Indefinite Derivative Research Articles

Stability analysis for nonautonomous impulsive hybrid stochastic delay systems

In this paper, the problems on the existence and uniqueness, and the input-to-state stability for the global solution of nonlinear nonautonomous impulsive stochastic delay systems with Markovian switching are considered. The existence and uniqueness for the global solution of such systems is studied by using the Lyapunov function, the theory of stochastic analysis, and the function of the impulsive density. In order to overcome the difficulty stemming from the simultaneous presence of the indefinite derivative for the Lyapunov function, the bounded external disturbances, and a nonnegative and bounded time-varying delay, one impulsive generalized Halanay inequality of integral version is established. The input-to-state stability, the integral input-to-state stability, the stochastic input-to-state stability and the eλt(λ>0)-weighted input-to-state stability for the global solution of nonlinear nonautonomous impulsive stochastic delay systems with Markovian switching are investigated, respectively. One example is provided to illustrate the effectiveness of the theoretical results obtained.

Exponential stability and boundedness of nonlinear perturbed systems by unbounded perturbation terms

We study the exponential stability and boundedness problem for perturbed nonlinear time-varying systems using Lyapunov Functions with indefinite derivatives. As the limiting function for the perturbation term, we use different forms and give stability and boundedness conditions in terms of the coefficients in these bounds. Contrary to most of the available conditions, we allow the coefficients to be unbounded. But instead, we put forward a condition that requires a series produced by coefficients to be limited and exponentially decaying. We perform our results on Linear time-varying systems and generalize many of the available results.

A new stability criterion and its application to robust stability analysis for linear systems with distributed delays

Recently, the possibility to verify positive definiteness of Lyapunov–Krasovskii functionals only on a specific Razumikhin-type set of functions to conclude on the stability of linear time delay systems was shown. In this paper, for a class of systems with multiple concentrated and distributed delays, we extend the result and prove that this specific set may be applied while verifying the negative definiteness of the functionals derivatives along the solutions as well. The result is applied in the robustness analysis with respect to uncertainties both in the system matrices and in the delays. It provides a simple way to prove the robust stability conditions due to obviating the need to work with indefinite derivatives of functionals along the solutions of a perturbed system. As a by-product, the estimates for an unstable eigenvalue of the system are derived, and the functionals defined on complex-valued initial functions are presented.

An improved fixed-time stabilization problem of delayed coupled memristor-based neural networks with pinning control and indefinite derivative approach

<abstract><p>In this brief, we propose a class of generalized memristor-based neural networks with nonlinear coupling. Based on the set-valued mapping theory, novel Lyapunov indefinite derivative and Memristor theory, the coupled memristor-based neural networks (CMNNs) can achieve fixed-time stabilization (FTS) by designing a proper pinning controller, which randomly controls a small number of neuron nodes. Different from the traditional Lyapunov method, this paper uses the implementation method of indefinite derivative to deal with the non-autonomous neural network system with nonlinear coupling topology between different neurons. The system can obtain stabilization in a fixed time and requires fewer conditions. Moreover, the fixed stable setting time estimation of the system is given through a few conditions, which can eliminate the dependence on the initial value. Finally, we give two numerical examples to verify the correctness of our results.</p></abstract>

Fixed-time synchronization problem of coupled delayed discontinuous neural networks via indefinite derivative method

<abstract><p>In this brief, we introduce a class of coupled delayed nonautonomous neural networks (CDNNs) with discontinuous activation function. Different from the conventional Lyapunov method, this brief uses the implementation of an indefinite derivative to deal with the nonautonomous system for the case that the topology between neurons is nonlinear coupling, and the system can achieve synchronization in fixed time by selecting the suitable control scheme. The settling time estimation of the system which can get rid of the dependence on the initial value is given. Finally, two examples are given to verify the correctness of the results in this paper.</p></abstract>

New Fixed-Time Stability Analysis of Delayed Discontinuous Systems via an Augmented Indefinite Lyapunov-Krasovskii Functional.

This article discusses the fixed-time stability (FTS) of a kind of delayed discontinuous system (DS) in Filippov sense. Based on the set-valued map, the FTS analysis of the general solution is first transformed into the zero solution of the differential inclusion. Second, the new criteria of the Lyapunov-Krasovskii functional (LKF) are given and LKF is proved to possess the indefinite derivatives by using the simple integral inequalities. In addition, the FTS of the considered delayed DS is achieved and the new settling time is estimated. Third, to demonstrate the applicability of the new FTS theorems, the FTS control of a class of discontinuous inertial neural networks (DINNs) with time-varying delays is solved. Finally, two numerical examples are given to examine the theoretical results and simulations are also provided to make some illustrations.

Finite-/Fixed-Time Stability of Nonautonomous Functional Differential Inclusion: Lyapunov Approach Involving Indefinite Derivative.

This article investigates a type of nonautonomous delayed differential equation (DDE) with discontinuity. Under the framework of the Filippov state solution, the finite-time stability (FNTS)/fixed-time stability (FXTS) problems of nonautonomous functional differential inclusion (FDI) are studied via the generalized Lyapunov functional method. The generalized Lyapunov functional used in this article is allowed to obtain an indefinite time derivative almost anywhere (a.a.) along the system's state solutions. Nevertheless, the classic Lyapunov functional requires that its time derivative retains seminegative/negative definiteness anywhere. As a result, novel FNTS and FXTS criteria of the trivial state solution for FDI are established. Moreover, the settling time (ST) of FNTS/FXTS is provided. Then, the developed Lyapunov functional approach is applied to realize the finite-/fixed-time stabilization control of delayed neuron networks (DNNs) possessing discontinuous activation and ball motion models. The proposed method and results concerning FNTS/FXTS are of great significance in neural network (NN)/mechanical control engineering applications.

Fixed-time synchronization criteria of fuzzy inertial neural networks via Lyapunov functions with indefinite derivatives and its application to image encryption

In this paper, we investigate the fixed-time synchronization for fuzzy inertial neural networks with time-varying coefficients and time delays. The fuzzy inertial neural networks are transformed into two forms of first-order differential systems, and then two kinds of different controllers of time-variable are designed. In these schemes, a series of novel criteria are proposed to ensure the fixed-time synchronization for the drive-response fuzzy inertial neural networks via state feedback control methods. Sufficient conditions of delay-dependent and delay-independent are obtained in terms of algebraic inequalities, not the matrix inequalities. Moreover, these new criteria allow the derivative of the Lyapunov function can be positive, but not negative definite as the earlier work. And the algebraic conditions become easy to be tested under the new criteria. Finally, two examples are added to illustrate effectiveness of the results we obtained. Meanwhile, a new image encryption algorithm is proposed based on the synchronization of drive-response networks, and the effectiveness of this algorithm is also demonstrated.

A novel fixed-time stability of nonlinear impulsive systems: a two-stage comparison principle method

In this article, we elaborated on the problem of fixed time stability (FxTS) of nonlinear impulsive systems. For handing this problem, we propose a new concept of fixed-time stable function set (FTSFS). By taking advantage of definitions of FTSFS and average impulsive interval, a unified FxTS criterion of nonlinear impulsive systems is presented, which is applicable to both destabilising impulses and stabilising impulses. Comparing with the available work, the proposed Lyapunov function is allowed to possess indefinite derivatives when deriving from the state trajectory of nonlinear impulsive systems. Finally, two numerical examples are provided to support the proposed criterion.

Indefinite Lyapunov–Razumikhin Functions-Based Stability and Event-Triggered Control of Switched Nonlinear Time-Delay Systems

This brief studies stability and event-triggered stabilization of switched nonlinear time-varying systems by virtue of Lyapunov-Razumikhin functions (LRFs), when the requirement on derivative of LRFs to non-positive is relaxed. A class of such LRFs is first constructed for the input-to-state stability (ISS) of nonlinear time-delay systems. Using multiple LRFs with indefinite derivatives, the ISS of switched nonlinear time-delay systems is presented under average dwell time switching. Then, a nonlinear event-triggered controller is designed for the underlying systems, and it is pointed out that the Zeno phenomenon can be avoided under the proposed controller. Moreover, an event-triggered control is introduced for the systems subject to disturbances to reach the ISS. Finally, one example is given to verify the validity of the proposed approaches.

Indefinite derivative for stability of time-varying nonlinear systems

Abstract This paper is concerned with stability analysis of nonlinear time-varying systems by using Lyapunov function based approach. The classical Lyapunov stability theorems are generalized in the sense that the time-derivative of the Lyapunov functions are allowed to be indefinite. Then, under quite general assumptions, we first present a new converse stability theorem for a large class of time-varying systems which will be used to prove certain stability properties of nonlinear systems with perturbation. Therefore, a new Lyapunov function is presented that guarantees global asymptotic stability under some restrictions on the perturbed system. Furthermore, some illustrative examples are presented.

Novel Fixed-Time Stability Criteria for Discontinuous Nonautonomous Systems: Lyapunov Method With Indefinite Derivative.

This article considers a general class of nonautonomous discontinuous ordinary differential equations (ODE). By constructing the Filippov multimap, the fixed-time stability (FTS) problem of discontinuous ODE is transformed into that of differential inclusion (DI). In order to establish the FTS criteria of the zero solution for DI, the generalized Lyapunov function (LF) method is developed. The generalized LF of this article is relaxed to have an indefinite derivative for almost everywhere along the state trajectories of the system. However, the traditional LF is required to possess negative definite or semi-negative definite derivative for everywhere. As a result, several novel sufficient conditions for FTS are given. Moreover, the settling time of FTS is provided. Then, the theoretical results are applied to solve the fixed-time stabilization control problems of ball motion model and neural networks (NNs) with discontinuities. The developed LF method of FTS is extremely significant in the field of control engineering.

Dynamics of Multi-Link Uncontrolled Wheeled Vehicle

In the paper, the dynamics is investigated of an uncontrolled multi-link wheeled vehicle consisting of trolleys with one wheel pair that are attached to the previous link with metal frames and are capable of rotating around their anchor points. The mechanical system under consideration is nonholonomic, since it has additional nonholonmic kinamic connections. A system of differential equations is obtained that describes the dynamics of motion of this wheeled vehicle using the joint solution of the equations of motion in the Lagrange form with indefinite factors and time derivatives of nonholonomic equations of constraints. The cases of constant and variable location of the center of mass of the leading link are investigated. In the case of a constant location of the center of mass of the leading link, the problem is reduced to the system reduced to the level of the integral of the kinetic energy of the system. The stability of the systems thus obtained is analyzed, and the graphs of the desired functions and phase portraits are constructed.

Stability analysis by a nonlinear upper bound on the derivative of Lyapunov function

In this work, we give results for asymptotic stability of nonlinear time varying systems using Lyapunov-like Functions with indefinite derivative. We put a nonlinear upper bound for the derivation of the Lyapunov Function and relate the asymptotic stability conditions with the coefficients of the terms of this bound. We also present a useful expression for a commonly used integral and this connects the stability problem and Lyapunov Method with the convergency of a series generated by coefficients of upper bound. This generalizes many works in the literature. Numerical examples demonstrate the efficiency of the given approach.

Periodicity and fixed-time periodic synchronization of discontinuous delayed quaternion neural networks

This paper focuses on the issue of periodic orbits analysis for discontinuous quaternion neural networks (QNNs) with time delay. First, based on topological degree theorem of set-valued analysis, a sufficient condition is obtained to ensure the periodicity of discontinuous delayed QNNs via decomposition method. Second, a new fixed-time convergence lemma is proposed to study fixed-time periodic synchronization of discontinuous delayed QNNs. Different from the existing fixed-time convergence lemma, the constructed Lyapunov function (LF) is allowed to possess indefinite derivative for almost everywhere. Moreover, a switching control algorithm is designed to realize the drive-response periodic synchronization in a fixed time. Finally, two examples are presented to substantiate the main results.

Input-to-state stability of systems with non-instantaneous impulses

In this paper, we investigate the input-to-state stability of systems with non-instantaneous impulses. By using the Lyapunov function with indefinite derivative and the average dwell time condition, we get a sufficient condition for the input-to-state stability of systems with noninstantaneous impulses, then we give an example to show the efficiency of the theoretical result obtained in this paper.

Input-to-state stability of systems with non-instantaneous impulses

In this paper, we investigate the input-to-state stability of systems with non-instantaneous impulses. By using the Lyapunov function with indefinite derivative and the average dwell time condition, we get a sufficient condition for the input-to-state stability of systems with noninstantaneous impulses, then we give an example to show the efficiency of the theoretical result obtained in this paper.

Input‐to‐state stability for non‐linear switched stochastic delayed systems with asynchronous switching

In this study, the input-to-state stability (ISS), stochastic-ISS and integral-ISS problems for a class of switched stochastic systems with time delays are studied. A continuously differentiable Lyapunov-Krasovskii function with an indefinite derivative is employed to derive the ISS-type properties of the systems. Two cases are considered: (i) synchronous switching, i.e. candidate controllers coincide with the switching of the system mode; (ii) asynchronous switching, i.e. the candidate controllers have a lag to the switching of the system modes. Furthermore, for asynchronous switching, the authors allow the Lyapunov-Krasovskii function to be time-varying and increasing, respectively, during the time when the active subsystem and the controller match each other. Then, by means of the average dwell-time method together with the Lyapunov-Krasovskii function, they can get the desired ISS-type results. Finally, two examples are given to show the validity of the results.

Stability analysis of linear time-varying time-delay systems by non-quadratic Lyapunov functions with indefinite derivatives

This paper presents a stability analysis of linear time-varying (LTV) time-delay systems by using non-quadratic Lyapunov functions and functionals. Two types of sufficient conditions are proposed for testing several different kinds of stability, such as asymptotic stability, exponential stability and uniformly exponential stability, for LTV systems without delay. Then, by constructing suitable non-quadratic Lyapunov functions (functionals) that are respectively the time-varying weighted L1 and L∞ norms of the state variables and by using properties of the uniformly asymptotically stable (UAS) function and the recently established improved Razumikhin and Krasovskii stability theorems, both delay-dependent and delay-independent conditions are proposed for testing uniformly exponential stability of a class of LTV time-delay systems. The time derivatives of the non-quadratic Lyapunov functions (functionals) along the solutions are allowed to be indefinite, namely, to take both negative and positive values. Numerical examples show that the non-quadratic Lyapunov functions (functionals) based methods are more efficient than the existing ones that are based on quadratic functions (functionals).

Input-to-state stability of nonlinear impulsive systems via Lyapunov method involving indefinite derivative

This paper investigates the input-to-state stability (ISS) and integral-input-to-state stability (iISS) of nonlinear impulsive systems. By using Lyapunov method involving indefinite derivative and average dwell-time (ADT) method, some sufficient conditions for ISS are obtained, where both types of impulses, stabilizing impulses and destabilizing impulses, are considered. In our approach, the time-derivative of the Lyapunov function is not necessarily negative definite, that allows wider applications than existing results in the literature. Several illustrative examples are presented, with their numerical simulations, to demonstrate the main results.