Robust exponential stability of impulsive switched systems with switching delays: A Razumikhin approach

In this paper, we focus on the robust exponential stability of a class of uncertain nonlinear impulsive switched systems with switching delays. We introduce a novel type of piecewise Lyapunov-Razumikhin functions. Such functions can efficiently eliminate the impulsive and switching jump of adjacent Lyapunov functions at impulsive switching instants. By Razumikhin technique, the delay-independent criteria of exponential stability are established on the minimum dwell time. Finally, an illustrative numerical example is presented to show the effectiveness of the obtained theoretical results.     

Finite-time L1 control for positive switched linear systems with time-varying delay

This paper is concerned with the problem of finite-time L₁ control for a class of positive switched linear systems with time-varying delay. Firstly, by using the average dwell time approach, sufficient conditions which can guarantee the L₁ finite-time boundedness of the underlying system are given. Then, in virtue of the results obtained, a state feedback controller is designed to ensure that the resulting closed-loop system is finite-time bounded with L₁-gain performance. All the obtained results are formulated in terms of linear matrix inequalities (LMIs), which can be solved conveniently. Finally, an example is given to illustrate the efficiency of the proposed method.     

Improved results on fixed-time stability of switched nonlinear time-varying systems

This paper considers the problem of fixed-time stability (FTS) for switched nonlinear time-varying (NTV) systems. Firstly, three sufficient conditions are proposed to verify the FTS of NTV systems by using the improved Lyapunov function, which has a tighter upper bound of time derivative. Then, two FTS conditions are given for the switched NTV system by extending the obtained results, moreover, a switching strategy is also provided by using the minimum dwell time method. Finally, the obtained results are extended to study the FTS of impulsive NTV systems. Comparing with the existing results, the obtained conditions have two improvements: (1) provides a more accurate estimate for the upper bound of settling time of NTV systems, and (2) allows the Lyapunov function to increase at the switching instant of switched NTV (or impulsive NTV) systems. Two numerical examples are given to illustrate the theoretical results.     

On robustly exponential stability of uncertain neutral systems with time-varying delays and nonlinear perturbations

The issue of robustly exponential stability for uncertain neutral-type systems is considered in this paper. The uncertainties are nonlinear and the delays are time-varying. In terms of a linear matrix inequality (LMI), the new sufficient stability condition with delay dependence is presented. The model transformation and bounding techniques for cross terms are avoided based on an integral inequality. Two illustrative examples are proposed to show the effectiveness of our method.     

Input-to-state stability and stabilization for switched nonlinear positive systems

Input-to-state stability (ISS) analysis and stabilization are concerned in this paper for switched nonlinear positive systems (SNPS), where the deterministic and random switching are both included. For general SNPS, switched affine nonlinear positive systems (SANPS) and switched linear positive systems (SLPS) with deterministic and some kinds of random ”slow” switching, some criterions on ISS are provided. From the criterions for SANPS and SLPS, the ISS properties can be judged just by the differential, algebraic and switching characteristics of the systems. Further, based on the criterions for SANPS and SLPS, some state feedback controllers are designed such that the closed-loop systems be positive, ISS or ISS in some stochastic senses. Four simulation examples verify the validity of our results.     

Exponential stability of discrete-time systems with slowly time-varying delays

Slowly time-varying delays are seldom, but do need to be, considered in the context of discrete-time systems. This paper addresses the exponential stability issue of discrete-time systems with slowly time-varying delays. The basic idea is to transform, by utilizing the switching transformation approach, the original system with slowly time-varying delays into an equivalent switched system with special switching signal. Different types of delays correspond to different types of switching signals, and the stability issue of the original system is converted into that of a switched system. It is the first time that the method of switched homogeneous polynomial Lyapunov function is applied to general delayed systems. Some sufficient exponential stability conditions for the original system are proposed in several situations. It is numerically shown that the conservativeness of the proposed conditions reduces as the degree of the switched homogeneous polynomial Lyapunov function increases.     

Exponential stability analysis for neutral switched systems with interval time-varying mixed delays and nonlinear perturbations

This paper is concerned with the problem of exponential stability for uncertain neutral switched systems with interval time-varying mixed delays and nonlinear perturbations. By using the average dwell time approach and the piecewise Lyapunov functional technique, some sufficient conditions are first proposed in terms of a set of linear matrix inequalities (LMIs), to guarantee the robustly exponential stability for the uncertain neutral switched systems, where the decay estimate is explicitly given to quantify the convergence rate. Three numerical examples are finally illustrated to show the effectiveness of the proposed method.     

Exponential stability and applications of switched positive linear impulsive systems with time-varying delays and all unstable subsystems

The global uniform exponential stability of switched positive linear impulsive systems with time-varying delays and all unstable subsystems is studied in this paper, which includes two types of distributed time-varying delays and discrete time-varying delays. Switching behaviors dominating the switched systems can be either stabilizing and destabilizing in the new designed switching sequence. We design new linear programming algorithm process to find the feasible ratio of stabilizing switching behaviors, which can be compensated by unstable subsystems, destabilizing switching behaviors, and impulses. Specically, we add a kind of nonnegative impulses which is consistent with the switching behaviors for the systems. Employing a multiple co-positive Lyapunov-Krasovskii functional, we present several new sufficient stability criteria and design new switching sequence. Then, we apply the obtained stability criteria to the exponential consensus of linear delayed multi-agent systems, and obtain the new exponential consensus criteria. Three simulations are provided to demonstrate the proposed stability criteria.     

A novel approach to exponential stability of nonlinear non-autonomous difference equations with variable delays

In this paper, by using a novel approach, we first prove a new generalization of discrete-type Halanay inequality. Based on our new generalized inequality, a novel criterion for the exponential stability of a certain class of nonlinear non-autonomous difference equations is proposed. Numerical examples are given to illustrate the effectiveness of the obtained results.     

Practical finite-time stability of switched nonlinear time-varying systems based on initial state-dependent dwell time methods

This paper considers the problem of practical finite-time stability (PFTS) for switched nonlinear time-varying (SNTV) systems. Starting with nonlinear time-varying (NTV) systems, a new sufficient condition is proposed to verify the PFTS of systems by using an improved Lyapunov function. Then, the results obtained are extended to study the PFTS of SNTV systems. Two stability conditions are proposed for SNTV systems under arbitrary switching, moreover, the time and region of convergence are also given. Furthermore, an initial state-dependent dwell time method is introduced to study the PFTS of SNTV systems. Three stability conditions are proposed by using the methods of initial state-dependent minimum dwell time (ISD-MDT) and initial state-dependent average dwell time (ISD-ADT), respectively. The comparisons between the obtained results and the existing results are also given, and the obtained results are extended to impulsive switched nonlinear time-varying (ISNTV) systems. Finally, a numerical example is provided to illustrate the theoretical results.     

Global exponential stability of high order recurrent neural network with time-varying delays

In this paper, global exponential stability of high order recurrent neural network with time-varying delay and bounded activation functions is investigated. Some improved conditions are obtained involving external input, connection weights, and time delays of recurrent neural network. Moreover, the location of the equilibrium point can be estimated. In addition, two examples are demonstrated to illustrate the effectiveness of the proposed criteria in comparison with some existing results.     

Global exponential stability of impulsive discrete-time neural networks with time-varying delays

This paper studies the problem of global exponential stability and exponential convergence rate for a class of impulsive discrete-time neural networks with time-varying delays. Firstly, by means of the Lyapunov stability theory, some inequality analysis techniques and a discrete-time Halanay-type inequality technique, sufficient conditions for ensuring global exponential stability of discrete-time neural networks are derived, and the estimated exponential convergence rate is provided as well. The obtained results are then applied to derive global exponential stability criteria and exponential convergence rate of impulsive discrete-time neural networks with time-varying delays. Finally, numerical examples are provided to illustrate the effectiveness and usefulness of the obtained criteria.     

On exponential stability analysis for neural networks with time-varying delays and general activation functions

This paper is concerned with the exponential stability analysis for a class of cellular neural networks with both interval time-varying delays and general activation functions. The boundedness assumption of the activation function is not required. The limitation on the derivative of time delay being less than one is relaxed and the lower bound of time-varying delay is not restricted to be zero. A new Lyapunov-Krasovskii functional involving more information on the state variables is established to derive a novel exponential stability criterion. The obtained condition shows potential advantages over the existing ones since no useful item is ignored throughout the estimate of upper bound of the derivative of Lyapunov functional. Finally, three numerical examples are included to illustrate the proposed design procedures and applications.     

Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays

By using the continuation theorem due to Mawhin and Gaines, some analysis techniques and the Lyapunov functional method, the sufficient conditions ensuring the existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays are established. The results are interesting and very different from previously known results [Xia and Wong, 2009 [7]; Tan and Tan, 2009 [19]; Huang et al., 2005 [20]; Liu and Huang, 2006 [22]]. Finally, applications and an example are given to illustrate the effectiveness of the results.     

Stability of perturbed switched nonlinear systems with delays

This paper addresses the stability problems of perturbed switched nonlinear systems with time-varying delays. It is assumed that the nominal switched nonlinear system (perturbation-free system) is uniformly exponentially stable and that the perturbations satisfy a linear growth bound condition. It is revealed that there exists an upper bound of perturbation guaranteeing that the perturbed system preserves the stability property of the nominal system, locally or globally, depending on both perturbations and the nominal system itself. An example is provided to illustrate the proposed theoretical results.     

Almost global stability of nonlinear switched systems with mode-dependent and edge-dependent average dwell time

It has recently been shown that almost global stability of nonlinear switched systems can be characterized using multiple Lyapunov densities. This has been accomplished for switched systems subject to a minimum dwell time or an average dwell time constraint. In this paper, as an extension of the aforementioned results, we provide a sufficient condition on mode-dependent and edge-dependent average dwell time to ensure almost global stability of a nonlinear switched system. The relations between average dwell time, mode-dependent, and edge-dependent average dwell time have been discussed. The obtained results for nonlinear switched systems imply the existing results for linear switched systems.