On the equivalence between global recurrence and the existence of a smooth Lyapunov function for hybrid systems

Abstract

We study a weak stability property called recurrence for a class of hybrid systems. An open set is recurrent if there are no finite escape times and every complete trajectory eventually reaches the set. Under sufficient regularity properties for the hybrid system we establish that the existence of a smooth, radially unbounded Lyapunov function that decreases along solutions outside an open, bounded set is a necessary and sufficient condition for recurrence of that set. Recurrence of open, bounded sets is robust to sufficiently small state dependent perturbations and this robustness property is crucial for establishing the existence of a Lyapunov function that is smooth. We also highlight some connections between recurrence and other well studied properties like asymptotic stability and ultimate boundedness. I. INTRODUCTION Hybrid systems are a class of dynamical systems that combine continuous-time evolution and discrete-time events. Several frameworks have been proposed in the literature for analysis of hybrid systems. We refer the reader to [1], [2] and [3] for details. Converse Lyapunov theorems are used to establish the equivalence between asymptotic stability properties and the existence of Lyapunov-like functions that satisfy certain decrease conditions along solutions. Applications of converse theorems in stabilization and robust stability analysis can be found in [4], [5] and [6]. For continuous-time systems converse theorems related to asymptotic stability are established in [6], [7] and [8]. See [9], [10] and [11] for similar results in the discrete-time case. Converse theorems for asymptotic stability of compact sets for a class of hybrid systems has been established in [12] and [13]. In this paper, we establish a converse theorem for a similar class of hybrid systems considered in [12] but for a weaker property called recurrence. Recurrence of an open, bounded set is a weak stability property that is frequently studied in the literature for stochastic systems. We refer the reader to [14], [15] and [16] for details. Loosely speaking, recurrence of an open, bounded set implies that solutions visit the set infinitely often with probability one. It is a weaker notion of stability compared to probabilistic notions of asymptotic stability but nevertheless useful in many applications. In particular, for the problem of RF-source seeking using Micro Aerial Vehicles (MAV) considered in [17] and [18], it is shown The authors are with the Electrical and

Stochastic hybrid systems: A modeling and stability theory tutorial

“…The symbol w represents a Brownian motion defined on the same probability space. Spontaneous transitions, like those considered in the first part of this tutorial, can be converted to forced transitions that fit the model (33). This is done by augmenting the state x with a timer variable τ P R, yielding a combined state px J , τ q J .…”

“…For the sake of a comprehensive and natural stability theory for (33), a solution candidate for (33) should have an appropriate causal dependence on the stochastic processes that drive the system. As spelled out in [5], this causal structure can be expressed in terms of a hybrid filtration tF t,j u pt,jqPRě0ˆZě0 of the underlying probability space, generated from the Brownian motion w and the random process tv i u 8 i"1 , that is right-continuous in t. We refer the reader to [5,§III] for more details.…”

“…As spelled out in [5], this causal structure can be expressed in terms of a hybrid filtration tF t,j u pt,jqPRě0ˆZě0 of the underlying probability space, generated from the Brownian motion w and the random process tv i u 8 i"1 , that is right-continuous in t. We refer the reader to [5,§III] for more details. A solution candidate for (33) is an adapted solution candidate if, for each pt, jq P R ě0ˆZě0 , the set-valued mapping ω Þ Ñ graphpxpωqq X pr0, tsˆt0, . .…”

“…For more details, we refer the reader to [5,§IV]. 7 We use S r pKq to denote the set of solutions to (33) with initial conditions belonging to K Ă R n . 6 As such, due to Item 2 above, x can be thought of as a mapping from Ω to the space of (not identically empty) outer semicontinuous set-valued mappings from R 2 to R n .…”

“…(The simplest example of such a system is the one-dimensional discrete-time system x`" v`, v " µp¨q, where µp¨q corresponds to a Gaussian distribution.) Recurrence of open, bounded sets for non-stochastic hybrid systems has been studied recently in [33], where an equivalent Lyapunov characterization is given.…”