Mohamad S. Alwan

On stability of linear and weakly nonlinear switched systems with time delay

This paper studies time-delayed switched systems that include both stable and unstable modes. By using multiple Lyapunov-functions technique and a dwell-time approach, several criteria on exponential stability for both linear and nonlinear systems are established. It is shown that by suitably controlling the switching between the stable and unstable modes, exponential stabilization of the switched system can be achieved. Some examples and numerical simulations are provided to illustrate our results.

Existence, continuation, and uniqueness problems of stochastic impulsive systems with time delay

This paper studies stochastic impulsive systems with time delay, where the impulse times are state-dependent. Using Ito calculus, we develop the essential foundation of the theory of the mentioned system. In particular, we establish results on local and global existence, forward continuation, and uniqueness of adapted solutions.

Comparison principle and stability of differential equations with piecewise constant arguments

Abstract This paper studies systems of nonlinear differential equations with piecewise constant arguments (EPCA). We develop a comparison principle for this system. Then, this result is used to establish some stability properties of the system. As for the stability results we employ the Lyapunov function approach. We also show that the piecewise (constant) arguments can play as a stabilizing role in some cases where the underlying systems are unstable. A class of linear retarded EPCA are also considered in this paper. Numerical examples and an application to a single-species logistic growth model with density-dependence harvesting are given to show the effectiveness of our theoretical results.

Recent results on stochastic hybrid dynamical systems

This paper addresses stochastic hybrid systems, particularly switched and impulsive systems. The systems under consideration are linear and non-linear continuous with/without time delay, and with/without external force. The importance of using such systems and the discrepancies between hybrid and non-hybrid systems are presented. The objective of this paper is to survey recent developments on some of the important qualitative properties such as stability, stabilisation, input-to-state stability and state estimation along with the challenges that might arise when taking hybridness into account. These properties are mainly developed by using the direct method of Lyapunov or Lyapunov–Razumikhin if time delay is taken into account. Several examples with simulations are presented to further clarify the results introduced in this paper.

Stability and input-to-state stability for stochastic systems and applications

This paper is concerned with establishing some stability and input-to-state stability (ISS) properties in terms of two different measures, h0 and h, for nonlinear systems of stochastic differential equations of Ito type. To analyze these properties, classical Lyapunovs method and a comparison principle are used. To justify the proposed theoretical results, applications to state estimating systems of Luenberger type and feedforward (or cascade) systems enhanced with numerical examples and simulations are presented.

Stability properties of nonlinear stochastic impulsive systems with time delay

Abstract 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.

Switched Singularly Perturbed Systems with Reliable Controllers

This paper addresses the problem of exponential stability for a class of switched control singularly perturbed systems (SCSPS) not only when all the control actuators are operational, but also when some of them experience failures. Multiple Lyapunov functions and average dwell-time switching signal approach are used to establish the stability criteria for the proposed systems. In this paper, we assume that a full access to all the system modes is available, though the mode-dependent, slow-state feedback controllers experience faulty actuators of an outage type. In the stability analysis, the system under study is viewed as an interconnected system that has been decomposed into isolated, lower order, slow and fast subsystems, and the interconnection between them. It has been observed that if the degree of stability of each isolated mode is greater than the interconnection between them, the interconnected mode is exponentially stable, and, then, the full order SCSPS is also exponentially stable for all admissible switching signals with average dwell-time. A numerical example with simulations is introduced to illustrate the validity of the proposed theoretical results.

Input-to-State Stability of Large-Scale Stochastic Impulsive Systems with Time Delay and Application to Control Systems

This chapter deals with large-scale nonlinear delay stochastic systems where the system states are subject to impulsive effects and perturbed by some disturbance input having bounded energy. The interest is to develop a comparison principle and establish input-to-state stability (ISS) in the mean square (m.s.) using vector Lyapunov function and Razumikhin technique. Impulses are being viewed as perturbation to stable systems, and they have a stabilizing role to unstable systems.

Input-to-State Stability and H∞ Performance for Stochastic Control Systems with Piece wise Constant Arguments

This paper addresses stochastic control system of differential equations with piecewise constant arguments (SEPCA ). The piecewise constant arguments are of delay type. The system is viewed as a hybrid (or particularly switched) system . This approach motivates the applicability of the classical theory of ordinary differential equations, but not of functional differential equations, and the design of a switching law. The main theme of this work is to establish the problems of input-to-state stabilization (ISS ), and H ∞ performance for a class of an uncertain control SEPCA. To analyze these result, a common Lyapunov function together with the techniques of differential inequalities and Razumikhin condition is used. A numerical example with simulations is presented to clarify the validity of the proposed theoretical approaches.

Stability Properties of Switched Singular Systems Subject to Impulsive Effects

This paper addresses impulsive switched singular systems with nonlinear perturbation term. The main theme is to establish exponential stability of the systems where the impulses are of fixed time type and treated as perturbation. We first establish the exponential stability of a single-mode impulsive systems using the Lyapunov method . We have observed that if the underlying continuous system is stable and the impulses are applied slowly, then it is guaranteed that the impulsive system maintains the stability property. Later, a switched system with impulsive effects is considered. The method of multiple Lyapunov function and average dwell time switching signal are used. We have noticed that if all subsystems are exponentially stable and the average dwell time is sufficiently large, then the impulsive switched system is exponentially stable. Numerical examples with simulations are given to illustrate the effectiveness of the proposed theoretical results.