LMI-Based Stability Criterion of Impulsive T-S Fuzzy Dynamic Equations via Fixed Point Theory

Abstract

By formulating a contraction mapping and the matrix exponential function, the authors apply linear matrix inequality (LMI) technique to investigate and obtain the LMI-based stability criterion of a class of time-delay Takagi-Sugeno (T-S) fuzzy differential equations. To the best of our knowledge, it is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. It is worth mentioning that LMI methods have high efficiency and other advantages in largescale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. At the end of this paper, a numerical example is presented to illustrate the effectiveness of the proposed methods.

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

“…Indeed, fixed point theories have always been considered by many authors. Burton [13,14], Rao and Pu [15], Jung [16], Luo [17], Zhang [18], and Wu et al [19] studied the stability by using the fixed point theory which solved the difficulties encountered in the study of stability by means of Lyapunov's direct method. Contraction mapping theorem was the usual method to study the stability of neural networks, except CGNNs.…”

“…Remark 6. As far as we can know, there is not any previous literature related to the fixed point theory where LMI-based stability criteria were presented, except for [15]. In this paper, the LMI-based stability criterion is the first time to be proposed for impulsive CGNNs via fixed point theorems.…”

Stability Analysis of Impulsive Stochastic Reaction-Diffusion Cellular Neural Network with Distributed Delay via Fixed Point Theory

“…Since our CNN model involves pulse and Laplacian operators, our model is different from the previous model ([ [15][16][17][18][19][20][21][22]), which also implies some difficulties in mathematical techniques. Motivated by the previous literature related to fixed point theory ( [15][16][17][18][19][20][21][22][25][26][27][28][29][30][31]), the authors employed Banach fixed point theorem, Hö lder inequality, Burkholder-Davis-Gundy inequality, and the continuous semigroup of Laplace operators to derive the stochastically exponential stability criterion of impulsive stochastic reaction-diffusion cellular neural networks with distributed delay.…”

“…Different methods lead to different criteria for stability criteria which may imply innovations. Fixed point theory and method is one of the alternative methods ( [15][16][17][18][19][20][21][22]). Unlike the known literature, we try to employ Banach fixed point theory in this paper to derive the stability of impulsive stochastic reactiondiffusion cellular neural networks with distributed delay.…”

Fixed Points and Exponential Stability for Impulsive Time-Delays BAM Neural Networks via LMI Approach and Contraction Mapping Principle

“…But every method may have its limitations. During the recent decades, other techniques have been developed to investigate the stability, in which the fixed point method is always one of those alternatives [21][22][23][24][25][26][27]. For example, in 2015, Zhou utilized Brouwer's fixed point theorem to prove the existence and uniqueness of equilibrium of the hybrid BAM neural networks with proportional delays and finally constructed appropriate delay differential inequalities to derive the stability of equilibrium [28].…”