Lyapunov stability for impulsive control affine systems

This article studies several notions of Lyapunov stability for impulsive control affine systems in the setting of nonautonomous dynamical systems. It presents some relations between the stability of an impulsive control affine system and the stability of its adjacent control system. Stability of compact sets and their components are specially investigated. Lyapunov functionals are employed to characterize each type of stability of closed sets.     

Stability analysis of impulsive functional systems of fractional order

In this paper, a class of impulsive fractional functional differential systems is investigated. Sufficient conditions for stability of the zero solution are proved, extending the corresponding theory of impulsive functional differential equations. The investigations are carried out by using the comparison principle, coupled with the Lyapunov function method. We apply our results to an impulsive single species model of Lotka–Volterra type.     

Stability analysis of impulsive switched systems with time delays

In this paper, exponential stability criteria of impulsive switched systems with variable delays are introduced. Based on some impulsive delay differential inequalities, some general criteria for the exponential stability are obtained. Finally, an example is given to illustrate the effectiveness of the theory.     

Stability and robustness of quasi-linear impulsive hybrid systems

Both hybrid dynamical systems and impulsive dynamical systems are studied extensively in the literature. However, impulsive hybrid systems are not yet well studied. Nonetheless, many physical systems exhibit both system switching and impulsive jump phenomena. This paper investigates stability and robust stability of a class of quasi-linear impulsive hybrid systems by using the methods of Lyapunov functions and Riccati inequalities. Sufficient conditions for stability and robust stability of those systems are established. Some examples are given to illustrate the applicability of our results.     

Stability of impulsive functional differential equations

In this paper the stability of impulsive functional differential equations in which the state variables on the impulses are related to the time delay is studied. By using Lyapunov functions and Razumikhin techniques, some criteria of stability, asymptotic stability and practical stability for impulsive functional differential equations in which the state variables on the impulses are related to the time delay are provided. Some examples are also presented to illustrate the efficiency of the results obtained.     

Stability properties of nonlinear stochastic impulsive systems with time delay

This article establishes mean square stability and stabilization for stochastic delay systems with impulses. Using Razumikhin methodology, two approaches, classical Lyapunov-based method and comparison principle, are proposed to develop sufficient conditions that guarantee the stability and stabilization properties. It is shown that if the continuous system is stable and the impulses are destabilizing, the impulses should not be applied frequently. On the other hand, if the continuous system is unstable, but the impulses are stabilizing, the impulses should occur frequently to compensate the continuous state growth. Numerical examples are also presented to clarify the proposed theoretical results.     

Modeling and analysis of a modified Leslie–Gower type three species food chain model with an impulsive control strategy

This paper describes a modified Leslie–Gower type three species food chain model with harvesting. We have incorporated impulsive control strategy to the system. Theories of impulsive differential equations, small amplitude perturbation skills and comparison technique are used to study dynamical behavior of the system. Sufficient conditions are derived to ensure global stability of the lowest-level prey and mid-level predator eradication periodic solution. Sufficient conditions are also derived to examine the permanence of the system. Numerical simulations are carried out to verify the analytical results, and the system is analyzed through graphical illustrations. It is observed that the stability of the system exhibits several states, ranging from stable situation to cyclic oscillatory behavior, under different favorable conditions. These results are useful to study the dynamic complexity of ecological systems. The computation of the largest Lyapunov exponent demonstrates the chaotic dynamic nature of the system. The qualitative nature of strange attractor is examined. It is to be noted that the harvesting effort can cause a stable equilibrium to become unstable and even a switching of stabilities.     

Stability of impulsive stochastic differential equations in terms of two measures via perturbing Lyapunov functions

In this paper, we study the stability of nonlinear impulsive stochastic differential equations in terms of two measures. The concept of perturbing Lyapunov functions is introduced to discuss stability properties of solutions of nonlinear impulsive stochastic differential equations in terms of two measures. By using perturbing Lyapunov functions and comparison method, some sufficient conditions for the above stability are given.     

Uniform eventual Lipschitz stability of impulsive systems on time scales

In this paper, we discuss the uniform eventual Lipschitz stability of impulsive system on time scales. By using comparison method, Lyapunov function and analysis technology, some criteria of such stability for system with impulses on time scales are obtained. An example is presented to illustrate the efficiency of proposed results.     

Input-to-state stability of impulsive systems and their networks

This paper considers input-to-state stability (ISS) characterization for a class of impulsive systems which jump map depends on time. We provide sufficient conditions in terms of exponential ISS-Lyapunov functions equipped with an appropriate dwell-time condition for establishing ISS property. Some modifications of dwell times which are more conservative, but easy to be verified are being introduced. We also show that impulsive system with multiple jump maps can naturally represent an interconnection of several impulsive systems with different impulse time sequences. Then we present a procedure to verify ISS of such networks.     

Hybrid event-triggered and impulsive control for time-delay systems

In this paper, we study the problem of hybrid event-triggered control for a class of nonlinear time-delay systems. Using a Razumikhin-type input-to-state stability result for time-delay systems, we design an event-triggered control algorithm to stabilize the given time-delay system. In order to exclude Zeno behavior, we combine the impulsive control mechanism with our event-triggered strategy. In this sense, the proposed algorithm is a hybrid impulsive and event-triggered strategy. Sufficient conditions for the stabilization of the nonlinear systems with time delay are obtained by using Lyapunov method and Razumikhin technique. Numerical simulations are provided to show the effectiveness of our theoretical results.     

Stability of singularly perturbed switched systems with time delay and impulsive effects

This paper studies the stability properties of singularly perturbed switched systems with time delay and impulsive effects. Such systems are assumed to consist of both unstable and stable subsystems. By using the multiple Lyapunov functions technique and the dwell time approach, some stability criteria are established. Our results show that impulses do contribute in order to obtain stability properties even when the system consists of only unstable subsystems. Numerical examples are presented to verify our theoretical results.     

Stability results for impulsive pantograph equations

By employing the Lyapunov functions and Razumikhin technique, some stability results are obtained for pantograph equations with impulses. Our results reveal the fact that certain impulses may make an unstable system stable and that the stability of pantograph equations may also be inherited by impulsive pantograph ones under appropriate impulsive perturbations.     

Stability analysis of impulsive fractional differential systems with delay

In this paper, a class of impulsive fractional differential systems with finite delay is considered. Some sufficient conditions for the finite-time stability of above systems are obtained by using generalized Bellman–Gronwall’s inequality, which extend some known results.     

Exponential stability analysis of impulsive stochastic functional differential systems with delayed impulses

This paper is concerned with the exponential stability analysis of impulsive stochastic functional differential systems with delayed impulses. Although the stability of impulsive stochastic functional differential systems have received considerable attention. However, relatively few works are concerned with the stability of systems with delayed impulses and our aim here is mainly to close the gap. Based on the Lyapunov functions and Razumikhin techniques, some exponential stability criteria are derived, which show that the system will stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous flows. The obtained results improve and complement ones from some recent works. Three examples are discussed to illustrate the effectiveness and the advantages of the results obtained.     

Causal modeling in impulsive systems: A new rigorous non-standard analysis approach

A new approach for modeling nonlinear impulsive system is suggested based on nonstandard analysis. Basic properties of the hyperreals in nonstandard analysis are revisited. Depending on the convergence rate of infinitesimals in hyperreals, a new extended real space is proposed, which extends the one dimensional real line to a countably infinite dimensional extended real space. Generalized functions are defined via a sequential approach on the extended space, which yields a class of Heaviside functions and singular functions. By using the extended functions, a causal way for characterizing jumps in discontinuous system follows. We illustrate the usefulness of the theoretical development by analyzing three simple cases of impulsive affine system: (1) scalar case, (2) multi-dimensional case, and (3) one dimensional horizontal bouncing ball. The results suggest not only the existence of such infinitesimal models within the jump but also how to detour the equilibrium point while connecting the discontinuous state at the impact.     

Feedback stabilization and adaptive stabilization of stochastic nonlinear systems by the control Lyapunov function

Our aims of this paper are twofold: On one hand, we study the asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. We derive sufficient conditions for the existence of control Lyapunov functions (CLFs) leading to the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium state. On the other hand, we consider the previous systems with an unknown constant parameters in the drift and introduce the concept of an adaptive CLF for stochastic system and use the stochastic version of Florchinger's control law to design an adaptive controller. In this framework, the problem of adaptive stabilization of nonlinear stochastic system is reduced to the problem of non-adaptive stabilization of a modified system.