Study On Stability Of Several Classes Of Dynamical Systems

It is well known that time delay is unavoidable in dynamical systems. Time delays may affect the stability of the system, even lead to instability, oscillation or chaos phenomena. Furthermore, in the applications and designs of networks, some unavoidable uncertainties which result from using an approximate system model for simplicity, external perturbations, parameter fluctuations, and data errors, etc, must be integrated into the system model. Such time delays, parametric uncertainties may significantly influence on the overall behavior of a dynamical system. Hence, it is significant and of prime importance to consider the effect of time delay and parametric uncertainties on the stability property of dynamical systems. In many practical systems, such as in the physical circuits, biological systems, chemical reaction process, stochastic disturbances in dynamical systems play a very important role. Therefore, the stability of dynamical system must take into account their effect. Recently, the stability analysis of dynamical systems has attracted a large number of researchers, and a series of significant results have been established.This dissertation focuses on the asymptotical and robust stability for several dynamical systems. The main contributions and originality contained in this dissertation are as follows:①Robust stability analysis of uncertain systems with two additive time-varying delay componentsSometimes in practical situations, for example, in networked control system, however, signals transmitted from one point to another may experience a few segments of networks, which can possibly induce successive delays with different properties due to the variable network transmission conditions. The problem of stability analysis for uncertain systems is concerned. The systems are based on a new time-delay model proposed recently, which contains multiple successive delay components in the state. We consider the case where only two successive delay components appear in the state. As a result, some less conservative stability criteria are established for systems with two successive delay components and parameter uncertainties. And the idea behind the proposed results can be easily extended to systems with multiple successive delay components.②Delay-range dependent stability of uncertain neural networks with interval time-varying delays In practical engineering systems, time-varying delay is a time delay that varies in an intervalτ≤τ(t )≤τ, in which the lower boundτis not restricted to be 0. In this thesis, the stability analysis for neural networks with interval time-varying delays and parameter uncertainties is addressed and some delay-derivative-independent stability criteria are established in term of linear matrix inequality (LMI).③Asymptotical stability analysis for static recurrent neural networks with time delay: delay fractioning approachThe asymptotical stability analysis for static recurrent neural networks with time delay is studied by means of a delay fractioning approach. Some delay-dependent asymptotical stability criteria for static recurrent neural networks with time delay are established. The obtained criteria not only have the advantage of simple form but also are less conservative than some existing ones in the literature. Experimental results also show that the delay fractioning approach is effective to expand the upper bound of time delay.④Stability analysis for genetic regulatory networks with interval time-varying delays: an LMI approachThe asymptotical and robust stability of genetic regulatory networks with interval time-varying delays and parameter uncertainties is investigated. First, by employing some free-weighting matrices and linear matrix inequalities, new delay-range-dependent and delay-derivative-dependent/independent stability criteria are derived. Then, the robust stability of genetic regulatory networks with interval time-varying delays and parameter uncertainties is addressed. Furthermore, the rigorous requirement of other literatures that the time derivatives of time-varying delays must be smaller than one is abandoned in the proposed scheme. As a result, the new criteria have wider range of applications and are applicable to both fast and slow time-varying delays. Since the criteria are obtained by LMIs, the results are easily verified⑤Stabilizing effects of stochastic noises in genetic regulatory networks with interval time-varying delaysAs molecular events in cells dominated by the thermal fluctuations and noise, gene expression can be regarded as a random process. Especially in low copy number of molecules, or a slower reaction rate, the role of this effect will be more prominent. Therefore, gene regulatory network should be described by more accurate models which include random noise. The stabilizing effects of stochastic noises in genetic regulatory networks with interval time-varying delays are concerned amd some new stability criteria are established to guarantee the delayed genetic regulatory networks to be robustly asymptotically stable in the mean square. The obtained criteria characterize the aggregated effects of the stochastic noises and time-varying delays on the stability of the considered genetic regulatory networks.