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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 → TN, (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) . and ij −1. Note that is the entity of the and [g ] = [gij ] gij (i, j) metric g. T F (X ) ◦ h = X (h ◦ F ), X ∈ T M,h ∈ C∞(N). x x x x x Definition 4 ([19], p. 72) The restricted exponential map is (2.5) defined by
expx : TxM → M, expx(v)= γv(1), v ∈ TxM, (2.12)
2.2 Dynamical systems on Riemannian manifolds where γv(1) is the unique maximal geodesic [19], P. 59, initiating from x with the velocity v up to one. This paper focuses on dynamical systems governed by dif- ferential equations on a connected n dimensional Rieman- Throughout, restricted exponential maps are referred to as nian manifold M. Locally these differential equations are exponential maps. An open ball of radius δ > 0 and cen- defined by (see [21]) tered at 0 ∈ T M in the tangent space at x is denoted by . x Bδ(0) = {v ∈ TxM | ||v||g < δ}. Similarly, the corre- X R x˙(t)= f(x(t),t), f ∈ (M × ), sponding closed ball is denoted by Bδ(0). Using the lo- x(0) = x0 ∈ M,t ∈ [t0,tf ]. (2.6) cal diffeomorphic property of exponential maps, the corre- sponding geodesic ball centered at x is dened as follows. The time dependent flow associated with a differentiable time dependent vector field f is a map Φf satisfying Lemma 1 ([19], Lemma 5.10) For any x ∈ M, there ex- ists a neighborhood Bδ(0) in TxM on which expx is a dif- Φf :[t0,tf ] × [t0,tf ] × M → M, feomorphism onto expx(Bδ(0)) ⊂ M. (s0,sf , x) 7→ Φf (sf ,s0, x) ∈ M, (2.7) Definition 5 ([19]) In a neighborhood of x ∈ M, where and expx is a local diffeomorphism (this neighborhood always exists by Lemma 1), a geodesic ball of radius δ > 0 is dΦf (s,s0, x) denoted by expx(Bδ(0)) ⊂ M. The corresponding closed = f(Φf (t,s0, x),t). (2.8) ds geodesic ball is denoted by exp (Bδ(0)). s=t x
3 Definition 6 For a vector space V , a star-shaped neighbor- hood of 0 ∈ V is any open set U such that if u ∈ U then αu ∈ U, α ∈ [0, 1].
Definition 7 ([19], p. 76) A normal neighborhood around x ∈ M is any open neighborhood of x which is a diffeo- morphic image of a star shaped neighborhood of 0 ∈ TxM under expx map.
Definition 8 The injectivity radius of M is . i(M) = inf i(x), (2.13) x∈M where
. R i(x) = sup{r ∈ ≥0| expx is diffeomorphic onto 1 1 expx(Br(0))}. Fig. 1. S and S \ {p} (2.14) unit circle S1 ⊂ R2 in Figure 1. A local coordinate system [20] for S1 is given by the local homeomorphism ψ : S1 → R (see also Figure 1) defined by Definition 9 The metric ball with respect to on is d (M,g) ψ defined by θ 7→ (sin(θ), cos(θ)) ∈ R2, θ ∈ (0, 2π) ⊂ R1. (3.17)
. 1 B(x, r) = {y ∈ M | d(x, y) < r}. (2.15) Inthe case of theremovalofa point p from S , the Euclidean distance between points converging in S1 \{p} to p ∈ S1 from either side converges to zero. However, at the same time, the Riemannian distance converges to 2π which is the 1 The following lemma reveals a relationship between normal largest distance on S between any pair of points. neighborhoods and metric balls on (M,g). We generalize the stability notion for dynamical systems on Riemannian manifolds as follows. Lemma 2 ([32], p. 122) Given any ǫ ∈ R>0 and x ∈ M, suppose that exp is a diffeomorphism on B (0) ⊂ T M, x ǫ x Definition 10 For the time-varying dynamical system x˙ = and B(x, r) ⊂ exp B (0) for some r ∈ R . Then x ǫ >0 f(x(t),t), f ∈ X(M × R), x¯ ∈ M is an equilibrium if exp Br(0) = B(x, r). (2.16) x Φf (t,t0, x¯)=¯x, t ∈ [t0, ∞), (3.18)
where Φf is the integral flow of f defined by (2.7).
We note that Bǫ(0) is the metric ball of radius ǫ with respect Definition 11 ([2,5,8,16]) For the dynamical system to the Riemannian metric g in TxM. x˙ = f(x(t),t), f ∈ X(M × R), an equilibrium x¯ ∈ M is
3 Lyapunov Analysis on Riemannian Manifolds (i) uniformly Lyapunov stable if for any neighborhood Ux¯ of x¯ ∈ M and any initial time t0 ∈ R, there exists a neigh- We extend the notion of stability to dynamical systems borhood Wx¯ of x¯, such that evolving on Riemannian manifolds. This problem has been addressed in [1,5,27] in a geometric framework. The main ∀x0 ∈Wx¯, Φf (t,t0, x0) ∈Ux¯, ∀t ∈ [t0, ∞). motivation here is to characterize the local properties of Lya- (3.19) punov functions based upon the Riemannian distance func- tion. These properties will be of great importance in analyz- ing a range of dynamical systems evolving on manifolds. (ii) uniformly locally asymptotically stable if it is Lyapunov stable and for any t0 ∈ R, there exists Ux¯ such that It is importantto note that, dependingon the geometry of the state space of a particular dynamical system, Riemannian ∀x0 ∈Ux¯, lim Φf (t,t0, x0)=¯x, i.e. distance might be significantly different than the Euclidean t→∞ lim d(Φf (t,t0, x0), x¯)=0, t ∈ [t0, ∞). (3.20) distance of embedded manifolds. As an example consider a t→∞
4 ∗ ∗R with dxχ ∈ Tx M as per (3.25), and dtχ ∈ Tt . (iii) uniformly globally asymptotically stable if it is Lya- R punov stable and for any t0 ∈ , Definition 13 ([1,5,16]) (Lyapunov Candidate Functions) A smooth function v : M × R → R is a Lyapunov function ∀x0 ∈ M, lim Φf (t,t0, x0)=¯x, t ∈ [t0, ∞). (3.21) X R t→∞ for the time-variant vector field f ∈ (M × ) if v is locally positive definite in a neighborhood of an equilibrium x¯ for (iv) uniformly locally exponentially stable if it is locally t ∈ [t0, ∞) and Lf v is locally negative semi-definite in a asymptotically stable and for any t0 ∈ R, there exist Ux¯ and neighborhood of x¯. K, λ ∈ R>0 such that Definition 14 The time-variant sublevel set Nb,t of a pos- ∀x ∈Ux, d(Φf (t,t , x ), x¯) ≤ itive semidefinite function v : M × R → R is defined as 0 ¯ 0 0 . Kd(x0, x¯) exp(−λ(t − t0)),t ∈ [t0, ∞). (3.22) Nb,t = {x ∈ M, v(x, t) ≤ b}. By Nb,t(¯x) we denote a connected sublevel set of M containing x¯ ∈ M. (v) globally exponentially stable if it is globally asymptot- ically stable and for any t0 ∈ R, there exist K, λ ∈ R>0, The following lemma shows that there exists a connected such that, compact sublevel set of an equilibrium point of a dynamical system on a Riemannian manifold. ∀x0 ∈ M, d(Φf (t,t0, x0), x¯) ≤ Kd(x0, x¯) exp(−λ(t − t0)),t ∈ [t0, ∞). (3.23) Lemma 3 Let x¯ ∈ M and v : M × R → R denote an equilibrium and a Lyapunov function respectively for system (2.6). Then, for any neighborhood Ux¯ of x¯ and any t ∈ R, there exists b ∈ R>0, such that Nb,t(¯x) is compact, We note that the convergenceon M is defined in the topology x¯ ∈ int(Nb,t(¯x)) and Nb,t(¯x) ⊂Ux¯, where int(·) gives the induced by d which is the same as the original topology of interior of a set. M by Theorem 2.
Definition 12 ([5,16]) A function χ : M → R is Proof. The proof is based on the proof given in [5], Lemma locally positive definite (positive semi-definite) in a R neighborhood of x¯ ∈ M if χ(¯x) = 0 and there 6.12. In this case we fix time t ∈ and consider v(·,t): M → R as a smooth time-invariant function. In this case, exists a neighborhood Ux¯ ⊂ M such that for all x ∈U \{x¯}, 0 <χ(x) (respectively 0 ≤ χ(x)). we apply the results of Lemma 6.12 in [5] to complete the x¯ proof. Given a smooth function χ : M → R, the Lie derivative of χ along a time invariant vector field f ∈ X(M) is defined by . To analyze the behavior of dynamical systems on manifolds Lf χ = dχ(f), (3.24) we employ the notion of comparison functions defined in [16]. where dχ : TM → R is the differential form of χ. In any neighbourhood of x ∈ M, dχ is given locally by Definition 15 ([16]) A continuous function α : [0,b) → R n ≥0 is of class K if it is strictly increasing and α(0) = 0, ∂χ ∗ and of class K∞ if b = ∞ and limr→∞ α(r)= ∞. dχ = dxi ∈ Tx M, (3.25) ∂xi i=1 X Definition 16 ([16]) A continuous function β : [0,b) × . ∗ R → R is of class KL if for each fixed s, β(·,s) ∈ K where n = dim(M) and Tx M is the cotangent space of M ≥0 ≥0 at x, see [21]. and for each fixed r ∈ [0,b), β(r, ·) is decreasing with lims→∞ β(r, s)=0. Remark 1 For time-varying dynamical systems evolving on M, the Lie derivative of a smooth time-varying function The following theorem provides K and KL comparison χ : M × R 7→ R is defined by function bounds for trajectories of uniformly stable dynam- ical systems evolving on Riemannian manifolds. . ∂ L χ = dχ ,f(x, t) , (3.26) f(x,t) ∂t Theorem 3 Any time-varying dynamical system of the form (2.6), evolving on a connected n dimensional Riemannian where manifold (M,g), satises the following properties:
∗ ∗R dχ = dxχ ⊕ dtχ ∈ Tx M ⊕ Tt , • If an equilibrium x¯ ∈ M is uniformly Lyapunov stable, (3.27) then there exists a class K function α and a neighborhood
5 Nx¯, such that have δ(r2) < δ(r1). Denote the associated neighborhoods r1 r2 of B(¯x, r1) and B(¯x, r2) by Wx¯ and Wx¯ respectively, see d(Φf (t,t0, x0), x¯) ≤ α(d(x0, x¯)), 3.30. Then δ(r2) <δ(r1) implies that x0 ∈ Nx¯,t ∈ [t0, ∞). (3.28) c c ∃x ∈ Wr1 , s.t. x ∈/ Wr2 , (3.32) • If x¯ is uniformly asymptotically stable then there exists a 0 x¯ 0 x¯ class KL function β and a neighborhood Nx¯, such that c c r1 where B(¯x, r1) ⊂ B(¯x, r2). However, x0 ∈ Wx¯ results in , which d(Φf (t,t0, x0), x¯) ≤ β(d(x0, x¯),t − t0), Φf (t,t0, x0) ∈ B(¯x, r1) ⊂ B(¯x, r2), t ∈ [t0, ∞) r2 x0 ∈ Nx¯,t ∈ [t0, ∞). (3.29) contradicts x0 ∈/ Wx¯ . Hence, δ(r1) ≤ δ(r2).c
Choose a ζ ∈ Kcsuch that ζ(r) ≤ δ(r), r ∈ R≥0 (this −1 Proof. Let us consider a neighborhoodUx¯ ⊂ expx¯ Bi(¯x)(0), is always possible since δ is non-decreasing), and ζ : R where i(¯x) is the injectivity radius at x¯ ∈ M and Bi(¯x)(0) ⊂ [0, supr∈[0,∞) ζ(r)) 7→ ≥0 is a K class function. Note that Tx¯M. Note that i(¯x) > 0, see Proposition 2.1.10 in [17]. In ζ is bounded by δ, hence, supr∈[0,∞) ζ(r) is bounded. Now order to prove the first assertion we note that the uniform choose . Nx¯ = expx¯ Bsupr∈[0,∞) ζ(r)(0) ⊂ expx¯ Bδ(i(¯x))(0) Lyapunov stability of x¯, implies that there exists Wx¯ ⊂ . −1 Then, r = ζ (d(x0, x¯)), x0 ∈ Nx¯, implies M, such that x0 ∈ Wx¯ results in Φf (t,t0, x0) ∈ Ux¯ for all t ∈ [t0, ∞). Hence, Wx¯ ⊆ Ux¯ ⊂ expx¯ Bi(¯x)(0) and d(x0, x¯)= ζ(r) ≤ δ(r), x0 ∈ Nx¯, (3.33) Φf (t,t0, x0) remains in a normal neighborhood of x¯.
Lemma 2 implies that exp B (0) = B(¯x, r), provided 0 < and hence, by (3.30) x¯ r . r ≤ i(¯x). Hence, for any Ux¯ = B(¯x, r), 0 6 1 neighborhood around x¯), such that there exists a KL func- n 2 2 where d(x(t0), x¯) = i=1 zi (t0) = ||z(t0)||e = tion β, which satisfies ||z(t0)||g. The last equality is due to the fact that in normal coordinates of x¯, the Riemannian P metric is given by d(Φf (t,t0, x0), x¯) ≤ β(d(x0, x¯),t − t0), x(t )= x ∈ N ,t ∈ [t , ∞). (3.35) 0 0 x¯ 0 ∂ ∂ g , = δ + O(r2), (3.40) ∂x ∂x ij Assume ||Txf(·,t)|| is uniformly bounded with respect to t i j on Nx¯, where ||.|| is the norm of the boundedlinear operator Tf : TM → TTM as per Definition 3. Then, for some where r is the distance and δ is the Kronecker delta, see n ij Ux¯ ⊂ Ux¯ , for all x(t0)= x0 ∈ Ux¯, there exist a Lyapunov ∂ ∂ [32], Chapter 5. Hence, gx¯( ∂x , ∂x )= δij and ||z(t0)||e = candidate function w : M ×R → R≥0 and α1, α2, α3, α4 ∈ i j ||z(t0)||g. Therefore, we have K, such that for all x ∈Ux¯ and t ∈ [t0, ∞), (i): α1 (d(x, x¯)) ≤ w(x, t) ≤ α2 (d(x, x¯)) , ||z(t)||g ≤ β(||z(t0)||g,t − t0), L (ii): f(x,t)w ≤−α3 (d(x, x¯)) , z(t0)= z0 ∈ Bǫ(0) ⊂ Tx¯M. (3.41) (iii): ||Txw|| ≤ α4 (d(x, x¯)) , (3.36) The uniform boundedness of Txf(·,t) with respect to t to- where d(·, ·) is the Riemannian metric, L is the Lie derivative ˆ gether with (3.37) and smoothness of −1 imply that ∂f and T w : TM → T R ≃ R × R is the pushforward of w. exp ∂z is uniformly bounded on Bǫ(0) ∈ Tx¯M. Hence, we can ap- ply Theorem 4.16 of [16] to demonstrate the existence of a Lyapunov function v : Tx¯M × R → R, satisfying Proof. By employing Lemma 1, consider expx¯ Bǫ(0) ⊂ M, 0 < ǫ, such that expx¯ is a diffeomorphism onto its image, then expx¯ is invertible and the inverse map is denoted (i): α1(||z||g) ≤ v(z,t) ≤ α2(||z||g), −1 by exp : M → Tx¯M. By Theorem2 the inducedtopology L x¯ (ii): fˆ z,t v ≤−α3(||z||g), of the distance function d is the same as the originaltopology ( ) ∂ of M and by Lemma 2 the metric balls and geodesic balls (iii): |Tzv( )|≤ α4(||z||g), (3.42) are identical. Hence, without loss of generality, we assume ∂z Nx¯ = expx¯ Bǫ(0), where expx¯ is a diffeomorphism onto expx¯ Bmax{ǫ,β(ǫ,0)}(0). where z ∈ Tx¯M,t ∈ [t0, ∞). Since expx¯ is a local dif- feomorphism by Lemma 1, for x ∈ Nx¯, we have x = −1 Since expx¯ is a diffeomorphism onto expx¯ Bǫ(0), then for expx¯ ◦ expx¯ x. Hence, by [21], Lemma 3.5 any x ∈ expx¯ Bǫ(0), there exists z ∈ Tx¯M such that x = −1 expx¯ z, or equivalently z = expx¯ x. Let us call the operator −1 −1 Id = Tx expx ◦ exp = T −1 expx ◦Tx exp ,(3.43) −1 ¯ x¯ expx¯ x ¯ x¯ expx¯ the geodesic lift. The time variation of z, as long as x stays in expx¯ Bǫ(0), is given by where Id is the identity map and T is the pushforward as −1 −1 per Definition 3. This shows z˙(t)= Tx expx¯ (f(x, t)) = Texpx¯ z expx¯ (f(expx¯ z,t)) . = fˆ(z,t), −1 (3.37) −1 Tx expx = T −1 expx . (3.44) ¯ expx¯ x ¯ where z˙(t) ∈ Tz(t)Tx¯M ≃ Tx¯M. We note that the equilib- rium x¯ of f(x, t) changes to z =0 ∈ Tx¯M for the dynam- The Lie derivative of with respect to ˆ is locally given by n v f ical equations in z coordinates. In the case M = R , we (3.26) as follows have Rn ∂ ∂ x = expx¯ z =x ¯ + z ∈ . (3.38) L v = dv( , fˆ(z,t)) = d v( )+ d v(fˆ(z,t)). fˆ(z,t) ∂t t ∂t z (3.45) For any x(t0) ∈ expx¯ Bǫ(0), we have x(t0) = expx¯ z(t0) for some z(t0) ∈ Bǫ(0). Now let us consider the geodesic ∂ ˆ . Since v is a scalar-valued function then dv( ∂t , f(z,t)) = curve γ : [0, 1] → M, γ(τ) = exp τz(t ). Employing the x¯ 0 T v( ∂ , fˆ(z,t)) = T v( ∂ )+ T (fˆ(z,t)). Employing exp , results of [19], Proposition 5.11, in the normal coordinates ∂t t ∂t z x¯ we define the following function on M: of x,¯ we have . −1 γ(τ) = (τz1(t0), ..., τzn(t0)), (3.39) vˆ(x, t) = v(expx¯ x, t), x ∈ expx¯ Bǫ(0). (3.46) 7 Then the Lie derivative of vˆ along f at state x and time t is subset of a normal neighborhood on an n dimensional Rie- mannian manifold (M,g). Assume ||Txf(·,t)|| is uniformly ∂ bounded, where ||.|| is the norm of the linear operator Tf : Lf(x,t)vˆ = dtvˆ( )+ Txvˆ(f(x, t)) n ∂t TM → TTM. Then, for some Ux¯ ⊂Ux¯ , for all x(t0) ∈Ux¯, there exist a Lyapunov function R R and ∂ −1 v : M × → ≥0 = dtvˆ( )+ Tzv Tx exp ◦Tz exp (fˆ(z,t)) R ∂t x¯ x¯ λ1, λ2, λ3, λ4 ∈ >0, such that for all x ∈Ux¯ ∂ −1 2 2 = dtvˆ( )+ Tzv Tx exp ◦ (i): λ d (x, x¯) ≤ v(x, t) ≤ λ d (x, x¯), ∂t x¯ 1 2 L 2 ˆ (ii): f(x,t)v ≤−λ3d (x, x¯), Texp−1 x expx¯(f(z,t)) x¯ (iii): ||Txv|| ≤ λ4d(x, x¯). (3.51) ∂ = d vˆ( )+ T v fˆ(z,t) by employing (3.44) t ∂t z ∂ Proof. Following the proof of Theorem 4, we employ the = dtv( )+ Tzv fˆ(z,t) by employing (3.46) −1 ∂t geodesic lift operator z = expx¯ x in a normal neighbor- L hood of x¯. Hence, we obtain the local exponential stability = fˆ(z,t)v. (3.47) of 0 ∈ Tx¯M, for the dynamical system z˙(t) = fˆ(z,t) = −1 ∂ T z exp (f(exp z,t)) as per the proof of Theorem 4. The same argument applies to T vˆ( ) and shows that expx¯ x¯ x¯ x ∂x The rest of the proof parallels the proof of Theorem 4 and the results of [16], Theorem 4.14. ∂ ∂ T vˆ( )= T v( ). (3.48) x ∂x z ∂z We note that by employing the normal coordinates used in the proof of Theorem 4, we have d(Φf (t,t0, x(t0)), x¯) ≤ As shown before we have d(x(t), x¯) = ||z(t)|| , hence, by g Kd(x(t0), x¯) exp(−λ(t − t0)) implies (3.42), vˆ locally satisfies (3.36). ||z(t)||g ≤ K exp(−λ(t − t0))||z(t0)||g which is required in the proof of Theorem 4.14 in [16]. Since the function constructed above is defined locally, it remains to extend the domain of its definition to M. For The Lyapunovfunctionsin Theorems4 and 5 are constructed , compactness of and smoothness δ ∈ (0,ǫ) Bδ(0) ⊂ Tx¯M in a normal neighborhood of an equilibrium where expx of expx¯ together imply that expx¯ Bδ(0) ⊂ expx¯ Bǫ(0) is a is a local diffeomorphism. Hence, the properties derived compact set in M. Choose a bump function ψ ∈ C∞(M), in Theorems 4 and 5 hold locally and the corresponding such that ψ ≡ 1 on exp B (0) and suppψ ⊂ exp B (0), neighborhoods are restricted by the injectivity radius of the . x¯ δ x¯ ǫ where suppψ = {x ∈ M s.t. ψ(x) 6=0}, for the definition equilibrium. Depending on the geometric features of M, of bump functions see [21]. As shown in [21], Proposition the injectivity radius of a particular point might be very 2.26, such bump functions always exist. Hence, we consider small. In this section we construct Lyapunov functions on . . R R Ux¯ = expx¯ Bδ(0) and w = ψ × vˆ : M × → . The Lie a compact subset of a local chart of an equilibrium of a derivative of w is given by dynamical system on M by scaling the Riemannian and Euclidean metrics. This is also a local method since we are Lf(x,t)w = Lf(x,t)ψ · vˆ = ψLf(x,t)vˆ +ˆvLf(x,t)ψ, (3.49) restricted to work within a local coordinate system. However, in some cases, it may provide much larger neighborhood on where on Ux¯ we have which Theorems 4 and 5 hold. Theorem 6 Let x¯ be an equilibrium for the dynamical sys- Lf(x,t)w = Lf(x,t)v.ˆ (3.50) tem (2.6) on a coordinate chart (U, φ) of x¯ as per Definition ∂ ∂ 1, such that there exists a KL function β, which satisfies Same argument shows that on Ux¯, Txw( ∂x ) = Txv( ∂x ), which completes the proof for the Lyapunov function w. d(Φf (t,t0, x0), x¯) ≤ β(d(x0, x¯),t − t0), x(t0)= x0 ∈ U. (3.52) Note that properties (ii) and (iii) are essential to obtain the Assume ||T f(·,t)|| is uniformly bounded with respect to t robustness results for perturbed dynamical systems, see [16], x on U, where ||.|| is the norm of the linear operator Tf(x, t): Chapters 9,10,11. The following theorem strengthens the hy- TM → TTM. Then, for some U ⊂ U, for all x(t ) = potheses of Theorem 3 to local exponential stability and de- x¯ 0 x ∈U , there exist a Lyapunov function w : M ×R → R rives the local properties of Lyapunov functions in a normal 0 x¯ ≥0 and α , α , α , α ∈ K, such that for all x ∈U neighborhood of equilibriums. 1 2 3 4 x¯ (i): α1 (d(x, x¯)) ≤ w(x, t) ≤ α2 (d(x, x¯)) , Theorem 5 Let x¯ be a uniformly exponentially stable equi- n n (ii): Lf(x,t)w ≤−α3 (d(x, x¯)) , librium of the dynamical system (2.6) on Nx¯ ⊂ Ux (Ux ¯ ¯ (3.53) is a normal neighborhood around x¯), where Nx¯ denotes a (iii): ||Txw|| ≤ α4 (d(x, x¯)) . 8 Proof. For the coordinate chart (U, φ), we have φ : M → As Be(R, 0) is a convex set, the line connecting x and x¯ is n −1 R . By definition, φ is a homeomorphism to an open set entirely in φ (Be(R, 0)). Hence, in Rn, see [20]. Without loss of generality, we assume . φ(¯x)=(0, ..., 0),otherwisewe can considerthe map η(x) = b b φ(x) − φ(¯x), where η is also a homeomorphism by defini- c1d(x, x¯) ≤ c1 ||γ˙ (s)||gds ≤ ||γ˙ (s)||eds tion. Denote Za Za = ||x − x¯||e. (3.59) . −1 R = max r, s.t. Be(r, 0) ⊂ φ(U), φ Be(r, 0) ⊂ U, Therefore, for x =Φf (t,t0, x0), we have r ∈ R>0, (3.54) 1 n where Be(r, 0) is the Euclidean ball of radius r. In R , ||x − x¯||e ≤ d(Φf (t,t0, x0), x¯) ≤ β(d(x0, x¯),t − t0) c2 Be(r, 0) is a compact set and since φ is a homeomorphism 1 −1 ≤ β( ||x − x¯||e,t − t0). (3.60) then, φ Be(r, 0) ⊂ M is a compact set. By (3.52),there c1 exists Wx¯⊂ M, such that for all x0 ∈ Wx¯, Φ(t,t0, x0) ∈ −1 1 . R Hence, ||x − x¯||e ≤ c2β( ||x − x¯||e,t − t0) = βˆ(||x − φ (Be( , 0)). c1 x¯||e,t − t0). Replace the Riemannian metric ||·||g by the Euclideanmetric −1 −1 The Euclidan induced norm of is defined by || · ||e on φ Be(R, 0) . Since φ Be(R, 0) is com- Txf(x, t) pact, there exists c ,c ∈ R , such that [19] 1 2 >0 ||T f(x, t)(X)|| ||T f(x, t)|| = sup x e x e ||X|| c1||X||g ≤ ||X||e ≤ c2||X||g, X∈TxM,X6=0 e −1 c2||Txf(x, t)(X)||g X ∈ TxM, x ∈ φ Be(R, 0) . (3.55) ≤ sup X∈T M,X6=0 c1||X||g x c1 Since the state trajectory is contained in φ−1 (B (R, 0)), by ≤ ||Txf(x, t)||g. (3.61) e c replacing the Riemannian metric with the Euclidean one, the 2 state trajectory will be bounded in B (R, 0). By employing e Hence, boundedness of implies the bounded- (3.55), the Euclidean distance function is bounded by the ||Txf(·,t)||g ness of . We apply the results of [16], Theorem Riemannian one as follows. Consider any piecewise smooth ||Txf(·,t)||e 4.16 to the dynamical system evolving on M, where ||·|| is curve γ :[a,b] → M connecting x ∈ φ−1(B (R, 0)) and g e replaced by || · || . Therefore, there exist a Lyapunov func- x¯, such that γ(a) = x and γ(b)=x ¯. If γ belongs to e tion v and functions α1, α2, α3, α4 ∈ K, such that −1 φ Be(R, 0) ⊂ M, then (i): α1(||x||e) ≤ v(x, t) ≤ α2(||x||e), b b (ii): Lf(x,t)v ≤−α3(||x||e), ||x − x¯||e ≤ ||γ˙ (s)||eds ≤ c2 ||γ˙ (s)||gds, (3.56) (iii): ||Txv||e ≤ α4(||x||e), x ∈ Be(R, 0), (3.62) Za Za where ||x − x¯||e is the Euclidean distance between x and where ||x||e = ||x − x¯||e. As a result of the scaling the −1 Riemannian and Euclidean norms, we have x¯. In case γ does not entirely belong to φ Be(R, 0) , then there exists a time t ∈ [a,b], such that γ(s) ∈ −1 (i): α1(c1d(x, x¯)) ≤ v(x, t) ≤ α2(c2d(x, x¯)), φ (Se(R, 0)) ,s ∈ [a,t] and ||x − γ(t)||e = R, (ii): Lf(x,t)v ≤−α3(c1d(x, x¯)), where Se(R, 0) = {x | ||x||e = R}. Hence, since −1 R c2 x ∈ φ (Be( , 0)), we have (iii): ||Txv|| ≤ α4(c2d(x, x¯)). (3.63) c1 t t ||x − x¯||e ≤ R ≤ ||γ˙ (s)||eds ≤ c2 ||γ˙ (s)||gds Followingthe last part of the proofof Theorem4, the domain a a of the definition of v can be extendedto M, which completes bZ Z the proof for Ux¯ = Wx¯. ≤ c2 ||γ˙ (s)||gds. (3.57) Za Therefore, for any piecewise smooth γ, ||x − x¯||e ≤ 4 Stability of Perturbed Dynamical Systems b c2 a ||γ˙ (s)||gds. Taking the infimum over all γ, (2.2) implies that The properties of the constructed Lyapunov functions in R Theorems 4 and 5 are employed to obtain the robust stabil- ||x − x¯||e ≤ c2d(x, x¯). (3.58) ity results for perturbed dynamical systems on Riemannian 9 manifolds. Consider the following perturbed dynamical sys- tem on (M,g). operator dw : TxM → R. Hence, for sufficiently small δ, L we have f+hw < 0, x ∈ Nb,t0 (¯x) (M −Wx¯). There- x˙(t)= f(x, t)+ h(x, t),f,h ∈ X(M × R). (4.64) fore, the state trajectory Φf+h(·,t0, x0) stays in Ux¯ for all T x0 ∈ int(Nb,t0 (¯x)). The term h can be considered as a perturbation of the nom- inal system f. As stated in [9,16,25], stability results for Without loss of generality, assume Ux¯ = expx¯ Br2 (0), r2 < (4.64) can be obtained based on technical assumptions on i(¯x). Then, by the results of Theorem 4, the variation of w the stability of the nominal system f and boundedness of h. along f + h is then given by The followingtheoremgivesthe stability of (4.64),where the Lf+hw = Lf w + Lhw ≤−α3(d(x, x¯)) + Lhw nominal system is locally uniformly asymptotically stable. = −α3(d(x, x¯)) + dw(h(x, t)) = −α (d(x, x¯)) + T w(h(x, t)) Theorem 7 Let x¯ be an equilibrium of dynamical system 3 x (2.6), which is locally uniformly asymptotically stable on a ≤−α3(d(x, x¯)) + ||Txw|| · ||h(x, t)||g normal neighborhood Nx¯. Assume the perturbed dynami- ≤−α3(d(x, x¯)) + δα4(d(x, x¯)) cal system (4.64) is complete and the Riemannian norm of ≤−(1 − θ)α3(d(x, x¯)) − θα3(d(x, x¯)) X R the perturbation h ∈ (M × ) is bounded on Nx¯, i.e. +δα4(d(x, x¯)) ≤−(1 − θ)α3(d(x, x¯)), ||h(x, t)||g ≤ δ, x ∈ Nx¯,t ∈ [t0, ∞). Then, for sufficiently −1 δα4(r1) small δ, there exists a neighborhood Ux¯ and a function if α3 ( ) ≤ d(x, x¯) ≤ r2, (4.68) ρ ∈ K, such that θ α3(r2) where r1 < r2, 0 <θ< 1 and δ ≤ θ . lim sup d(Φf+h(t,t0, x0), x¯) ≤ ρ(δ), x0 ∈ Ux¯. (4.65) α4(r1) t→∞ . −1 δα4(r1) Define η = α2(α3 ( θ )), then {x ∈ M | d(x, x¯) ≤ −1 δα4(r1) Proof. Following the proof of Theorem 4, there exists Ux¯ ⊂ α3 ( θ )} ⊂ Nt,η = {x ∈ M| w(x, t) ≤ η} ⊂ Nx¯, such that (3.36) holds for a Lyapunov function w. First {x ∈ M| α1(d(x, x¯)) ≤ η}. Hence, solutions initial- we show that the neighborhood Ux¯ in Theorem 4 can be ized in {x ∈ M | d(x, x¯) ≤ α−1( δα4(r1) )} remain shrunk, such that Φ (·,t , x ) ∈ U provided x ∈ U . 3 θ f+h 0 0 x¯ 0 x¯ in {x ∈ M| α1(d(x, x¯)) ≤ η} since w˙ < 0 for By Lemma 3 there exists Nb,t0 (¯x) and α3 ∈ K, such that −1 δα4(r1) x ∈ Nt,η −{x ∈ M | d(x, x¯) ≤ α3 ( θ )}. This proves Lf+hw = Lf w + Lhw ≤−α3(d(x, x¯)) + Lhw, δα4(r1) x ∈ int(Nb,t (¯x)). (4.66) −1 −1 0 lim sup d(Φf+h(t,t0, x0), x¯) ≤ α1 α3 t→∞ θ . By the Shrinking Lemma [20] there exists a precompact = ρ(δ), (4.69) neighborhood Wx¯, such that, Wx¯ ⊂ int(Nb,t0 (¯x)) ⊂ δα r Nb,t0 (¯x), see [20]. Hence, M − Wx¯ is a closed set and . −1 4( 1) for any x0 ∈ Ux¯ = {x ∈ M | d(x, x¯) < α3 ( θ )} Nb,t0 (¯x) (M −Wx¯) is a compact set (closed subsets of int(Nb,t0 (ˆx)). compact sets are compact). The continuity of α3 and d(·, x¯) T T together with the compactness of Nb,t0 (¯x) (M − Wx¯) imply the existence of the following parameter M, T In the following theorem we strengthen the uniform asymp- . M = sup −α3(d(x, x¯)) < 0. (4.67) totic stability to the uniform exponential stability for the x∈Nb,t0 (¯x) (M−Wx¯) nominal system x˙ = f(x, t). It will be shown that the state trajectory of the perturbed system stays close to the equilib- T rium of the nominal system when some specific conditions Note that α3 ∈ K, x ∈ Nb,t (¯x) (M −Wx¯) and since 0 are satisfied. Wx¯ is a neighborhood of x¯ then d(x, x¯) > 0, x ∈ TM Nb,t0 (¯x) (M − Wx¯). Therefore, < 0. Using (3.26) implies that Lhw = dw(h) ≤ ||dw|| · ||h||g ≤ δ||dw||, Theorem 8 Let x¯ be an equilibrium of (2.6), which is lo- where ||dwT || is the induced norm of the linear operator cally uniformly exponentially stable on a normal neigh- dw : TM → R. The smoothness of w and compact- borhood Ux¯. Assume the nominal and perturbed dynami- ness of Nb,t0 (¯x) together imply ||dw|| < ∞. It is im- cal systems are both complete and the Riemannian norm portant to note that ||dw|| is closely related to ||T w|| of the perturbation h ∈ X(M × R) is bounded on Ux¯, i.e. through the component of the Riemannian metric g. As is ||h(x, t)||g ≤ δ, x ∈Ux¯,t ∈ [t0, ∞). Also assume ||f||g and shown by Theorem 4, ||Txw|| ≤ α4(d(x, x¯)). Hence, the ||Tf|| are uniformly bounded with respect to t on compact smoothness of M and compactness of Nb,t0 (¯x) imply that subsets of M, where Tf : TM → TTM as per Definition ||dw|| < ∞. Note that ||dw|| is the norm of the linear 3. Then, for sufficiently small δ, there exists positive con- 10 stants ζ,γ and k, such that 5 Conclusion d(Φf+h(t,t0, x0), x¯) ≤ k exp(−γ(t − t0))d(x0, x¯)+ ζδ. In this paper we have presented the stability results for dy- (4.70) namical systems evolving on Riemannian manifolds. We have obtained converseLyapunovtheorems for nonlineardy- namical systems defined on smooth connected Riemannian Proof. By Theorem 5 there exists a Lyapunov candi- manifolds and have characterized properties of Lyapunov functions with respect to the Riemannian distance function. date function v which satisfies (3.51). Hence, following the proof of Theorem 7, there exists a connected com- The results are given by using the geometrical concepts such as normal neighborhoods, injectivity radius and bump func- pact sublevel set of v, such that N (¯x) ⊂ U , where b,t0 x¯ tions on Riemannian manifolds. Φf+h(t,t0, xˆ0) ∈ Nb,t0 (¯x), xˆ0 ∈ int(Nb,t0 (¯x)),t ∈ [t0, ∞). Since int(Nb,t0 (¯x)) is an open set, for a given x0 ∈ int(Nb,t0 (¯x)), we can choose xˆ0 sufficiently close to References x0, such that xˆ0 ∈ int(Nb,t0 (¯x)). [1] R. Abraham, J. E. Marsden, and T. S. Ratiu. Manifolds, Tensor By employing the results of Theorem 5, the variation of v Analysis, and Applications. 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