LMI-Based Stability Criterion for Impulsive CGNNs via Fixed Point Theory

Abstract

Linear matrices inequalities (LMIs) method and the contraction mapping theorem were employed to prove the existence of globally exponentially stable trivial solution for impulsive Cohen-Grossberg neural networks (CGNNs). It is worth mentioning that it is the first time to use the contraction mapping theorem to prove the stability for CGNNs while only the Leray-Schauder fixed point theorem was applied in previous related literature. An example is given to illustrate the effectiveness of the proposed methods due to the large allowable variation range of impulse.

Contraction Mapping Theory and Approach to LMI-Based Stability Criteria of T-S Fuzzy Impulsive Time-Delays Integrodifferential Equations

“…One can also understand from Remark 1 that we proposed in this paper a really feasible new condition on amplification function and behavior function . This implies that our result and methods are novelty versus [18,Theorem 4]. Besides, our conditions and result are also different from those of existing literature [17,[19][20][21][22]] (see below " Table 1" and "Remark 13").…”

“…Remark 5. Since (H4) is replaced with (A3), the methods of [18] cannot be applied to this paper, and we have to formulate new contraction mapping, different from [18]. Definition 6.…”

“…Brouwer fixed point theorem and Schauder fixed point theorem were always applied to stability analysis of various CGNNs models [14][15][16][17]. Of course, Banach fixed point theorem was also employed to derive the stability criteria of various CGNNs models [17][18][19][20][21][22]. But for most of existing literature, either using the same inverse function translates CGNNs model into another model similar as follows (see, e.g., [17,[19][20][21]), ( ) = − ( −1 ( ( ))) + ∑ =1 [ ( ) ( −1 ( ( ))) + ( −1 ( ( − ( ))))] , (1) or the stability criteria are too complex [22,Theorem 3.1], which cannot be programmable for computer software while practical engineering is often involved in large-scale computing.…”

“…But, in order to innovate, we have to find 2 Journal of Function Spaces another way. In [18], a LMI-based stability criterion was given for the following CGNNs: (2)…”

“…Theorem A (see [18,Theorem 4]). If (H1)-(H4) are satisfied and there exists a positive constant < 1 such that the following LMI condition holds,…”