Converse Lyapunov Theorem Research Articles

Lyapunov's second method for nonautonomous differential equations

Converse Lyapunov theorems are presented for nonautonomous systems modelled as skew product flows. These characterize various types of stability of invariant sets and pullback, forward and uniform attractors in such nonautonomous systems.

Stability Criteria for Switched and Hybrid Systems

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

A converse Lyapunov theorem for almost sure stabilizability

We prove a converse Lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions: given a stochastic system a.s. stochastic open-loop stabilizable at the origin, we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium. This result provides, with the direct Lyapunov theorems proved in a companion paper, a complete Lyapunov-like characterization of the a.s. stabilizability.

A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations

An ordinary differential equation's (ODE) equilibrium is asymptotically stable, if and only if the ODE possesses a Lyapunov function, that is, an energy-like function decreasing along any trajectory of the ODE and with exactly one local minimum. Theorems regarding the `only if' part are called converse theorems. Recently, the author presented a linear programming problem, of which every feasible solution parameterizes a Lyapunov function for the nonlinear autonomous ODE in question. In 2004 the author proved the first general constructive converse theorem by showing that if the equilibrium of the ODE is exponentially stable, then the linear programming problem possesses a feasible solution. In this paper we prove a constructive converse theorem on asymptotic stability for nonlinear autonomous ODEs and so improve the 2004 results. The only restriction on the ODE = f(x) is that f is a class function. Note, that these results imply that the algorithm presented by the author in 2002 is capable of constructing a Lyapunov function for all nonlinear systems, of which the equilibrium is asymptotically stable.

Input-to-state stability for discrete-time time-varying systems with applications to robust stabilization of systems in power form

Input-to-state stability (ISS) of a parameterized family of discrete-time time-varying nonlinear systems is investigated. A converse Lyapunov theorem for such systems is developed. We consider parameterized families of discrete-time systems and concentrate on a semiglobal practical type of stability that naturally arises when an approximate discrete-time model is used to design a controller for a sampled-data system. An application of our main result to time-varying periodic systems is presented, and this is used to solve a robust stabilization problem, namely to design a control law for systems in power form yielding semiglobal practical ISS (SP-ISS).

A Converse Lyapunov Theorem for Linear Parameter-Varying and Linear Switching Systems

We study families of linear time-varying systems, where time variations have to satisfy restrictions on the dwell time, that is, on the minimum distance between discontinuities, as well as on the derivative in between discontinuities. Such classes of systems may be formulated as linear flows on vector bundles. The main objective of this paper is to construct parameter-dependent Lyapunov functions, which characterize the exponential growth rate. This is possible in the generic irreducible case. As an application the Gelfand formula is generalized to the class of systems studied here. In other words, the maximal exponential growth rate may be approximated by only considering the periodic systems in the family of time-varying systems. A perspective on the question of continuous dependence of the exponential growth rate on the data is given.

Nonlinear small-gain theorems for discrete-time feedback systems and applications☆

We derive in this work a local nonlinear small-gain theorem in the framework of input-to-state stability for discrete time systems. Our primary objective is to show that, as in the continuous-time context, these discrete-time nonlinear small-gain theorems are very effective in stability analysis and synthesis for various classes of discrete-time control systems. Two converse Lyapunov theorems for discrete exponential stability are developed to assist these applications. New results in stability and stabilization presented in this paper are significant extensions of previous work by other authors (IEEE Trans. Automat. Control 38 (1993) 1398; 39 (1994) 2340; 33 (1988) 1082).

Common Lyapunov functions for families of commuting nonlinear systems

We present constructions of a local and global common Lyapunov function for a finite family of pairwise commuting globally asymptotically stable nonlinear systems. The constructions are based on an iterative procedure, which at each step invokes a converse Lyapunov theorem for one of the individual systems. Our results extend a previously available one which relies on exponential stability of the vector fields.

Nonlinear small-gain theorems for discrete-time feedback systems and applications

We derive in this work a local nonlinear small-gain theorem in the framework of input-to-state stability for discrete time systems. Our primary objective is to show that, as in the continuous-time context, these discrete-time nonlinear small-gain theorems are very effective in stability analysis and synthesis for various classes of discrete-time control systems. Two converse Lyapunov theorems for discrete exponential stability are developed to assist these applications. New results in stability and stabilization presented in this paper are significant extensions of previous work by other authors (IEEE Trans. Automat. Control 38 (1993) 1398; 39 (1994) 2340; 33 (1988) 1082).

ON LIPSCHITZ CONTINUITY OF THE TOP LYAPUNOV EXPONENT OF LINEAR PARAMETER VARYING AND LINEAR SWITCHING SYSTEMS

We study families of time-varying linear systems, where time-variations have to satisfy restrictions on the dwell time, that is, on the minimum distance between discontinuities, as well as on the derivative in between discontinuities. For this class of systems we study continuity properties of the growth rate as a function of the systems' data. It is shown by example, that a straightforward topology on the space of systems does not yield the desired continuity result. A new natural metric is introduced and a continuity result is obtained. Furthermore, local Lipschitz continuity may be shown for the (generic) case of irreducible systems. The methods rely heavily on a recent converse Lyapunov theorem for the class under consideration.

Weak Converse Lyapunov Theorems and Control-Lyapunov Functions

Given a weakly uniformly globally asymptotically stable closed (not necessarily compact) set ${\cal A}$ for a differential inclusion that is defined on $\mathbb{R}^n$, is locally Lipschitz on $\mathbb{R}^n \backslash {\cal A}$, and satisfies other basic conditions, we construct a weak Lyapunov function that is locally Lipschitz on $\mathbb{R}^n$. Using this result, we show that uniform global asymptotic controllability to a closed (not necessarily compact) set for a locally Lipschitz nonlinear control system implies the existence of a locally Lipschitz control-Lyapunov function, and from this control-Lyapunov function we construct a feedback that is robust to measurement noise.

A constructive converse Lyapunov theorem on exponential stability

Closed physical systems eventually come to rest, the reason beingthat due to friction of some kind they continuously lose energy.The mathematical extension of this principle is the concept of aLyapunov function. A Lyapunov function for a dynamical system, ofwhich the dynamics are modelled by an ordinary differentialequation (ODE), is a function that is decreasing along anytrajectory of the system and with exactly one local minimum. Thisimplies that the system must eventually come to rest at thisminimum. Although it has been known for over 50 years that theasymptotic stability of an ODE's equilibrium isequivalent to the existence of a Lyapunov function for the ODE,there has been no constructive method for non-local Lyapunovfunctions, except in special cases. Recently, a novel method toconstruct Lyapunov functions for ODEs via linear programming waspresented [5], [6], which includes an algorithmicdescription of how to derive a linear program for a continuousautonomous ODE, such that a Lyapunov function can be constructedfrom any feasible solution of this linear program. We will showhow to choose the free parameters of this linear program,dependent on the ODE in question, so that it will have a feasiblesolution if the equilibrium at the origin is exponentially stable.This leads to the first constructive converse Lyapunov theorem inthe theory of dynamical systems/ODEs.

A Converse Lyapunov Theorem for Nonuniform in Time Global Asymptotic Stability and Its Application to Feedback Stabilization

Lyapunov-like characterizations for the concepts of nonuniform in time robust global asymptotic stability and input-to-state stability for time-varying systems are established. The main result of our work enables us to derive (1) necessary and sufficient conditions for feedback stabilization for affine in the control systems and (2) sufficient conditions for the propagation of the input-to-state stability property through integrators.

An invariance principle for discrete-time nonlinear systems

In this paper, we derive an invariance principle generalizing LaSalle's invariance principle for discrete-time nonlinear systems. Next, using the invariance principle, we develop a series of results relating stability, observability, and converse Lyapunov theorems for discrete-time nonlinear systems.

A converse Lyapunov theorem for, non-uniform in time, global exponential robust stability

Lyapunov-like characterizations are established for the concepts of, non-uniform in time, global exponential robust stability and input-to-state stability for time-varying control systems.

A converse Lyapunov theorem for discrete-time systems with disturbances

This paper presents a converse Lyapunov theorem for discrete-time systems with disturbances taking values in compact sets. Among several new stability results, it is shown that a smooth Lyapunov function exists for a family of time-varying discrete systems if these systems are robustly globally asymptotically stable.

Converse Lyapunov Theorems for Non-Uniform in Time Global Asymptotic Stability and Stabilization by Means of Time-Varying Feedback

Two converse Lyapunov Theorems are presented concerning non-uniform in time global asymptotic stability. A necessary and sufficient condition for stabilization of affine in the control systems by means of a time-varying feedback is established.

On Stability and Stabilizability of Nonlinear Inter-Connected Discrete Time Systems

In this work we discuss applications of nonlinear small-gain theorems in the framework of input-to-state stability for discrete time systems. Our primary objective is to show that, as in the continuous-time context, the discrete-time nonlinear small-gain theorems are very effective in the stability analysis and synthesis for interconnected systems. A converse Lyapunov theorem for discrete exponential stability is developed to assist these applications.

On converse Lyapunov theorems for ISS and iISS switched nonlinear systems

In this paper we present converse Lyapunov theorems for ISS and integral input to state stable (iISS) switched nonlinear systems. Their proofs are based on existing converse Lyapunov theorems for input–output to state stable and iISS nonlinear systems, and on the association of the switched system with a nonlinear system with inputs and disturbances that take values in a compact set.

A converse Lyapunov theorem for nonlinear switched systems

In this paper we present a converse Lyapunov theorem for uniform asymptotic stability of switched nonlinear systems. Its proof is a simple consequence of some results on converse Lyapunov theorems for systems with bounded disturbances obtained by Lin et al. (SIAM J. Control Optim. 34 (1996) 124–160), once an association of the switched system with a nonlinear system with disturbances is established.