Stability analysis of switched homogeneous time-delay systems under synchronous and asynchronous commutation

In this work, stability analysis for a class of switched nonlinear time-delay systems is performed by applying Lyapunov–Krasovskii and Lyapunov–Razumikhin approaches. It is assumed that each subsystem in the family is homogeneous (of positive or negative degree) and asymptotically stable in the delay-free setting. The cases of existence of a common or multiple Lyapunov–Krasovskii functionals and a common Lyapunov–Razumikhin function are explored. The scenarios with synchronous and asynchronous switching are considered, and it is demonstrated that depending on the kind of commutation, one of the frameworks for stability analysis outperforms another, but finally leading to similar restrictions for both types of switching (despite the asynchronous one seems to be more demanded). The obtained results are applied to mechanical systems having restoring forces with real-valued powers.     

Global feedback stabilization of multi-input bilinear systems

This paper mainly investigates the problem of stabilization of homogeneous bilinear systems with multiple inputs. Explicit state feedback laws are given to stabilize the bilinear systems. Meanwhile, an estimate of the convergence speed is obtained under the given feedback laws. Besides, sufficient conditions, which are easy to be verified, are presented for the stabilization of the bilinear systems.     

Nonlinear stability in reaction-diffusion systems via optimal Lyapunov functions

We define optimal Lyapunov functions to study nonlinear stability of constant solutions to reaction-diffusion systems. A computable and finite radius of attraction for the initial data is obtained. Applications are given to the well-known Brusselator model and a three-species model for the spatial spread of rabies among foxes.     

Robust stability of discrete-time state-delayed systems employing generalized overflow nonlinearities

This paper considers the problem of global asymptotic stability of a class of nonlinear uncertain discrete-time state-delayed systems. The class of systems under investigation involves multiple state delays, norm-bounded parameter uncertainties, and generalized overflow nonlinearities which cover usual types of overflow arithmetic used in practice. A new criterion for the global asymptotic stability of such systems is presented. A numerical example is given to illustrate the applicability of the criterion presented.     

Lyapunov theorems for systems described by retarded functional differential equations

Lyapunov-like characterizations for non-uniform in time and uniform robust global asymptotic stability of uncertain systems described by retarded functional differential equations are provided.     

The variational Lyapunov function and strict stability theory for differential systems

In this paper, we investigate strict stability of differential systems by variational Lyapunov function. We obtain some sufficient conditions and comparison theorems.     

Asymptotic stability of differential systems of neutral type

We offer sufficient conditions for the asymptotic stability of the equilibrium point of linear neutral differential systems. An application of our results to a family of artificial neural networks of neutral type is also illustrated.     

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.     

Guaranteed asymptotic stability for a class of uncertain linear dynamical systems

We consider a class of linear dynamical systems with bounded, Lebesgue-measurable uncertainties in the system and input matrices as well as in the input itself. A state feedback control is derived, which guarantees global, uniform asymptotic stability of the zero state; this control is continuous, except at the zero state.This paper is based in part on research supported by the National Science Foundation.     

A semi-algebraic approach for asymptotic stability analysis

This paper deals with the problem of computing Lyapunov functions for asymptotic stability analysis of autonomous polynomial systems of differential equations. We propose a new semi-algebraic approach by making advantage of the local property of the Lyapunov function as well as its derivative. This is done by first constructing a semi-algebraic system and then solving this semi-algebraic system in an adaptive way. Experiment results show that our semi-algebraic approach is more efficient in practice, especially for low-order systems.     

Aggregation and stability analysis of nonlinear complex systems

The problem of stability of large-scale systems in critical cases is investigated. New form of aggregation for essentially nonlinear complex systems is suggested. With the help of this form the sufficient conditions of asymptotic stability are determined. The results obtained are used for the stability analysis of complex systems by the nonlinear approximation and for the investigation of absolute stability conditions for a certain class of nonlinear systems.     

Asymptotic stability of switched linear time-varying systems based on top-floor function

This paper considers the problem of asymptotic stability for switched linear time-varying (SLTV) systems. First, some stability conditions for SLTV systems are given by using infinite integrals. Then, based on the results obtained, two stability conditions are proposed by combining the methods of top-floor function and average dwell time. Moreover, using strict top-floor function, a stability condition is also provided when some subsystems are unstable. With the help of top-floor function, the stability problem of SLTV systems can be simplified and solved by using the technique of linear matrix inequalities (LMIs). Finally, a numerical example is given to illustrate the theoretical results.     

Lyapunov stability of discontinuous dynamic systems

This paper derives necessary and sufficient conditions for thestability of dynamic systems in the sense of Lyapunov. Theseconditions are used to study the stability of discontinuousdynamic systems, which include fuzzy systems, hybrid systems,and impulsive differentia^! systems.     

Conley index condition for asymptotic stability

In this paper, we use Conley index theory to develop necessary conditions for stability of equilibrium and periodic solutions of nonlinear continuous-time systems. The Conley index is a topological generalization of the Morse theory which has been developed to analyze dynamical systems using topological methods. In particular, the Conley index of an invariant set with respect to a dynamical system is defined as the relative homology of an index pair for the invariant set. The Conley index can then be used to examine the structure of the system invariant set as well as the system dynamics within the invariant set, including system stability properties. Efficient numerical algorithms using homology theory have been developed in the literature to compute the Conley index and can be used to deduce the stability properties of nonlinear dynamical systems.     

EVENTUAL STABILITY OF IMPULSIVE DIFFERENTIAL SYSTEMS

In this article, criteria of eventual stability are established for impulsive differential systems using piecewise continuous Lyapunov functions. The sufficient conditions that are obtained significantly depend on the moments of impulses. An example is discussed to illustrate the theorem.     

Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems

In this note, we study the exponential stability of impulsive functional differential systems with infinite delays by using the Razumikhin technique and Lyapunov functions. Several Razumikhin-type theorems on exponential stability are obtained, which shows that certain impulsive perturbations may make unstable systems exponentially stable. Some examples are discussed to illustrate our results.     

时滞脉冲切换系统的实用稳定性-带Razumikhin条件的Lyapunov函数方法

用带Razumikhin条件的Lyapunov函数方法研究了一般形式的时滞脉冲切换系统的实用稳定性,得到了时滞脉冲切换系统实用稳定、一致实用稳定的充分条件.最后,给出了具体的例子及其数值模拟.     

Lyapunov redesign for a class of uncertain systems with delays

We address the robust stabilization problem of a general class of uncertain linear systems with multiple delays from a Lyapunov redesign (LR) methodology that relies on Sliding Mode Control (SMC) theory. The proposed approach, which is based on a well-known (yet not used for LR) class of Lyapunov–Krasovskii functionals, allows us to enlarge the type of uncertain systems with delays that can be stabilized. The uncertainties of the system are compensated within a delay-free sliding manifold that is taken from the time-derivative of the considered functional.