Non-uniform Time Domain Research Articles

Analysis of Hyers–Ulam stability and controllability of non-linear switched impulsive systems with delays on time scales

In this article, we have used the concepts of time scales theory to discuss nonlinear switched impulsive systems with delay. Our main objective is to determine the Hyers–Ulam stability and controllability of nonlinear switched impulsive systems with delay on non-uniform time domains. To obtain the necessary and sufficient conditions for existence, Hyers–Ulam stability, and controllability, we utilize the Banach fixed-point theorem and Krasnoselskii’s fixed-point theorem. In order to demonstrate our conclusions, we have discussed some simulation-based examples along with the three tank liquid control problem and a potential practical situation related to the infectious disease with switching rules. The results of this manuscript provide all the necessary and sufficient conditions for Hyers–Ulam stability and controllability that are true for discrete, continuous as well as unified time domains simultaneously.

An analysis of hybrid impulsive prey-predator-mutualist system on nonuniform time domains

Abstract In this work, we propose a hybrid impulsive prey-predator-mutualist model on nonuniform time domains. We have investigated the permanence/persistence results for the proposed model using the comparison theorems of impulsive differential equations and some dynamic inequality on the nonuniform time domains. In addition, we have established certain necessary requirements for the uniform asymptotic stability of the almost periodic solution and global attractivity of the proposed model. Furthermore, we provide several numerical examples on nonuniform time domains with computer simulation to demonstrate the viability of the results of the acquired analytical work.

Existence, controllability and Hyers–Ulam stability of a hybrid neutral switched system with impulsive effects

In this study, our main focus is on the analysis of the existence, controllability, and Hyers–Ulam stability of a neutral switched system with impulsive effects on non-uniform time domains. First, we use the Banach fixed point theorem and the time scales calculus to explore the existence, Ulam's type stability, and controllability of a considered switched system. In the end, we examine the controllability results through numerical simulations. This article aims to identify the essential requirements for achieving controllability and stability on different time scales.

IMPACT OF HUNTING COOPERATION AND FEEDBACK CONTROL FOR A NONLINEAR HYBRID LESLIE–GOWER PREDATOR-PREY SYSTEM ON NONUNIFORM TIME DOMAIN

We propose a hybrid Leslie–Gower predator-prey system with feedback control and hunting cooperation among predators on nonuniform time domain. Our proposed model commutes between a continuous-time dynamical system and discrete-time dynamical system. Using comparison theorems and some dynamic inequality on the time scale, we study the permanence results for the proposed system. Also, with the help of a suitable Lyapunov function, we establish some sufficient conditions that will ensure the global attractivity of the proposed model. Furthermore, simulation studies are provided to verify the theoretical results, as well as to further investigate the effects of feedback control strategy and hunting cooperation on the predator-prey system. These findings suggest that hunting cooperation and feedback control strategy have a substantial influence on controlling the population of a predator-prey dynamical system.

Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.

Exponential Lag Synchronization of Cohen–Grossberg Neural Networks with Discrete and Distributed Delays on Time Scales

AbstractIn this article, we investigate exponential lag synchronization results for the Cohen–Grossberg neural networks with discrete and distributed delays on an arbitrary time domain by applying feedback control. We formulate the problem by using the time scales theory so that the results can be applied to any uniform or non-uniform time domains. Also, we provide a comparison of results that shows that obtained results are unified and generalize the existing results. Mainly, we use the unified matrix-measure theory and Halanay inequality to establish these results. In the last section, we provide two simulated examples for different time domains to show the effectiveness and generality of the obtained analytical results.

Existence and Stability Results for Coupled Fractional Dynamic System with Impulses over Non-uniform Time Domains

Abstract In this manuscript, we examine the existence, uniqueness and stability results for a coupled fractional dynamical system with impulsive and initial-boundary (IB) conditions on non-uniform time domains by implying the theory of time scales. Firstly, we find the solution of the consider problem, then we examine the existence and stability results. Mainly, we use the Banach and Leray-Schauder’s alternative type fixed point theorems to establish the main results. At last, we present an example to illustrate the utilization of these analytical outcomes.

Time scale reachability and controllability of time‐varying linear systems

AbstractIn this paper, we investigate the analysis of reachability and controllability for time‐varying linear systems evolving on nonuniform time domains also known as arbitrary time scales. Based on appropriate operators, some necessary and sufficient conditions are derived to guarantee the reachability for this class of systems using linear algebra tools. Furthermore, a link between optimality and the proposed operators is given. Then, the connection between reachability and controllability analysis on arbitrary time scales is discussed. Through the paper, some examples illustrate the theoretical results.

Time scale observability and constructibility of linear dynamic equations

This paper investigates the observability and constructibility problems of time-varying linear dynamic equations using time scale theory. First, we define observability, reachability and constructibility operators on time scales. Some necessary and sufficient conditions are proposed to ensure the observability on non-uniform time domains based on some linear algebra tools. Then, constructibility is also examined using the same approach. Moreover, the link between observability and constructibility concepts on arbitrary time sets is discussed. Further, the observability and reachability duality relationship for time-varying linear systems on time scales is established. The current work unifies and extends some existing results given for standard cases (i.e. the continuous line and the discrete time domain) to non-uniform time domains. Finally, the obtained results are described with an illustrative example.

Time scale state feedback h-stabilisation of linear systems under Lipschitz-type disturbances

This paper studies the h-stabilisation problem of certain classes of perturbed systems on time scales. Sufficient conditions for the control law design are proposed to ensure the h-stability of the closed-loop dynamical system under Lipschitz-type disturbances. Using the Gronwall inequality approach with time scale theory, the h-stability of the closed-loop system is investigated in non-uniform time domains with bounded graininess. Some numerical examples are provided to show the feasibility of the obtained results using the proposed approach for systems evolving on some arbitrary time scales.

Extensions of Razumikhin-type stability theorems for nonlinear time-delay systems on time scales

This paper focuses on the stability analysis of nonlinear time-delay systems on time scales, which are of generality as they can include not only the traditional continuous and discrete ones, but also some other cases, such as systems on general uniform or non-uniform time domains. One existing classical method for analyzing stability of such systems is the Razumikhin-type theorem with requiring the time derivative of relating Lyapunov function to be non-positive for uniform stability and negative for uniform asymptotic or exponential stability, which are difficult to be satisfied. To relax these restrictions, by introducing the time-scale type uniformly stable function and uniformly asymptotically stable function, this paper presents several less conservative stability criteria, in which the time-scale time derivative of Lyapunov function can be positive or non-negative. To demonstrate the effectiveness of the theoretic results, a numerical example about non-continuous and non-discrete time-delay systems is given.

Stability analysis of switched systems on time scales with all modes unstable

In this paper, global uniform asymptotic stability problems of switched systems on time scales with all modes unstable are considered. First, a more general stability criterion is proposed for nonlinear switched systems on time scales, in which the value of Lyapunov function at some switching instants may not necessarily decrease. Second, based on the discretized Lyapunov function technique, a sufficient condition is derived for stability analysis of linear switched systems with all subsystems unstable on a special kind of time scale (the graininess function is constant). It is shown that this approach does not require the system matrix of one subsystem must commute with the one of the other subsystem, which is essential for stability analysis of linear switched systems on non-uniform time domains by the eigenvalue based approach in the existing results. Moreover, the idea of this paper not only provides a unified approach to study continuous switched systems and their discrete counterparts simultaneously, but also is effective to investigate stability problems of other systems on uniform or non-uniform time domains. Two examples are given for illustrating the effectiveness of the proposed results.

Stability analysis of a class of switched nonlinear systems using the time scale theory

This paper investigates stability for a class of switched nonlinear systems evolving on a non-uniform time domain. The studied class of systems switch between a nonlinear continuous-time and a nonlinear discrete-time subsystem. This problem is formulated using time scale theory where the latter is formed by a union of disjoint intervals with variable lengths and variable gaps. Using Gronwall’s inequality and properties of the time scale exponential function, sufficient conditions are derived to ensure exponential stability. The results can be applied to cases where the continuous-time subsystem or the discrete-time subsystem is not necessarily stable, and the state matrices of each subsystem are not necessarily pairwise commuting. To illustrate the effectiveness of the proposed scheme, the consensus problem for general linear multi-agent systems under intermittent information transmissions is studied under this framework.

Stability of switched systems on non-uniform time domains with non commuting matrices

The time scale theory is introduced to study the stability of a class of switched linear systems on non-uniform time domains, where the dynamical system commutes between a continuous-time linear subsystem and a discrete-time linear subsystem during a certain period of time. Without assuming that matrices are pairwise commuting, some conditions are derived to guarantee the exponential stability of the switched system by considering that the continuous-time subsystem is stable and the discrete-time subsystem can be stable or unstable. Some examples show the effectiveness of the proposed scheme.

Positive Reachability and Observability of Linear Positive Time-Variant Systems on Non-Uniform Time Domains

Positive reachability and positive observability of positive linear time-variant systems on non-uniform time domains are studied. Such systems are treated as particular cases of linear systems on arbitrary time scales. Differential and integral calculus on time scales allows for unified treatment of continuous- and discrete-time systems. Positive reachability and observability are characterized with the aid of associated reachability and observability matrices. It is shown that the properties are dual to each other. Two types of discretizations of continuous-time systems are considered: Euler discretization and sampling. It is shown that both discretizations preserve positive reachability.

Stability analysis of switched linear systems on non-uniform time domain with unstable subsystems

This paper deals with the stability analysis of a class of switched linear systems evolving on an arbitrary time domain. This class consists of a set of unstable linear continuous-time and unstable linear discrete-time subsystems. Using the time scale theory, some sufficient conditions are derived to guarantee the exponential stability of this class of systems with all modes unstable. Simulation results show the effectiveness of the proposed scheme.

State feedback stabilization of a class of uncertain nonlinear systems on non-uniform time domains

This paper deals with the stabilization problem for a class of uncertain nonlinear systems on non-uniform time domains. Some sufficient conditions are derived to design a state feedback controller for a class of uncertain nonlinear time-varying systems under vanishing perturbations. Using some Gronwall’s integral inequalities, the uniform exponential stability of the closed-loop systems on arbitrary time scales, is guaranteed. Then, based on the Lyapunov theory, new sufficient conditions are proposed to derive the controller which ensures the practical stability of the closed-loop time-invariant nonlinear uncertain system under non-vanishing perturbations. Some examples illustrate these results.

Region of exponential stability of switched linear systems on time scales

This paper deals with the stability analysis of a class of linear time-invariant switched systems on non-uniform time domains. The considered class consists of a set of linear continuous-time and linear discrete-time subsystems. Some sufficient and necessary conditions are derived to guarantee the exponential stability of this class of systems on time scales with bounded graininess function, introducing a region of exponential stability. Some examples illustrate these results.

Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions

This paper deals with the stability analysis of a class of uncertain switched systems on non-uniform time domains. The considered class consists of dynamical systems which commute between an uncertain continuous-time subsystem and an uncertain discrete-time subsystem during a certain period of time. The theory of dynamic equations on time scale is used to study the stability of these systems on non-uniform time domains formed by a union of disjoint intervals with variable length and variable gap. Using the concept of common Lyapunov function, sufficient conditions are derived to guarantee the asymptotic stability of this class of systems on time scale with bounded graininess function. The proposed scheme is used to study the leader–follower consensus problem under intermittent information transmissions.

Stability analysis of a class of switched linear systems on non-uniform time domains

This paper deals with the stability analysis of a class of switched linear systems on non-uniform time domains. The considered class consists of a set of linear continuous-time and linear discrete-time subsystems. First, some conditions are derived to guarantee the exponential stability of this class of systems on time scales with bounded graininess function when the subsystems are exponentially stable. These results are extended when considering an unstable discrete time subsystem or an unstable continuous-time subsystem. Some examples illustrate these results.