Stability Of Delay-differential Systems Research Articles

Stability of delay differential systems under impulsive control suffered by logic choice

ABSTRACT This paper focuses on the stability of delay differential systems under impulsive control suffered by logic choice. Firstly, an impulsive control suffered by logic choice is constructed for a delay differential system to make it more suitable for some practical problems. Secondly, by using a semi-tensor product, an effective method for qualitative analysis, such as stability analysis, of a differential delay system under impulsive control suffered by logic choice is proposed. That is, the properties of a differential delay system under impulsive control suffered by logic choice can be obtained by the properties of a corresponding non-impulsive differential delay system. Thirdly, some stability criteria are obtained. Finally, three illustrative examples are also discussed.

A Theoretical Study of Second Lyapunov Method on Stability of Zero Solution for a Delay Differential System

This study examined the asymptotic stability of delay differential systems using Lyapunov’s second method. It showed that the zero solutions are asymptotically stable when satisfying some necessary and sufficient conditions. The solution origin of the given system is asymptotically stable for small perturbations, which has no effect on equilibrium system if some conditions are held and it is asymptotically stable for large perturbations that moving system of its equilibrium then its comeback to it if other conditions are held. The proof of theorems in this article results by using and applying Gronwell-Bellman inequality which leads us to main result.

Stochastic suppression and stabilization of delay differential systems

AbstractIn this paper, we investigate stochastic suppression and stabilization for nonlinear delay differential system \documentclass{article}\usepackage{amssymb}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}${\dot{x}}(t)=f(x(t),x(t-\delta(t)),t)$\end{document}, where δ(t) is the variable delay and f satisfies the one‐sided polynomial growth condition. Since f may defy the linear growth condition or the one‐sided linear growth condition, this system may explode in a finite time. To stabilize this system by Brownian noises, we stochastically perturb this system into the nonlinear stochastic differential system dx(t)=f(x(t), x(t−δ(t)), t)dt+qx(t)dw1(t)+σ|x(t)|βx(t)dw2(t) by introducing two independent Brownian motions w1(t) and w2(t). This paper shows that the Brownian motion w2(t) may suppress the potential explosion of the solution of this stochastic system for appropriate choice of β under the condition σ≠0. Moreover, for sufficiently large q, the Brownian motion w1(t) may exponentially stabilize this system. Copyright © 2010 John Wiley & Sons, Ltd.

Impulsive stabilization of delay differential systems via the Lyapunov–Razumikhin method

This work studies global exponential stability of impulsive delay differential systems. By employing the Razumikhin technique and Lyapunov functions, several global exponential stability criteria are established for general impulsive delay differential equations. Our results show that delay differential equations may be exponentially stabilized by impulses. An example and its simulation are also given to illustrate our results.

On the stability of delay-differential systems in the behavioral framework

In the behavioral approach, a linear time-invariant (LTI) delay-differential system is naturally introduced by Glusing-Luerssen (1997) as a continuous-time system whose dynamics are governed by a set of delay-differential equations, involving both painted and distributed delay operators. For this class of systems the notion of autonomy is introduced and characterized. Finally, asymptotic stability of autonomous delay-differential systems is investigated.

Adaptive stabilization of delay differential systems with unknown uncertainty bounds

Based on output measurements, two feedback control schemes are developed for robust stabilization of a wide class of non-linear uncertain delay differential systems with time-varying state delay. The parameters of the adaptive controller are updated robustly when the bounds of the uncertainties are unknown. Both schemes are proved to render the controlled system globally practically stable. The developed controller is demonstrated in a water-quality model for the purpose of preserving standards of water constituents in streams.

Stability and stabilization of delay differential systems

The delay systems considered here are represented by linear delay differential equations. The system parameters and the delays are assumed to be imperfectly known. The instantaneous state vector is perturbed by a bounded external disturbance vector. The problem addressed is that of characterizing conditions that guarantee that the trajectory of the instantaneous state vector remains in a domain defined by a set of symmetrical linear constraints. It is shown that the positive invariance property can be used to solve this problem, and that positive invariance of a compact domain of the instantaneous state space implies delay-independent asymptotic stability of the associated deterministic system. The possible use of these results for the control of a multiple-delay MIMO differential model is then presented. Finally, an example is given.

Stability and Stabilization of Delay Differential Systems

A delay system is represented by a linear delay differential equation. The system parameters and the delays are assumed to be unperfectly known. The instantaneous state vector is perturbed by a bounded external disturbance vector. The addressed problem is to characterize conditions which guarantee that the trajectory of the instantaneous state vector remains in a domain defined by a set of symmetrical linear constraints. It is shown that positive invariance property can be used to solve this problem, and that positive invariance of a compact domain of the instantaneous state space implies delay independent asymptotic stability of the associated deterministic system. The possible use of these results for controlling a multiple delay MIMO differential model is then presented. An example is finally proposed.