 3.35 | 3.21 | 3.20 | 3.19 | |

3.77 | 3.41 | 3.38 | 3.37 | ||

Theorem | 3.40 | 3.32 | 3.31 | 3.24 | |

3.86 | 3.49 | 3.42 | 3.40 |

Consider the delayed neural networks (

Calculated MAUBs

Methods | [ | [0.4, 0.8] | [0.8, 0.9] | [0.9, 1.1 ] | |

Li et al. [ | 0.8712 | 0.8257 | 0.9327 | 0.9805 | |

1.1872 | 1.1815 | 1.2657 | 1.3028 | ||

Theorem | 0.9221 | 0.9115 | 0.9995 | 1.1134 | |

1.2315 | 1.2189 | 1.3012 | 1.3898 |

Based on Table

This paper has investigated the asymptotical stability for DNNs with continuously distributed delay. Through employing one improved delay-partitioning idea and combining reciprocal convex technique with convex combination one, one stability criterion with significantly reduced conservatism has been established in terms of LMIs. The proposed stability condition benefits from the partition of delay intervals and reciprocal convex technique. Three numerical examples have been given to demonstrate the effectiveness of the presented criteria and the improvements over some existent ones. Finally, it should be worth noting that the delay-partitioning idea presented in this work is widely applicable in many cases.

This work is supported by the National Natural Science Foundation of China no. 60875035, no. 60904020, no. 61004064, no. 61004032, and the Special Foundation of China Postdoctoral Science Foundation Projects no. 201003546.