Stability Of Invariant Sets Research Articles

Investigation of Invariant Sets of Itô Stochastic Systems with the Use of Lyapunov Functions

By using Lyapunov functions, we obtain conditions for the invariance and stochastic stability of invariant sets of Ito-type systems.

Non-linear impulsive dynamical systems. Part I: Stability and dissipativity

In this paper we develop Lyapunov and invariant set stability theorems for non-linear impulsive dynamical systems. Furthermore, we generalize dissipativity theory to non-linear dynamical systems with impulsive effects. Specifically, the classical concepts of system storage functions and supply rates are extended to impulsive dynamical systems providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous-time system dynamics and dissipated energy over the resetting instants. Furthermore, extended Kalman‐Yakubovich‐Popov conditions in terms of the impulsive system dynamics characterizing dissipativeness via system storage functions are derived. Finally, the framework is specialized to passive and non-expansive impulsive systems to provide a generalization of the classical notions of passivity and non-expansivity for non-linear impulsive systems. These results are used in the second part of this paper to develop extensions of the small gain and positivity theorems for feedback impulsive systems as well as to develop optimal hybrid feedback controllers.

Non-linear impulsive dynamical systems. Part II: Stability of feedback interconnections and optimality

In a companion paper (Nonlinear Impulsive Dynamical Systems. Part I: Stability and Dissipativity) Lyapunov and invariant set stability theorems and dissipativity theory were developed for non-linear impulsive dynamical systems. In this paper we build on these results to develop general stability criteria for feedback interconnections of non-linear impulsive systems. In addition, a unified framework for hybrid feedback optimal and inverse optimal control involving a hybrid non-linear-non-quadratic performance functional is developed. It is shown that the hybrid cost functional can be evaluated in closed-form as long as the cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability of the non-linear closed-loop impulsive system. Furthermore, the Lyapunov function is shown to be a solution of a steady-state, hybrid Hamilton‐Jacobi‐Bellman equation.

Stabilization of invariant sets for nonlinear non-affine systems

The problem of the global and local stabilization of invariant sets for general nonlinear controlled systems is considered. New state feedback stabilizing controllers and sufficient conditions of asymptotic stability of a goal set with the specified region of attraction are proposed. The proofs of the obtained results are based on the detailed analysis of the ω-limit sets of the closed-loop system with the speed–gradient (or “Jurdjevic–Quinn”) controllers for “affinized” controlled system. Two illustrating examples are given.

The notion of V-detectability and stabilization of invariant sets of nonlinear systems

The problem of global (local) stabilizability of an invariant set of passive nonlinear controlled system is considered. New notion of detectability of the zero value set of a storage function is introduced. It is shown that this property plays a fundamental role in solving of invariant set stabilization problems and in defining the global (local) stabilizing regulator. Sufficient conditions, implying that passive nonlinear system has a detectable zero value set for a smooth storage function, are proposed.

Stabilization of Invariant Sets in Nonlinear Systems: Chetaev's Bundles and Speed-Gradient

Abstract The current results related to stabilization of sets based on SpeedGradient method and the notion of V -detectability are overviewed. Extensions to nonaffine and cascaded systems are presented.

On the stability of invariant sets of a system of autonomous differential equations under constantly acting perturbations

We consider a system of ordinary autonomous differential equations that has an invariant set. We obtain sufficient conditions for the stability of this system under constantly acting perturbations.

Approximation Dynamics and the Stability of Invariant Sets

We introduce the concept of a weakly, normally hyperbolic set for a system of ordinary differential equations. This concept includes the notion of a hyperbolic flow, as well as that of a normally hyperbolic invariant manifold. Moreover, it has the property that it is closed under finite set products. Consequently, the theory presented here can be used for the study of perturbations of the dynamics of coupled systems of weakly, normally hyperbolic sets. Our main objective is to show that under a smallC1-perturbation, a weakly, normally hyperbolic setKis preserved by a homeomorphism, where the imageKYis a compact invariant set, with a related hyperbolic structure, for the perturbed equation. In addition, the homeomorphism is close to the identity inC0,1and the perturbed dynamics onKYare close to the original dynamics onK.

Asymptotic Stability of Invariant Sets of Nonlinear Differential Inclusions

Asymptotic Stability of Invariant Sets of Nonlinear Differential Inclusions

Modeling and Qualitative Theory for General Hybrid Dynamical and Control Systems

Hybrid dynamical systems which are capable of exhibiting simultaneously several kinds of dynamic behavior, such as continuous-time dynamics, discrete-time dynamics, jump phenomena, switching and logic commands, discrete events, and the like, are of great current interest. In the present paper we employ a general model of dynamical system suitable in the qualitative analysis of such systems. This model of dynamical system allows discontinuous motions, and convergence of motions is relative to generalized time. For the model of hybrid dynamical systems described above, results are established for the principal Lyapunov stability of invariant sets and the principal Lagrange stability of motions. Some of the results of the present paper are applied in the analysis of a specific class of systems.

SYMMETRY BREAKING BIFURCATIONS OF CHAOTIC ATTRACTORS

In an array of coupled oscillators, synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries, and this leads to the idea of symmetry breaking bifurcations of chaotic attractors. Finally, the results and ideas are illustrated for several systems of coupled oscillators.

Invariant set stability in discrete non-linear systems

Sufficient conditions for local stability of invariant sets in non-linear discrete systems with time-varying parameters and difference inclusions are investigated. The results obtained are a step towards establishing an analogue of Lyapunov's First Method for the systems under consideration.

The existence and stability of invariant sets of dynamical systems

The possibility of using Lyapunov functions to construct invariant sets of dynamical systems is discussed. The investigations presented herein are based on certain ideas known from the literature /1–11/ and culminate in a generalization of Routh's Theorem and its modification /1–6, 12, 13/.

Density of the set of attractive compacta

A compact set K in a smooth closed manifold M is said to be attractive, if on M there exists a system of differential equations, for which K is an asymptotically stable invariant set. It is proved that the set of attractive compacta is dense and its complement contains a dense set of type Gδ in the space of all compacta of the manifold M endowed with two natural topologies.

Singular perturbations, large-scale systems and asymptotic stability of invariant sets

The paper deals with asymptotic stability of invariant sets of large-scale non-linear time-varying systems subjected to singular perturbations and possibly to external disturbances and arbitrary structural variations. New aggregation of the systems proposed in the paper reduces the asymptotic stability test to verification of negative definiteness of lower order symmetric aggregation matrices that are independent of small parameters.

Iterations of continuous mappings on metric spaces Asymptotic stability and Lyapunov functions

The stability of difference equations, which represent discrete-time motions, is studied on general metric spaces. An analogous theorem to the equivalence between the asymptotic stability of invariant sets and the existence of Lyapunov functions for continuous-time motions is proved. One consequence of the results is the reduction of the asymptotic stability to an invariance condition.