Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation

A nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients is considered. It is shown that the sufficient conditions for exponential mean square stability of the linear part of the considered nonlinear equation also are sufficient conditions for stability in probability of the initial nonlinear equation. Some new sufficient condition of stability in probability for the zero solution of the considered nonlinear non-autonomous stochastic differential equation is obtained which can be considered as a multi-condition of stability because it allows to get for one considered equation at once several different complementary of each other sufficient stability conditions. The obtained results are illustrated with numerical simulations and figures.Comment: Published at https://doi.org/10.15559/18-VMSTA110 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/

New advances in H∞ control and filtering for nonlinear systems

The main objective of this special issue is to summarise recent advances in H∞ control and filtering for nonlinear systems, including time-delay, hybrid and stochastic systems. The published papers provide new ideas and approaches, clearly indicating the advances made in problem statements, methodologies or applications with respect to the existing results. The special issue also includes papers focusing on advanced and non-traditional methods and presenting considerable novelties in theoretical background or experimental setup. Some papers present applications to newly emerging fields, such as network-based control and estimation

Path Integral Approach to Random Neural Networks

In this work we study of the dynamics of large size random neural networks. Different methods have been developed to analyse their behavior, most of them rely on heuristic methods based on Gaussian assumptions regarding the fluctuations in the limit of infinite sizes. These approaches, however, do not justify the underlying assumptions systematically. Furthermore, they are incapable of deriving in general the stability of the derived mean field equations, and they are not amenable to analysis of finite size corrections. Here we present a systematic method based on Path Integrals which overcomes these limitations. We apply the method to a large non-linear rate based neural network with random asymmetric connectivity matrix. We derive the Dynamic Mean Field (DMF) equations for the system, and derive the Lyapunov exponent of the system. Although the main results are well known, here for the first time, we calculate the spectrum of fluctuations around the mean field equations from which we derive the general stability conditions for the DMF states. The methods presented here, can be applied to neural networks with more complex dynamics and architectures. In addition, the theory can be used to compute systematic finite size corrections to the mean field equations.Comment: 20 pages, 5 figure

On delayed genetic regulatory networks with polytopic uncertainties: Robust stability analysis

Copyright [2008] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, we investigate the robust asymptotic stability problem of genetic regulatory networks with time-varying delays and polytopic parameter uncertainties. Both cases of differentiable and nondifferentiable time-delays are considered, and the convex polytopic description is utilized to characterize the genetic network model uncertainties. By using a Lyapunov functional approach and linear matrix inequality (LMI) techniques, the stability criteria for the uncertain delayed genetic networks are established in the form of LMIs, which can be readily verified by using standard numerical software. An important feature of the results reported here is that all the stability conditions are dependent on the upper and lower bounds of the delays, which is made possible by using up-to-date techniques for achieving delay dependence. Another feature of the results lies in that a novel Lyapunov functional dependent on the uncertain parameters is utilized, which renders the results to be potentially less conservative than those obtained via a fixed Lyapunov functional for the entire uncertainty domain. A genetic network example is employed to illustrate the applicability and usefulness of the developed theoretical results