Existence of Exponentialp-Stability Nonconstant Equilibrium of Markovian Jumping Nonlinear Diffusion Equations via Ekeland Variational Principle

Abstract

The authors obtained a delay-dependent exponential -stability criterion for a class of Markovian jumping nonlinear diffusion equations by employing the Lyapunov stability theory and some variational methods. As far as we know, it is the first time to apply Ekeland variational principle to obtain the existence of exponential stability equilibrium of -Laplacian dynamic system so that some methods used in this paper are different from those methods of many previous related literatures. In addition, the obtained existence criterion is only involved in the activation functions so that the criterion is simpler and easier than other existence criteria to be verified in practical application. Moreover, a numerical example shows the effectiveness of the proposed methods owing to the large allowable variation range of time-delay.

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

“…However, these successful applications always depend on the stability of the equilibrium solution for CGNNs. All the time, the method of Lyapunov theory has usually been employed to solve the stability problem of dynamical systems [1][2][3][4][5][6][7][8][9][10][11]. In studying the stability of neural networks, Lyapunov-Krasovskii functional method can always be combined with other methods in a perfect way, such as the linear matrix inequality (LMI) optimization approach, -Matrix theory, and nonsmooth analysis technique (see, e.g., [6][7][8]).…”

New Stability Criterion for Takagi‐Sugeno Fuzzy Cohen‐Grossberg Neural Networks with Probabilistic Time‐Varying Delays

“…Since the stability of nonlinear -Laplacian diffusion neural networks was originally investigated in [2], various -Laplacian diffusion neural networks have attracted a lot of interest ( [6,17,34,39,44]). As pointed out in Discussion 1, some new conditions and methods may not be applicable to CGNNs model with nonlinear -Laplacian diffusion.…”

“…But the success of these applications largely depends on whether the system has some stability, and so people began to be interested in the stability analysis of the system. In recent decades, reaction-diffusion neural networks have received much attention ( [7][8][9][10][11][12][13]), including various Laplacian diffusion ( [6,[14][15][16][17][18][19][20]). Besides, people are paying more and more attention to fuzzy neural network system ( [21][22][23][24][25][26][27][28][29][30][31][32][33][34]), due to encountering always some inconveniences such as the complicity, the uncertainty, and vagueness ( [27,[35][36][37]).…”

pth Moment Stability of a Stationary Solution for a Reaction Diffusion System with Distributed Delays

“…However, in practical engineering, the time delay is unavoidable, which may lead to chaos and instability of the system [16][17][18][19][20][21][22][23][24]. Thus, in this paper, we are to investigate the delayed p-Laplacian reaction-diffusion dynamic system.…”

“…Inspired by some methods of [24], the authors in [25] employed an impulsive differential inequality lemma to further study the time-delay neural networks with pulse perturbation. In [16], Ruofeng Rao and Shouming Zhong employed the Ekeland variational principle and Lyapunov stability theory to derive a globally exponential pth moment stability criterion for a Markovian jumping T-S fuzzy diffusion system with time-delays:…”