Impulsive systems with hybrid delayed impulses: Input-to-state stability

In this paper, we introduce a new type of stability for nonlinear impulsive systems of differential equations, namely practical h-stability. By using the Lyapunov stability theory, some sufficient conditions which guarantee practical h-stability are established. Our original results generalize well-known fundamental stability results, practical stability, practical exponential stability and practical asymptotic stability for nonlinear time-varying impulsive systems. Then two classes of nonlinear impulsive systems, namely perturbed and cascaded impulsive systems, are discussed. Furthermore, the problem of practical h-stabilization for certain classes of nonlinear impulsive systems is considered. Finally, two numerical examples are given to show the effectiveness of our theoretical results.

Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays

Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects

In this paper we investigate the problem of input-to-state stability (ISS) of nonlinear delayed impulsive systems. Both the continuous dynamics and the discrete dynamics of nonlinear impulsive systems are subjected to external input. By virtue of the Razumikhin technique in combination with Lyapunov functions, we obtain some Razumikhin-type theorems that warrant input-to-state stability of nonlinear impulsive systems with time-delays. The Razumikhin-type input-to-state stability theorems cover the cases where the delayed continuous dynamics are input-to-state stable or destabilizing and the discrete dynamics are input-to-state stable or destabilizing, such that nonlinear delayed impulsive systems are able to retain input-to-state stability under certain conditions. The applicability of the derived Razumikhin-type theorems is illustrated by numerical results.

Control vector Lyapunov functions for large-scale impulsive dynamical systems

Exponential stability of infinite-dimensional impulsive stochastic systems with Poisson jumps under aperiodically intermittent control.

Exponential stability of non-linear stochastic delay differential system with generalized delay-dependent impulsive points

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybrid comparison dynamics that are dependent on comparison system states as well as the nonlinear impulsive dynamical system states. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions and hybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamical systems can be addressed via vector Lyapunov functions. Furthermore, we extend the novel notion of control vector Lyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we present a universal decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system. These results are then used to develop decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling uncertainty

This article addresses a class of large-scale stochastic impulsive systems with time delay and time-varying input disturbance having bounded magnitude. The main interest is to develop sufficient conditions for the input-to-state stability (ISS) and stabilization in the presence of impulsive effects. The method of Razumikhin–Lyapunov function is used to develop the ISS and stabilization properties. Later, these results are applied to a class of control systems where the controller actuators are susceptible to failures. It should be noted that our results are delay independent, and the designed reliable controller is robust with respect to the actuator failures and to the system uncertainties. It is also observed that if the isolated continuous system is ISS and subjected to bounded impulsive effects, then the resulting impulsive system preserves the ISS property. Moreover, if the isolated continuous subsystems are all ISS and the interconnection amongst them is bounded from above, then the impulsive interconnected system is ISS provided that the degree of stability of each subsystem is larger than the magnitude of interconnection. If the underlying continuous system is unstable, then the input-to-state stabilization of the impulsive system is guaranteed if the stabilizing impulses are applied to the system frequently. As an implication to these results, if the input disturbance is zero, then the input-to-state stability (or stabilization) reduces to the stability (or stabilization) of the equilibrium state of the underlying disturbance-free system. A numerical example and simulations are provided to illustrate the proposed results.

This paper introduces appropriate concepts of input-to state stability (ISS) for nonlinear impulsive hybrid systems(NIHS). First we study the case in which one kind of dynamics system without external inputs has a stability property, then by applying the ISS-Lyapunov function theory, several sufficient conditions are derived for the ISS property of the whole NIHS with inputs. The results of ISS are used to study the robustly globally uniformly exponential stability for impulsive hybrid systems. The impulsive hybrid system can keep ISS no matter how often the impulses occur. Our proposed results are evaluated using illustrative example to show their effectiveness.

In this paper, we present a practical exponential stability result for impulsive dynamic systems depending on a parameter. Stability theorem and converse stability theorem are established by employing the second Lyapunov method. These theorems are used to analyze the practical exponential stability of the solution of perturbed impulsive systems and cascaded impulsive systems, depending on a parameter. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper finite-time stabilization of nonlinear affine systems, is considered. First we provide a formula for a finite-time stabilizing state feedback control law, for the last class of system. Then we extend the result to the case of a norm-bounded uncertain system and a polytopic uncertain system stabilization are given. Some numerical examples are given to illustrate the main result.

This paper studies the exponential stability of impulsive delay systems. By employing the Razumikhin technique and Lyapunov functions, several exponential stability criteria are established for both linear and non-linear impulsive delay systems. Some examples are also worked through to illustrate our results.

In this paper, a computational method is developed for the bifurcation dynamics of periodic motions in impulsive nonlinear dynamical systems. Such a method based on the implicit mapping method is to discretize the nonlinear dynamical systems in the time interval of two impulses. The impulsive pendulum is studied as an example herein. Using specific mapping structures, periodic motions in the impulsive dynamical systems are determined, and the corresponding stability and bifurcation analysis of the periodic motions are carried out. The impulsive homoclinic orbits related to the different periodic motions are achieved, and the homoclinic bifurcations of periodic motions are determined. The sampled periodic motions and impulsive homoclinic orbits are illustrated for a better understanding of the bifurcation dynamics of impulsive nonlinear systems. Such a method can be applied to other impulsive nonlinear systems.

Aperiodic intermittent control for exponential input-to-state stabilization of nonlinear impulsive systems