A local characterizationof Lyapunovfunctions on Riemannian manifolds ⋆ Farzin Taringoo a, Peter M. Dower a, Dragan Nesiˇ c´ a, Ying Tan a aElectrical and Electronic Engineering Department, The University of Melbourne, Victoria, 3010, Australia Abstract This paper proposes converse Lyapunov theorems for nonlinear dynamical systems defined on smooth connected Riemannian manifolds and characterizes properties of Lyapunov functions with respect to the Riemannian distance function. We extend classical Lyapunov converse theorems for dynamical systems in Rn to dynamical systems evolving on Riemannian manifolds. This is performed by restricting our analysis to the so called normal neighborhoods of equilibriums on Riemannian manifolds. By employing the derived properties of Lyapunov functions, we obtain the stability of perturbed dynamical systems on Riemannian manifolds. Key words: Dynamical systems, Riemannian manifolds, Geodesic curves. 1 Introduction manifolds do not necessarily possess the vector space prop- erties of Euclidean spaces. Consequently, a generalizationof Many systems include dynamics that naturally evolve on the traditional framework of the stability theory is inevitable. Riemannian manifolds, see for example [1,3–5,33], with their analysis requires the application of differential geomet- Stability analysis for systems evolving on manifolds is a new ric tools. Examples of such systems can be found in many area of research, see for example [24,26]. Recent results mechanical settings, see [3–5]. concerning the existence and properties of Lyapunov func- tions are documented in [7,10,11,13,15,22,29,31,34,36,38]. Stability theory is an important topic in control theory. This In particular, in [7,36], the existence of complete Lyapunov theory addresses the stability of trajectories of dynamical functions for dynamical systems on compact metric spaces systems as solutions of differential equations or differential is derived. In general, Riemannian manifolds can be consid- inclusions, see [16,33,34]. Lyapunov stability theory is the ered as metric spaces by employing the notion of Rieman- core mathematical tool for analyzing and characterizing the nian distance function, see [19]. stability of equilibria. Stability in the sense of Lyapunov has been extensively analyzed in the literature, see for example In this paper, we present several converse Lyapunov theo- rems for dynamical systems evolving on Riemannian man- arXiv:1311.0078v2 [math.OC] 24 Jun 2014 [16,18,23]. ifolds and prove some local properties of such Lyapunov Traditionally, the development of stability theory has fo- functions. To this end, we define Lyapunov stability of dy- cused on dynamical systems evolving on Euclidean spaces. namical systems on Riemannian manifolds based on the There, the application of the attendant vector space proper- Riemannian distance function. We employ the notion of ties of Euclidean spaces leads to significant simplifications geodesics on Riemannian manifolds and apply the stability n in the analysis, due to coordinate transformations available results for dynamical systems on R to obtain the existence to shift equilibria to the origin. Consequently, stability anal- of Lyapunov functions for dynamical systems defined on ysis can be reduced to the analysis of a equilibria located Riemannian manifolds. Using a version of stability theory at the origin. However, many dynamical systems defined on for systems evolving on Riemannian manifolds, see [2,5,8], the stability results for dynamical systems evolving on Eu- ⋆ This work was supported by the Australian Research Council clidean spaces [12,16,37] are extended to those evolving on Discovery Project DP120101144. Riemannian manifolds. We introduce a lift operator to con- Email addresses: [email protected] (Farzin vert the dynamical equations on a Riemannian manifold to Taringoo), [email protected] (Peter M. Dower), a dynamical system on the tangent space of an equilibrium, [email protected] (Dragan Neˇsi´c), see [14,19,32], and invoke some of the standard results of [email protected] (Ying Tan). the stability theory presented in [12,16]. It is shown that Preprint submitted to Automatica 20 October 2018 in a normal neighborhood [19] of an equilibrium of a dy- Table 1 namical system, the constructed Lyapunov functions satisfy Symbols and Their Descriptions certain properties which can be used to analyze the stability Symbol Description and robustness of the underlying dynamical system. These results are extended and applied to study perturbed dynamic M Riemannian manifold systems. Geometric features of the normal neighborhoods, X(M) space of smooth time invariant such as existence of unique length minimizing geodesics and vector fields on M their local representations enable us to closely relate the sta- X R bility results obtained for dynamical systems in Rn to those (M × ) space of smooth time varying defined on Riemannian manifolds. vector fields on M TxM tangent space at x ∈ M In terms of exposition, Section 2 presents some mathemati- ∗ cal preliminaries needed for the subsequent analysis. Section Tx M cotangent space at x ∈ M 3 presents the main results for the existence of Lyapunov T M tangent bundle of M functions for dynamical systems evolving on Riemannian T ∗M cotangent bundle of M manifolds. These results are employed in Section 4 to derive ∂ basis tangent vectors at x ∈ M the stability of perturbed dynamical systems on Riemannian ∂xi manifolds. The paper concludes with some closing remarks dxi basis cotangent vectors at x ∈ M in Section 5. f(x,t) time-varying vector fields on M || · ||g Riemannian norm 2 Preliminaries || · ||e Euclidean norm || · || induced norm In this section we provide the differential geometric material which is necessary for the analysis presented in the rest of g(·, ·) Riemannian metric on M the paper. We define some of the frequently used symbols d(·, ·) Riemannian distance on M of this paper in Table 1. Φf flow associated with f T F pushforward of F Definition 1 Let M be a an n dimensional manifold. A co- TxF pushforward of F at x ordinate chart on M is (U, φ), where U is an open set in M R>0 (0, ∞) and φ is a homomorphism from U to φ(U) ⊂ Rn, see [21]. R≥0 [0, ∞) C∞(M) space of smooth functions on M 2.1 Riemannian manifolds ≃ isomorphism B(x,r) metric ball centered at x with radius r Definition 2 (see [21], Chapter 3) A Riemannianmanifold Br(0) Ball with radius r in tangent spaces (M,g) is a differentiable manifold M together with a Rie- mannian metric g, where g is defined for each x ∈ M via an inner product gx : TxM × TxM → R on the tan- can be connected via a path γ ∈ P(x, y), where gent space TxM (to M at x) such that the function defined by x 7→ gx(X(x), Y (x)) is smooth for any vector fields X, Y ∈ X(M). In addition, . γ piecewise smooth, P(x, y) = γ :[a,b] → M (2.1) (i) (M,g) is n dimensional if M is n dimensional; ( γ(a)= x, γ(b)= y ) (ii) (M,g) is connected if for any x, y ∈ M, there exists a piecewise smooth curve that connects x to y. Theorem 1 ([19], P. 94) Suppose (M,g) is an n dimen- sional connected Riemannian manifold. Then, for any x, y ∈ M, there exists a piecewise smooth path γ ∈ P(x, y) that . Note that in the special case where M = Rn, the connects x to y. Riemannian metric g is defined everywhere by gx = n i,j=1 gij (x)dxi ⊗ dxj , where ⊗ is the tensor product on The existence of connecting paths (via Theorem 1) between ∗ ∗ Tx M × Tx M, see [21]. pairs of elementsof an n dimensional connected Riemannian P manifold (M,g) facilitates the definition of a corresponding As formalized in Definition 2, connected Riemannian man- Riemannian distance. In particular, the Riemannian distance ifolds possess the property that any pair of points x, y ∈ M d : M × M → R is defined by the infimal path length 2 between any two elements of M, with One may show, for a smooth vector field f, the integral flow Φf (s,t0, ·): M → M is a local diffeomorphism , . b see [21]. Here we assume that the vector field f is smooth d(x, y) = inf g (γ ˙ (t), γ˙ (t)) dt . (2.2) γ(t) and complete, i.e. Φf exists for all t ∈ (t0, ∞). γ∈P(x,y) a Z q . Note that in the special case where M = Rn, the Rieman- 2.3 Geodesic Curves nian distance (2.2) simplifies to d(x, y)= kx − yke. Geodesics are defined [14] as length minimizing curves on Using the definition of Riemannian distance d of (2.2), Riemannian manifolds which satisfy (M, d) defines a metric space as formalized by the follow- ing theorem. ∇γ˙ (t)γ˙ (t)=0, (2.9) Theorem 2 ([19], P. 94) Any n dimensional connected Rie- where γ(·) is a geodesic curve on (M,g) and ∇ is the Levi- mannian manifold (M,g) defines a metric space (M, d) via Civita connection on M, see [19]. The solution of the Euler- the Riemannian distance d of (2.2). Furthermore, the in- Lagrange variational problem associated with the length duced topology of (M, d) is the same as the manifold topol- minimizing problem shows that all the geodesics on an n ogy of (M,g). dimensional Riemannian manifold (M,g) must satisfy the following system of ordinary differential equations: Next, the crucial pushforward operator is introduced. n i Definition 3 For a given smooth mapping F : M → N γ¨i(s)+ Γj,kγ˙j (s)γ ˙ k(s)=0, i =1, ..., n, (2.10) from manifold to manifold the pushforward is j,k=1 M N T F X defined as a generalization of the Jacobian of smooth maps between Euclidean spaces as follows: where n T F : TM → T N, (2.3) i 1 il ∂gjl Γj,k = g (gjl,k + gkl,j − gjk,l), gjl,k = , 2 ∂xk where l=1 X (2.11) T F : T M → T N, (2.4) x x F (x) where all the indexes i,j,k,l run from 1 up to n = dim(M) .