Forty-Fourth Annual Allerton Conference WID.150 Allerton House, UIUC, Illinois, USA Sept 27-29, 2006 Duality in Stability Theory: Lyapunov function and Lyapunov measure Umesh Vaidya Abstract— In this paper, we introduce Lyapunov measure to to each other. While the dynamical system describes the evo- verify weak (a.e.) notions of stability for an invariant set of lution of an initial condition, the P-F operator describes the a nonlinear dynamical system. Using certain linear operators evolution of uncertainty in initial conditions. Under suitable from Ergodic theory, we derive Lyapunov results to deduce a.e. stability. In order to highlight the linearity of our approach, we technical conditions, spectrum of these operator on the unit also provide explicit formulas for obtaining Lyapunov function circle provides information about the asymptotic behavior of and Lyapunov measure in terms of the resolvent of the two the system [DJ99], [MB04]. In this paper, spectral analysis linear operators. of the stochastic operators is used to study the stability properties of the invariant sets of deterministic dynamical I. INTRODUCTION systems. In particular, we introduce Lyapunov measure as For nonlinear dynamical systems, Lyapunov function a dual to Lyapunov function. Lyapunov measure is closely based methods play a central role in both stability anal- related to Rantzer’s density function, and like its counterpart ysis and control synthesis [Vid02]. Given the complexity it is shown to capture the weaker a.e. notion of stability. of dynamical behavior possible even in low dimensions Just as invariant measure is a stochastic counterpart of the [ER85], these methods are powerful because they provide invariant set, existence of Lyapunov measure is shown to an analysis and design approach for global stability of an give a stochastic conclusion on the stability of the invariant equilibrium solution. However, unlike linear systems, there measure. The key advantage of relating Lyapunov measure is no constructive procedure to obtain a Lyapunov function to the P-F operator is that the relationship serves to provide for general nonlinear systems. This lack of constructive explicit formulas of the Lyapunov measure. procedure for Lyapunov function is an important barrier in For stable linear dynamical systems, the Lyapunov func- nonlinear control theory. For nonlinear ODEs, two ideas have tion can be obtained as a positive solution of the so-called appeared in recent literature towards overcoming this barrier. Lyapunov equation. The equation is linear and the Lyapunov In [Ran01], Rantzer introduced a dual to the Lyapunov function is efficiently computed and can even be expressed function, referred to by the author as a density function, to analytically as an infinite-matrix-series expansion. For the define and study weaker notion of stability of an equilibrium series to converge, there exists a spectral condition on the solution of nonlinear ODEs. The author shows that the exis- linear dynamical system (ρ(A) < 1). The P-F formulation tence of a density function guarantees asymptotic stability in allows one to generalize these results to the study of sta- an almost everywhere sense, i.e., with respect to any set of bility of invariant and possibly chaotic attractor sets of initial conditions in the phase space with a positive Lebesgue nonlinear dynamical systems. More importantly, it provides measure. The second idea involves computation of Lyapunov a framework that allows one to carry over the intuition functions using SOS polynomials. This idea appears in the of the linear dynamical systems. For instance, the spectral work of Parrilo [Par00], where the construction of Lyapunov condition is now expressed in terms of the P-F operator. function is cast as a linear problem with suitable choice The Lyapunov measure is shown to be a solution of a linear of polynomials (monomials) serving as a basis. In a recent resolvent operator and admits an infinite-series expansion. paper by [PPR04], these two ideas have been combined to The stability result, however, is typically weaker and one show that density formulation together with its discretization can only conclude stability in measure-theoretic (such as a.e.) using SOS methods leads to a convex and linear problem for sense. the joint design of the density function and state feedback The outline of this paper is as follows. In Section II, controller. preliminaries and notation are summarized. In Section III, We show that the duality expressed in the paper of [Ran01] Lyapunov measure is introduced and related to the stochastic and linearity expressed in the paper of [PPR04] is well- operators and appropriately defined notions of stability of understood using the methods of Ergodic theory. Given a an attractor set. Finally, we summarize some conclusions in dynamical system, one can associate two different linear Section IV. operators known as Koopman and Frobenius-Perron (P-F) operator [LM94], [Man87]. These two operators are adjoint II. PRELIMINARIES AND NOTATION This work was supported by research grants from the Iowa State Univer- In this paper, discrete dynamical systems or mappings of sity at Ames (UV) the form U. Vaidya is with the Department of Electrical & Computer Engineering, Iowa State University, Ames, IA 50011 [email protected] xn+1 = T(xn) (1) 185 WID.150 are considered. T : X → X is in general assumed to be only This definition stresses the statistical behavior of the continuous and non-singular with X ⊂ Rn, a compact set. attractor set. Although rigorous results have been shown A mapping T is said to be non-singular with respect to only for special cases, it is generally believed that physical a measure µ if µ(T −1B) = 0 for all B ∈ B(X) such that measures exists in physical dynamical systems [You02]. This µ(B) = 0. B(X) denotes the Borel σ-algebra on X, M (X) is also supported by numerical evidence with set-oriented the vector space of real valued measures on B(X). Even methods for approximation of physical measures; cf., [DJ00]. though, deterministic dynamics are considered, stochastic A locally stable equilibrium is the simplest example of an approach is employed for their analysis. The stochastic attractor set and the physical measure corresponding to it is Perron-Frobenius (P-F) operator for a mapping T : X → X the Dirac-delta measure supported on the equilibrium point. is given by The existence of physical measure is related to the notion P[µ](A) = µ(T −1(A)), (2) of a.e. stability. In the following definitions and subsequent sections, we use U(ε) to denote an ε-neighborhood of an M X A B X where µ ∈ ( ) and ∈ ( ); cf., [LM94], [DJ00]. The invariant set A, and Ac := X \ A to denote the complement invariant measure are the fixed points of the P-F operator set. P that are additionally probability measures. From Ergodic theory, an invariant measure is always known to exist under Definition 2 (ω-limit set) A point y ∈ X is called a ω- limit the assumption that the mapping T is at least continuous and point for a point x ∈ X if there exists a sequence of integers X is compact; cf., [KH95]. n {nk} such that T k (x) → y as k → ∞. The set of all ω-limit For Eq. (1), the operator point for x is denoted by ω(x) and is called its ω- limit set. U f (x) = f (Tx) (3) Definition 3 (Almost everywhere stable) An invariant set is called the Koopman operator for f ∈C0(X). The Koopman A for the dynamical system T : X → X is said to be stable operator is a dual to the P-F operator, where the duality is almost everywhere (a.e.) with respect to a finite measure m ∈ expressed by the following M (Ac) if c < U f , µ >= U f (x)dµ(x) = f (x)dPµ(x) =< f ,Pµ > . m{x ∈ A : ω(x) 6⊆ A} = 0 (8) ZX ZX (4) A. Physical measure and almost everywhere stability For the special case of a.e. stability of an equilibrium point x0 A set A ⊂ X is called T-invariant if with respect to the Lebesgue measure, the definition reduces to n T(A) = A. (5) m{x ∈ X : lim T (x) 6= x0} = 0, (9) n→∞ In this paper, global stability properties of T-invariant sets where m in this case is the Lebesgue measure. that are additionally “minimal” in some sense will be investi- Motivated by the familiar notion of point-wise exponential gated. Using Eq. (2), a measure µ is said to be a T-invariant stability in phase space, we introduce a stronger notion measure if of stability in the measure space. This stronger notion of −1 µ(B) = µ(T (B)) (6) stability captures a geometric decay rate of convergence. for all B ∈ B(A). A T-invariant measure in Eq. (6) is Definition 4 (Stable a.e. with geometric decay) The a stochastic counterpart of the T-invariant set in Eq. (5) invariant set A X for the dynamical system T : X X is [ER85], [KH95]. For typical dynamical systems, the set A ⊂ → said to be stable almost everywhere with geometric decay equals the support of its invariant measure µ. To make the w.r.t. to a finite measure m M Ac if given ε 0, there correspondence precise, an attractor is defined as any set ∈ ( ) > exists K ε ∞ and β 1 such that that satisfies the following two properties: ( ) < < 1) Set A is a closed and invariant, i.e., T(A) = A, m{x ∈ Ac : T n(x) ∈ B} < Kβ n ∀ n ≥ 0 (10) 2) There is a unique physical invariant measure, defined for all sets B ∈ B(X \U(ε)). below, supported on A.