THE property of attractiveness of a (non-trivial) invariant

Transcription

1 3396 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 61, NO 11, NOVEMBER 16 Transverse Exponential Stability and Applications Vincent Andrieu, Bayu Jayawardhana, Senior Member, IEEE, and Laurent Praly Abstract We investigate how the following properties are related to each other: i A manifold is transversally exponentially stable; ii the transverse linearization along any solution in the manifold is exponentially stable; and iii there exists a field of positive definite quadratic forms whose restrictions to the directions transversal to the manifold are decreasing along the flow We illustrate their relevance with the study of exponential incremental stability Finally, we apply these results to two control design problems, nonlinear observer design and synchronization In particular, we provide necessary and sufficient conditions for the design of nonlinear observer and of nonlinear synchronizer with exponential convergence property Index Terms Contraction, linearization, Lyapunov methods, nonlinear control systems, synchronization I INTRODUCTION THE property of attractiveness of a non-trivial invariant manifold is often sought in many control design problems In the classical internal model-based output regulation 16], it is nown that the closed-loop system must have an attractive invariant manifold, on which, the tracing error is equal to zero In the Immersion & Invariance 6], in the sliding-mode control approaches, or observer designs 3], obtaining an attractive manifold is an integral part of the design procedure Many multi-agent system problems such as formation control, consensus, and synchronization problems, are also closely related to the analysis and design of an attractive invariant manifold, see, for example, 8], 9], 34] The study of stability and/or attractiveness of invariant manifolds and more generally of sets has a long history See, for instance, 35, Section 16] and the references therein In this paper, we focus on the exponential convergence property by studying the system linearized transversally to the manifold We show that this attractiveness property is equivalent to the existence of positive definite quadratic forms which are decreasing along the flow of the transversally linear system For constant quadratic forms and when the system has some specific structure, the Manuscript received May 13, 15; revised November, 15, December 1, 15, and December, 15; accepted January 4, 16 Date of publication February 11, 16; date of current version October 5, 16 Recommended by Associate Editor L Zaccarian V Andrieu is with Université Lyon 1 CNRS UMR 57 LAGEP, France Fachbereich C-Mathemati und Naturwissenschaften, Bergische Universität Wuppertal, D-4119 Wuppertal, Germany vincentandrieu@gmail com B Jayawardhana is with ENTEG, Faculty of Mathematics and Natural Sciences, University of Groningen, 9747AG Groningen, The Netherlands bayujw@ieeeorg; bjayawardhana@rugnl L Praly is with MINES ParisTech, PSL Research University, Centre for Automatic Control and Systems Theory, 757 Paris, France Laurent Praly@mines-paristechfr Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 1119/TAC16585 latter becomes the Demidovich criterion which is a sufficient, but not yet necessary, condition for convergent systems ], 3] and 5] On the other hand, if we consider the standard output regulation theory as pursued in 16], the attractiveness of the invariant manifold is established using the center manifold theorem which corresponds to the stability property of the linearized system at an equilibrium point Due to the lac of characterization of an attractive invariant manifold, most of the literature on constructive design for nonlinear output regulator is based on various different sufficient conditions that can be very conservative In this regard, our main results can potentially provide a new framewor for control designs aiming at maing an invariant manifold attractive The paper is divided into two parts In Section II-A, we study a dynamical system that admits a transverse exponentially stable invariant manifold In particular, we establish equivalent relations between: i the transverse exponential stability of an invariant manifold; ii the exponential stability of the transverse linearized system; and iii the existence of field of positive definite quadratic forms the restrictions to the transverse direction to the manifold of which are decreasing along the flow We illustrate these results by considering a particular case of exponential incremental stable systems in Section II-B Here, incremental stability refers to the property where the distance between any two trajectories converges to zero see, for example, 4], 5] and 1] For such systems, the property i iii is used to prove that the exponential incremental stability property is equivalent to the existence of a Riemannian distance which is contracted by the flow In the second part of the paper, we apply the equivalence results to two different control problems: nonlinear observer design and synchronization of nonlinear multi-agent systems In both problems, a necessary condition is obtained Based on this necessary condition, we propose a novel design for an observer, in Section III-A, and for a synchronizer, in Section III-B In Section III-A, we reinterpret the three properties i, ii, and iii in the context of observer design This allows us to revisit some of the results obtained recently in 1] and 7] and, more importantly, to show that the sufficient condition given in 7] is actually also a necessary condition to design an exponential local full-order observer Finally, in Section III-B, we solve a nonlinear synchronization problem In particular, we give some necessary and sufficient conditions to achieve local exponential synchronization of nonlinear multi-agent systems involving more than two agents This result generalizes our preliminary wor in ] Moreover, under an extra assumption, we show how to obtain a global synchronization for the two agents case IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

2 ANDRIEU et al: TRANSVERSE EXPONENTIAL STABILITY AND APPLICATIONS 3397 It is worth noting that our main results are applicable to other control problems beyond the two control problems mentioned before In a recent paper by Wang, Ortega & Su 33], our results have been applied to solve an adaptive control problem via the Immersion & Invariance principle II MAIN RESULT A Transversally Exponentially Stable Manifold Throughout this section, we consider a system in the form ė = F e, x, ẋ = Ge, x 1 where e is in R n e, x is in R n x and the functions F : R n e R n x R n e and G : R n e R n x are C We denote by Ee,x,t,Xe,x,t the unique solution which goes through e,x in R n e at t = We assume it is defined for all positive times, ie, the system is forward complete The system 1 above can be used, for example, to study the behavior of two distinct solutions Xx 1,t and Xx,t of the system defined on R n by ẋ = fx Indeed, we obtain an e, x-system of the type 1 with F e, x =fx + e fx, Ge, x =fx 3 This is the context of incremental stability that we will use throughout this section to illustrate our main results In the following, to simplify our notations, we denote by B e a the open ball of radius a centered at the origin in R n e We study the lins between the following three notions TULES-NL Transversal uniform local exponential stability The system 1 is forward complete and there exist strictly positive real numbers r,,andλ such that we have, for all e,x,t in B e r R, Ee,x,t e exp λt 4 UES-TL Uniform exponential stability for the transversally linear system The system x = G x :=G, x 5 is forward complete and there exist strictly positive real numbers and λ such that any solution Ẽẽ, x,t, X x,t of the transversally linear system ẽ = F e, xẽ, x = G x 6 satisfies, for all ẽ, x,t in R n e R, Ẽẽ, x, t exp λt ẽ 7 ULMTE Uniform Lyapunov Matrix Transversal Equation For all positive definite matrix Q, there exists a continuous function P : R n x R n e n e and strictly positive real numbers p and p such that P has a derivative d GP along G in the following sense P X x, h P x d GP x := lim 8 h h and we have, for all x in R n x, d GP x+p x F, x+ F e e, x P x Q 9 pi P x pi 1 In other words, the system 1 is said to be TULES-NL if the manifold E := {e, x :e =} is exponentially stable for the system 1, locally in e and uniformly in x; and it is said to be UES-TL if the manifold Ẽ := { x, ẽ :ẽ =} of the linearized system transversal to E in 6 is exponentially stable uniformly in x Concerning the ULMTE property, condition 9 is related to the notion of horizontal contraction introduced in 11, Section VII] However, a ey difference is that we do not require the monotonicity condition 9 to hold in the whole manifold R n e but only along the invariant submanifold E In this case the corresponding horizontal Finsler-Lyapunov function V :R n x R n e R n e that we get taes the form V x, e, δ x,δ e = δ e P xδ e In the case where the manifold is reduced to a single point, ie, when the system 1 is simply ė = F e with an equilibrium point at the origin ie, F = then the TULES-NL property can be understood as the local exponential stability of the origin; the UES-TL notion translates to the exponential stability of the linear system ẽ = F/ eẽ; and the ULMTE concept is about the existence of a positive definite matrix P solution to the Lyapunov equation P F/ e + F/ e P = Q where Q is an arbitrary positive definite matrix In this particular case, it is well nown that these three properties are equivalent For the example of incremental stability, as mentioned before, the three properties of TULES-NL, UES-TL, and ULMTE can be understood globally as follows: P1 TULES-NL System is globally exponentially incrementally stable Namely, there exist two strictly positive real numbers and λ such that for all x 1,x in R n R n we have, for all t in R, Xx 1,t Xx,t x 1 x exp λt 11 P UES-TL The manifold E = {x, e,e=} is globally exponentially stable for the system ė = f xe, ẋ = fx 1 x Namely, there exist two strictly positive real numbers e and λ e such that for all e, x in R n R n, the corresponding solution of 1 satisfies Ee, x, t e e exp λ e t, t R

3 3398 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 61, NO 11, NOVEMBER 16 P3 ULMTE There exists a positive definite matrix Q in R n n,ac function P : R n R n n and strictly positive real numbers p and p such that P has a derivative d f P along f in the sense of, and satisfie and 19 In this context, it is nown that P3 P1 Actually, asymptotic incremental stability for which Property P1 is a particular case is nown to be equivalent to the existence of an appropriate Lyapunov function This has been established in 4], 3], 36] or 5], for instance In our context, this Lyapunov function is given as a Riemannian distance We shall show below that, as for the case of an equilibrium point, we have also P1 P P3, see Proposition 4, namely, incremental exponential stability implies the existence of a Riemannian distance for which the flow is contracting In studying the equivalence relation between TULES-NL, UES-TL, and ULMTE, we are not interested in the possibility of a solution near the invariant manifold to inherit some properties of solutions in this manifold, such as the asymptotic phase, the shadowing property, the reduction principle, etc, nor in the existence of some special coordinates allowing us to exhibit some invariant splitting in the dynamics exponential dichotomy This is the reason that, besides forward completeness, we assume nothing for the in-manifold dynamics given in 5 So, for not misleading our reader, we prefer to use the word transversal instead of normal as seen for instance in the various definitions of normally hyperbolic submanifolds given in 14, Section 1] In order to simplify the exposition of our results and to concentrate our attention on the main ideas, we assume everything is global and/or uniform, including restrictive bounds Most of this can be relaxed with woring on open or compact sets, but then with restricting the results to time intervals where a solution remains in such a particular set 1 TULES-NL UES-TL: In the spirit of Lyapunov first method, we have the following result Proposition 1: If Property TULES-NL holds and there exist positive real numbers ρ, μ,andc such that, for all x in R n x, F e,x μ, G x,x ρ 13 and, for all e, x in B e r, F e, x e e c, F e, x x e c, G e, x e c 14 then Property UES-TL holds The proof of this proposition is given in Appendix A Roughly speaing, it is based on the comparison between a given e-component of a solution Ẽẽ, x,t of 6 with pieces of e-component of solutions Eẽ i, x i,t t i of solutions of 1, where ẽ i, x i are sequences of points defined on Ẽẽ, x,t Thans to the bound3 and 14, it is possible to show that Ẽ and E remain sufficiently closed so that Ẽ inherit the convergence property of the solution E As a consequence, in the particular case in which F does not depend on x, thetwo functions E and Ẽ do not depend on x either and the bounds on the derivatives of the G function are useless UES-TL ULMTE: Analogous to the property of existence of a solution to the Lyapunov matrix equation, we have the following proposition on the lin between UES-TL and ULMTE notions Proposition : If Property UES-TL holds and there exists a positive real number μ such that F e,x μ x Rn x 15 then Property ULMTE holds The proof of this proposition is given in Appendix B The idea is to show that, for every symmetric positive definite matrix Q, the function P : R n x R n e n e given by P x = T lim, x, s Q, x, sd6 T + is well defined, continuous, and satisfies all the requirements of the property ULMTE The assumption 15 is used to show that P satisfies the left inequality in 1 Nevertheless, this inequality holds without 15 provided the function s /, x, s does not go too fast to zero 3 ULMTE TULES-NL: Proposition 3: If Property ULMTE holds and there exist positive real numbers η and c such that, for all e, x in B e η, P x x c 17 F e e e,x c, F x e e,x c, G e e,x c 18 then Property TULES-NL holds The proof of this proposition can be found in Appendix C This is a direct consequence of the use of V e, x =e P xe as a Lyapunov function The bound7 and 18 are used to show that, with 9, the time derivative of this Lyapunov function is negative in a uniform tubular neighborhood of the manifold {e, x,e=} B Revisiting the Exponential Incremental Stable Systems Incremental stability of an autonomous system is the property that a distance between any two solutions of converges asymptotically to zero The characterization of it has been studied thoroughly, for example, in 4], 5] and 1] In 4], 5], a Lyapunov characterization of incremental stability δ-gas for autonomous systems and δ-iss for nonautonomous ones is given based on the Euclidean distance between two states that evolve in an identical system A variant of this notion is that of convergent systems discussed in ], 5] All these studies are based on the notion of contracting flows which has been widely studied in the literature and for a long time, see, for example, 9], 11], 13], 18] 1] These flows generate trajectories between which an appropriately defined distance is monotonically decreasing with increasing time See 17] for a historical discussion on the contraction analysis and 3] for a partial survey

4 ANDRIEU et al: TRANSVERSE EXPONENTIAL STABILITY AND APPLICATIONS 3399 The big issue in this view point is to find the appropriate distance which may be a difficult tas The results in Section II may help in this regard with providing an explicit construction of a Riemannian distance Precisely, let P be a C function defined on R n the values of which are symmetric matrices satisfying pi P x pi x R n 19 The length of any piece-wise C 1 path γ :,s ] R n between two arbitrary points x 1 = γ and x = γs in R n is defined as: Lγ s s = dγ ds σ P γσ dγ σdσ ds By minimizing along all such paths, we get the distance dx 1,x Then, thans to the well-established relation between geodesically monotone vector field semi-group generator operator and contracting non-expansive flow semi-group see 7], 13], 15], 18]and many others, we now that this distance between any two solutions of is exponentially decreasing to as time moves forward if we have d f P x+px f f x+ x x x P x Q x R n 1 where Q is a positive definite symmetric matrix and P Xx, h P x d f P x = lim h h For a proof, see, for example 18, Theorem 1] or 15, Theorems 57 and 533] or 4, Lemma 33] replacing fx by x + hfx] In this context, using the main results of our previous section, we can show that, if we have exponential incremental stability, then there exists a function P meeting the above requirements Specifically, we have the following proposition Proposition 4 Incremental Stability: Assume the system is forward complete with a function f which is C 3 with bounded first, second, and third derivatives Let Xx, t denotes its solutions Then, we have P1 P P3 and, therefore, P1 P P3 In other words, exponential incremental stability property is equivalent to the existence of a Riemannian distance which is contracted by the flow and can be used as a δ-gas Lyapunov function Note also that, despite the fact that the main results in Section II are local, when we restrict ourselves to the incremental stability problem, we can obtain a global result Proof P1 P P3: Consider the system 3 and let n x = n e = n The boundedness of the first derivative of f implies the forward completeness of the corresponding systems 1 and 5 Moreover, the inequalitie3 15 with r =+ follow from the assumption of boundedness of the derivatives of f As a consequence, P1 P follows from Proposition 1 and P P3 from Proposition Note, however, that it remains to show that P defined in 16 is C This is obtained employing the boundedness of the first, second, and third derivatives of f Indeed, note that we have for all t, x/ e,x,t= X/ xx, t So, to show that P is C 1, it suffices to show that the mapping t X/ x i xx, t goes exponentially to zero as time goes to infinity Note that this is indeed the case since, given a vector v in R n and i in {1,,n}, the mapping νt = X/ x i xx, tv is the solution to ν = f n x Xx, t ν+ f x j x Xx, t X j x,t X x i x x,tv j=1 Hence, from 1, 19, and the fact that f has bounded second derivatives, it yields ß the existence of a positive real number c such that ν P Xx, t ν ν Qν + c ν X x, t x Since t X/ xx, t exponentially goes to zero as time goes to infinity, it implies that ν exponentially goes to zero Hence, P is C 1 Employing the bound on the third derivative and following the same route, it follows that P is C III APPLICATIONS In this section, we apply Proposition,, and 3 in two different contexts: full order observer and synchronization A Nonlinear Observer Design Consider a system ẋ = fx, y = hx 3 with state x in R n and output y in R p, augmented with a state observer of the particular form with state ˆx in R n and where ˆx = fˆx+ky, ˆx 4 Khx,x= x 5 Assuming the functions f, h,andk are C, we are interested in having the manifold {x, ˆx :x =ˆx} exponentially stable for the overall system ẋ = fx, ˆx = fˆx+ky, ˆx 6 When specified to this context, the properties TULES-NL, UES-TL, and ULMTE are Exponentially convergent observer TULES-NL: The system 6 is forward complete and there exist strictly positive real numbers r,, andλ such that we have, for all x, ˆx, t in R n R satisfying x ˆx r, we have Xx,t ˆXx, ˆx,t x ˆx exp λt 7

5 34 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 61, NO 11, NOVEMBER 16 UES-TL FOR OBSERVER The system ẋ = fx is forward complete and there exist strictly positive real numbers and λ such that any solution Ẽẽ,x,t,Xx,t of the transversally linear system ] f ẽ = x x+ K h hx,x y x x ẽ, ẋ = fx 8 satisfies, for all ẽ,x,t in R n R n R, Ẽẽ,x,t exp λt ẽ 9 ULMTE FOR OBSERVER: For all positive definite matrix Q, there exists a continuous function P : R n R n n and strictly positive real numbers p and p such that we have, for all x in R n, pi P x pi and f d f P x+sym P x x x + K y hx,x h x x ] Q 3 where SymA =A + A Proposition,, and 3 give conditions under which these properties are equivalent But these properties assume the data of the correction term K Hence, by rewriting UES-TEL and TULES-NL in a way in which the design parameter K disappears, these propositions give necessary conditions for the existence of an exponentially convergent observer Property UES-TL involves the existence of an observer with correction term depending on x for the time-varying linear system resulting from the linearization along a solution to the system 3, ie, ẽ = f x xẽ, ỹ = h xẽ 31 x seeing ỹ as output As a consequence of Proposition 1, a necessary condition for Property UES-TL to hold and further, when some derivatives are bounded, for the existence of an exponentially convergent observer is that the system 3 be infinitesimally detectable in the following sense 1 Infinitesimal detectability We say that the system 3 is infinitesimally detectable if every solution of ẋ = fx, ẽ = f x xẽ, h xẽ = x defined on, + satisfies lim t + Ẽe, x, t = A similar necessary condition has been established in 1] for a larger class of observers, but under an extra assumption the existence of a locally quadratic Lyapunov function Example 1: Consider the planar system ẋ 1 = x 3, ẋ = x 1, y = x 1 3 We wish to now whether or not it is possible to design an exponentially convergent observer for this nonlinear oscillator in the form of 4 The linearized system is ẽ 1 =3x ẽ, ẽ = ẽ 1, ỹ = ẽ 1 33 This system is not detectable when the solution, along which we linearize, is the origin which is an equilibrium of 3 Consequently, the system 3 is not infinitesimally detectable on R, and so there is no exponentially convergent observer on R Fortunately, the subset {x R :x 1/ + x 4 /4 ɛ}, with ɛ>, is invariant and 3 is infinitesimally detectable in it To design a correction term K for an exponentially convergent observer, we use the property that Ly, x = K y, x y should be an observer gain for the linear system 33 So, we start our design by selecting LWepic This gives see 8] Ly, x = ] 3x 3x +1 Ax = f x x+ K h hx,x y x x =3x ] The transition matrix generated by AXx, t when Xx, t is a solution of 3 is ] 1 It Φt, = exp It 1 where t It =3 X x, s ds Since X x, t is periodic, 9 holds when the initial condition x is in the compact invariant subset { C = x R : 1 } ε x 1 + x4 4 ɛ 34 Then, according to Propositions and 3, and in view of 5, we obtain an exponentially convergent observer by choosing K as Ky, x = y x 1 Ls, xds = ] 3x 3x +1 y x 1 Similarly, Property ULMTE involves the existence of P and K such that 3 holds By restricting this inequality on quadratic forms to vectors which are in the ernel of h/ x, we obtain as a consequence of Proposition and, that a necessary condition for Property ULMTE to hold and further, when some derivatives are bounded, for the existence of an exponentially convergent observer is that the system 3 be R-detectable R for Riemann in the following sense R-Detectability We say that the system 3 is R-detectable if there exist a continuous function P : R n R n n and positive real numbers <p pand <qsuch that P has a derivative d f P along f in the sense of and we have pi P x pi x R n 35 v d f P xv +v P x f x xv qv P xv 36 holds for all x, v in R n R n satisfying h/ xxv =

6 ANDRIEU et al: TRANSVERSE EXPONENTIAL STABILITY AND APPLICATIONS 341 A similar necessary condition has been established in 7], where only asymptotic and not exponential convergence is assumed In that case, the condition allows p and q to be zero Further, it is established in 8] that when the R-detectability holds then 1 h Ky, x =Px x xt y hx gives, for large enough, a locally exponentially convergent observer Example 1 continued: For the system 3, the necessary P11 P R-Detectability condition is the existenceof P = 1 P 1 P satisfying, in particular, 36 which is see 34] P x 1 xx 3 P x xx 1 +6P 1 xx < qp x x C 37 We view this as a condition on P only, since whatever P 1 is, we can always pic P 11 to satisfy 35 Note also that we can tae care of any term with x in factor by selecting P 1 appropriately With this, it can be shown that it is sufficient to pic P in the form rx P x =rx+ x 1x rx + x 3rx where the presence of r defined below is justified by homogeneity considerations 4ε rx = x 1 + x4 4 ε This motivates us to design x P 1 x = 5 1 rx 4 rx 3 In this case, the left hand side in the inequality 37 is 1 +x 1 x rxx r 4 rx 1 rx ε 4 rx rx P x 1 6 P x Finally, by choosing x 1 x 4 P 11 x =+ P 1 x P x rx 4 it can be shown that we obtain { εi min 1, rx } I P x 3max{1,rx} I 1 4 ε Hence, 35 holds on C From this, the correction term P x P x Ky, x = P 11 P x P 1 x y x P 1 x 1 gives a locally convergent observer on C B Exponential Synchronization Finally, we revisit the synchronization problem as another class of control problems that can be dealt with the results in Section II We consider here the synchronization of m identical systems given by ẇ i = fw i +gw i u i, i =1,,m 38 In this setting, all systems have the same drift vector field f and the same control vector field g : R n R n p, but not the same controls in R p The state of the whole system is denoted w =w 1,w m in R mn We also define the diagonal subset of R mn D = {w 1,,w m R mn,w 1 = w = w m } Given w in R mn, we denote the Euclidean distance to the set D as w D The synchronization problem that we consider in this section is as follows Definition 1: The control laws u i = φ i w, w =w 1,, w m, i =1,msolve the local uniform exponential synchronization problem for 38 if the following holds: 1 φ is invariant by permutation of agents More precisely, given a permutation π : {1,m} {1,m} φ is zero on D: φ πi w 1,,w m =φ i w π1,,w πm φw = w D 39 3 and the set D is uniformly exponentially stable for the closed-loop system, ie, there exist positive real numbers r w, and λ> such that, for all w in R mn satisfying w D <r w, W w, t D exp λt w D 4 holds for all t in the domain of existence of the solutions W w, t going through w at t = When r w =, it is called the global uniform exponential synchronization problem In this context, we assume that every agent shares an information which will be designed later to all other agents in which case, it forms a complete graph and it has local access to its state variables It is possible to rewrite the property of having the manifold D exponentially stable as property TULES-NL As it has been done in the observer design context, employing Proposition and and by rewriting properties UES-TL and ULMTE it is possible to give equivalent characterization of the synchronization property By rewriting these conditions in a way in which the control law disappears, these properties give necessary conditions to achieve exponential synchronization Proposition 5 Necessary Condition: Consider the systems in 38 and assume the existence of control laws u i = φ i w, i =1,mthat solve the uniform exponential synchronization of 38 Assume moreover that g is bounded and f, g and the φ i s have bounded first and second derivatives Then, the following two properties hold

7 34 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 61, NO 11, NOVEMBER 16 Q1: The origin of the transversally linear system ẽ = f x xẽ + g xu, x = f x 41 is stabilizable by a linear in ẽ state feedbac Q: For every positive definite matrix Q, there exist a continuous function P : R n R n n and positive real numbers p and p such that inequalitie9 are satisfied, P has a derivative d f P along f in the sense of, and d f v P xv +v P x f x xv v Qv 4 holds for all v, x in R n satisfying v P xgx = Proof: First of all, note that the vector fields having bounded first derivatives, it implies that the system is complete Consider i, j,, l, 4 integers in {1,,m} and consider a permutation π : {1,,m} {1,,m} such that π i = and π j = l Note that = l if and only if i = j Note that the invariance by permutation implies Hence, it follows that φ w =φ i w π1,,w πm φ w l w = φ i w j w π1,,w πm and if we consider w in D, this implies φ w l w = φ i w j w, i j, l φ j w j w = φ i w i w By denoting e =e,e m with e i = w i w 1, i =,m, and x = w 1, we obtain an e, x-system of the type 1 with F e, x =F i e, x i=,m 43 F i e, x =fx + e i fx + gx + e i φ i e, x gx φ 1 e, x 44 Ge, x =fx+gx φ 1 e, x 45 wherewehaveusedthenotation Note that we have φ i e, x =φ i x, x + e,,x+ e n e m 1 w D 46 w m D e +m 1 i=1 w 1 w i m 1+ m 1 m e 47 Hence, 4 implies for all e, x with e mr w / m + m 1 Ee, x, t m 1 1+ m 1 m exp λt e It follows from the assumptions of the proposition that Property TULES-NL is satisfied with r = mr w / m + m 1 and that inequalitie3 and 14 hold We conclude with Proposition 1 that Property UES-TL is satisfied also So, in particular, there exist positive real numbers and λ such that any e i component of Ẽẽ, x,t, X x,t solution of 6 satisfies, for all ẽ, x,t in R mn R mn R, Ẽiẽ, x, t exp λt ẽ 48 On another hand, with 39, we obtain: F i, x = f e i x x+g x φi, x φ ] 1, x 49 e i e i And, when j i, it yields F i φi, x =g x, x φ ] 1, x e j e j e j Consequently, any solution of the system f ẽ i = x x+g x φi, x φ ]] 1, x e i e i = 5 and x = f x can be expressed as an e i component of Ẽẽ, x,t, X x,t solution of 6 Since these solutions satisfy 48, Property Q1 does hold Finally, we consider the system with state e i,x in R n ė i = F i e i,x, ẋ = Ge i,x=fx 51 with F i e i,x=f i,e i,,x The previous property and Proposition imply that Property ULMTE is satisfied for system 51 So, in particular, we have a function P satisfying the properties in Q and such that we have, for all v, x in R n R n, f v d f P xv+v P x x x+g x φi, x φ ] 1, x v e i e i ẽ i v Qv which implies 4 when v P xgx = Example : As an illustrative example consider the case in which the system is given by by m agents w i in R with individual dynamics ẇ i1 = w i +sinw i, ẇ i = a + u i 5 where a is a real number Because of a singularity when 1+cosw i =, this system is not feedbac linearizable per se Hence, the design of a synchronizing controller may be involved

8 ANDRIEU et al: TRANSVERSE EXPONENTIAL STABILITY AND APPLICATIONS 343 In order to chec if local synchronization in the sense of Definition 1 is possible, the necessary conditions of Proposition 5 may be tested The transversally linear system is 1+cosx + at ẽ = ] ẽ + 1] u 53 When a =and x =π/3, this system is not stabilizable by any feedbac law Hence, in this case, with Proposition 5, there is no exponentially synchronizing control law in the sense of Definition 1 satisfying 39 in particular Similar to the analysis of incremental stability in the previous section and observer design in 7], by using a function P satisfying the property Q in Proposition 5, we can obtain sufficient conditions for the solvability of uniform exponential synchronization of 38 We do this under an extra assumption which is that, up to a scaling factor, the control vector field g is a gradient field with P as Riemannian metric Proposition 6 Local Sufficient Condition: Assume f has bounded first and second derivatives, and g is bounded and has bounded first and second derivatives Moreover, assume that 1 there exist a C function U : R n R and a bounded C function α : R n R p which has bounded first and second derivative such that x x = P xgxαx 54 holds for all x in R n ;and there exist a positive definite matrix Q,aC function P : R n R n n with bounded derivative, and positive real numbers p, p and ρ> such that 19 is satisfied and v d f P xv +v P x f x xv ρ x xv holds for all x, v in R n R n v Qv 55 Then, there exist a real number l such that with the control laws u i = φ i w given by m φ i w =lαw i Uw j m Uw i 56 j=1 and l l and if the closed loop system is complete, then the local uniform exponential synchronization of 38 is solved Note that, for its implementation, the control law 56 requires that each agent i communicates Uw i to all the other agents Proof: First of all, note that the control law φ i is invariant by permutation due to its structure Let e =e,,e m with e i = x 1 x i and x = x 1 We obtain an e, x-system of the type 1 with F and G as defined in with φ as control input For this system, we will show that property ULMTE is satisfied Consider the function P m : R n R m 1n m 1n defined as a bloc diagonal matrix composed of m 1 matrices P, ie, P m x =DiagP x,,px Note that with property 49 and 5, it yields that F/ e, x is also m 1 bloc diagonal Hence, we have d GP m x+p m x F, x+ F e e, x P m x = Diag {R x,,r x} where f R x =d f P xv + P x x lg xα x x ] x x ] f + x lg xα x x x x P x With 55, this gives v R xv v Qv + l x xv for all x, v in R n R n Hence, picing l>/, inequality 9 holds To apply proposition 3, it remains to show that inequalities 15, 17 and 18 are satisfied Note that employing the bounds on the functions P, f, g, α, and their derivatives, it is possible to get a positive real number c depending on lsuch that for all i in,,mand all e, x F i e, x e i c m Ux + e i Ux + e i m j=1 + c F i e, x e i x c m Ux + e i Ux + e i m j=1 + c F i e, x e i e j c m Ux + e i Ux + e i m j=1 + c m Ux + e i Ux + e i m c e, e, x j=1 So, we fix η positive and pic c = c η + c The above shows that inequalitie7 and 18 are satisfied With Proposition 3, we conclude that Property TULES-NL holds Hence, e =is locally exponentially stable manifold With inequalities 46 and 47, this implies that inequality 4 holds In this result, it is important to remar that there is no guarantee that the control law given here ensures completeness of the solution Note, however, that on the manifold w D =, the trajectories satisfy ẋ = fx which is a complete system Example continued: We come bac to the example 5 in the case where a =1 We note that the linear system 53 is stabilizable by a feedbac in the form u = 1+cosx + t 3]ẽ Indeed, the solution of 53 with the previous feedbac satisfies ] 1 ẽ =1+cosx + t ẽ 3

9 344 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 61, NO 11, NOVEMBER 16 Hence, its solutions are Ẽe, x, t =ψx, tẽ, whereψ is the generator of this time varying linear system given as ] 1 ψx, t =exp t +sinx + t 3 ] 1+ϕx, t 1+ϕx, t = ϕx, t 1 ϕx, t ϕx, t with ϕx, t =e t+ sint+x Consequently, we get that ẽ goes exponentially to zero Hence, Property Q 1 is satisfied We can then introduce the matrix P solution to Q and given in 16 as P x = + ψx, s ψx, sds 57 This matrix is positive definite and satisfies property Q So, we may want to use it for designing an exponentially synchronizing control law With decomposing the matrixp as P w i = P11 w i ] P 1 w i P 1 w i P w i we obtain P wgw =P 1 w i P w i ] Note that it can be shown numerically that P 1 w i = 4 ϕw i,t 9 ϕw i,t dt > ϕw i,t 4 It follows the that function αw =1/P 1 w i is well defined and setting Uw i =w i1 + w i P s P 1 s ds property 54 is satisfied Hence, for this example, picing l a sufficiently large real number, the control law 56 ensures local exponential synchronization of m agents We have checed this via simulation for the case m =5, l =3 The time evolution of the solution with w i, i =1,,5 chosen randomly according to a uniform distribution on, 1] is shown in Fig 1a for w i1 and Fig 1b for w i As in the context of the observer design given in 7], a global result can be obtained by imposing a further constraint on P Specifically, the notion we need to introduce is the following Definition Totally Geodesically Set: Given a C function P defined on R n the values of which are symmetric positive definite matrices, a C 1 function ϕ : R n R + and a real number ϕ, the level set S = {x R n,ϕx = ϕ} is said to be totally geodesic with respect to P if, for any x, v in S R n such that ϕ x xv =,v P xv =1 Fig 1 Numerical simulation results of the interconnected five agents as considered in Example 1 The left-figure shows the trajectories of the first state variable of each agent, w i1,i =15 while the right-figure shows those of the second state variable of each agent w i,i=15 The simulation results show that the proposed contraction-based distributed control law is able to synchronize the states w i, i =15 atheplotofw i1, i =1,5; b the plot of w i, i =1,5 any geodesic γ, ie, a solution of d dγ ds ds s P γ s l = 1 s dγ x ds s P x dγ ds x=γ s 58 with γ=x and dγ/ds=v satisfies ϕ/ xγsdγ/ dss =for all s For the case of two agents only, we have the following Proposition 7 Global Sufficient Condition for m =: Assume 1 there exist a C 3 function U : R n R which has bounded first and second derivatives, and a C 1 function α : R n R p such that, for all x in R n, 54 is satisfied;

10 ANDRIEU et al: TRANSVERSE EXPONENTIAL STABILITY AND APPLICATIONS 345 there exist a positive real number λ, ac 3 function P : R n R n n and positive real numbers p and p, such that inequalitie9 hold and we have, for all x, v in R n R n such that / xx v = 1 v d f P xv + v P x f x xv λv P xv 59 3 For all Ū in R, the set S = {x R n,ux =Ū} is totally geodesic with respect to P Then, there exists a function l : R n R +, invariant by permutation such that, with the controls given by φ i w =lwαw i Uw j Uw i with i, j {1,,, 1}, the following holds and for all w in R n, W w, t D w D exp λ t 6 where t is any positive real number in the time domain of definition of the closed loop solution The proof of this result is given in Appendix D It borrows some ideas of 7] However, different from 7], we have here a global convergence result This follows from the fact that in the high gain parameter l, the norm of the full state space can be used and not only the norm of the estimate as in the observer case Note that nothing is said about the domain of existence of the solution IV CONCLUSION We have studied the relationship between the exponential stability of an invariant manifold and the existence of a Riemannian metric for which the flow is transversally contracting It was shown that the following properties are equivalent 1 A manifold is transversally exponentially stable The transverse linearization along any solution in the manifold is exponentially stable 3 There exists a field of positive definite quadratic forms whose restrictions to the transverse direction to the manifold are decreasing along the flow As an illustrative example for these equivalence results, we have revisited the property of exponential incremental stability where we can obtain a global result The characterization of transverse exponential stability has allowed us to investigate a necessary condition for two different control problems of nonlinear observer design and of synchronization of nonlinear multi-agent systems which leads to a novel constructive design for each problem Recent results by others has also shown the applicability of our results beyond these two control problems Although the main results hold for local uniform transverse exponential stability, we show that global results can also be obtained in some particular cases The extension of all the results to the global case is currently under study APPENDIX A Proof of Proposition 1 Proof: Let us start with some estimations Let z = e ẽ Along solutions of 1 and 6, we have ż = F e, x F F, xẽ =, xz +Δx, e, x e e with the notation Δe, x, x =F e, x F, xe e =F e, x F e, x] + F e, x F, x F ], xe e Note that the manifold E := {e, x :e =} being invariant, it yields F,x= x 61 With Hadamard s lemma, 61 and 14, we obtain the existence of positive real numbers c 1 and c such that, for all e, x, x in B e r, Δe, x, x c 1 e + c e x x This, with 15, gives, for all e, ẽ, x, x in B e r R n e R n x, 1 õ z μ z + c1 e + c e x x 6 Similarly, 1, 6, and 14 give, for all e, x, x in B e r R n x ú, x x Ge, x G,x + G,x G, x c e + ρ x x 63 Now, let r be a positive real number smaller than r and S be a positive real number both to be made precise later on Let ẽ in B e r and x in R n x be arbitrary and let Ẽẽ, x,t, X x,t be the corresponding solution of 6 Because of the completeness assumption on 1, the linearity of 6 and the fact that, X x,t is solution of both 1 and 6, Ẽ, X is defined on, + We denote: ẽ i = Ẽẽ, x,is, x i = X x,is i N and consider the corresponding solutions Eẽ i, x i,s,xẽ i, x i,s of 1 By assumption, they are defined on, + and, 1 Here, the notation ô z is abusive The function x x is not C 1,but only Lipschitz Nevertheless, given a vector field f an upper right Dini Lie derivative, ie, lim sup h + x + hfx x /h does exist and, by the triangle inequality, we have fx lim sup h + x + hfx x h fx So, here and in the following, ô x denotes this upper right Dini Lie derivative

11 346 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 61, NO 11, NOVEMBER 16 because of 4, if ẽ i is in B e r, theneẽ i, x i,s is in B e r for all positive times s, maing possible the use of inequalities 6and 63Finally,for each integeri, we define the following time functions on,s] Z i s = Eẽ i, x i,s Ẽẽ, x,s+ is W i s = Xẽ i, x i,s X x,s+ is Note that we have Z i = W i = From the inequalities 63 and 7, we get, for each integer i such that ẽ i is in B e r, andforalls in,s], W i s c c s s exp ρs σ Eẽ i, x i,s dσ exp ρs σ exp λσdσ ẽ i c exp λs exp ρ + λs 1 ẽ i ρ + λ Similarly, using 6 and Grönwall inequality we get Z i s c s expμs σ Eẽ i, x i,σ Eẽ i, x i,σ +W i σdσ γs ẽ i s,s] where we have used the notation, s γs =c exp μs σ exp λσ exp ρ + λσ 1 + c exp λσ dσ ρ + λ With all this, we have obtained that, if we have ẽ j in B e r for all j in {,,i},thenwehavealso,forallsin,s] and all j in {,,i}, Ẽẽ, x,s+ js = Ẽẽ j, x j,s Eẽ j, x j,s + Z j s exp λs+γs ẽ j ] ẽ j Now, given a real number ε in,1, we select S and r to satisfy: min{, 1 ε} exp λs { } min{, 1 ε} r min r, sup s,s] γs Then, for all ẽ j smaller in norm than r, wehave Ẽẽ, x,s+ js 1 ɛ ẽ j So, since ẽ is in B e r, it follows by induction that we have: ẽ i = Ẽẽ, x,is 1 ε i r r i N Since, with 15, we have also õ ẽ μ ẽ, we have established, for all s in,s] and all i in N, Ẽẽ, x,s+ is expμs1 ɛ i ẽ and therefore, for all ẽ, x,t in B e a R, Ẽẽ, x,t expμs1 ɛ t S S ẽ By rearranging this inequality and taing advantage of the homogeneity of the system 6 in the ẽ component, we have obtained 7 with =expμs/1 ɛ and λ = ln1 ɛ/s B Proof of Proposition Proof: Let Ẽẽ, x,t, X x,t be the solution of 6 passing through an arbitrary pair ẽ, x in R n e By assumption, it is defined on, + For any v in R n e,wehave t, x,tv = F, e X x Ẽ,t, x,tv Uniqueness of solutions then implies, for all ẽ, x,t in R n e R, Ẽẽ, x,t=/, x,tẽ and our assumption 7 gives, for all ẽ, x,t in R n e R, /, x,tẽ ẽ exp λt, and therefore, x,t exp λt x,t R n x R This allows us to claim that, for every symmetric positive definite matrix Q, the function P : R n x R n e n e given by 16 is well defined, continuous, and satisfies λ max {P x} Moreover, we have ] 1 v t, x,t λ λ max{q} = p x R n x ] 1 = v F, x,t, e X x,t

12 ANDRIEU et al: TRANSVERSE EXPONENTIAL STABILITY AND APPLICATIONS 347 With 15, this yields v /, x,t] 1 expμt v and implies v v] v ] 1, x,t, x,tv 1 λ min {Q} v, x,t ] 1 v, x,t Q, x,tv v expμt λ min {Q} This gives v, x,t Q, x,tv p = 1 μ λ min{q} λ min {P x} x R n x Finally, to get 9, let us exploit the semi group property of the solutions We have for all ẽ, x in R n x R n e and all t, r in R Ẽ Ẽẽ, x, t, X x, t,r = Ẽẽ, x, t + r Differentiating with respect to ẽ, the previous equality yields Ẽẽ, x, t, X x, t,r Ẽ ẽ, x, t = ẽ, x, t + r Setting in the previous equality ẽ, x :=, X x, h,h:= t, s := t + r we get for all x in R n x and all s, h in R, x, s + h, X x, h, h =, X x, h,s Consequently, this yields P X x, h = lim T T + = lim T + T, e X x, h,s, X x, h, h Q, X x, h,s ds, x, s + h Q, x, s + hds, X x, h, h But we have: lim h lim h T s =, X x, h, h h, x, s + h h T +, x, s I = F, x e, x, s = s, x, s ds Q, x, s Q s Q, x, T, x, T, x, s, x, s ds Q Since lim T and lim h commute because of the exponential convergence to of /, x, s, we conclude that the derivative 8 does exist and satisfies 9 C Proof of Proposition 3 Proof: Consider the function V e, x =e P xe Using 9, the time derivative of V along the solutions of the system 1 is given, for all e, x, by e, ûv x =e P xf e, x+ e P e xge, x x e Qe +e P x F e, x F ] e,xe + e P e xge, x G,x] x On the other hand, using Hadamard s Lemma and 18, we get: F F e, x e,xe c e Ge, x G,x c e e, x B e η These inequalities together with 1 and 17 imply, for all e, x in B e η, λmin {Q} e, ûv x c1 + c p ] p p e V e, x It shows immediately that 4 holds with r, and λ satisfying: r< p } {η, p min λ min {Q} pc1 + c p = p λmin {Q} λ = rc1 + c p ] p p

13 348 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 61, NO 11, NOVEMBER 16 D Proof of Proposition 7 The result holds when w is in D or when U is constant since 59 holds for all v So, in view of 7, Proposition A1], we can assume without loss of generality that / w has nowhere a zero norm and, in the following, we restrict our attention to R n \DInR n \Dthe dynamics of w is ẇ i = fw i +lwgw i αw i Uw j Uw i ] j=1 With the C matrix function P we define the Riemannian length of a piece-wise C 1 path γ :,s ] R n, between w 1 = γ and w = γs as in and the corresponding distance dw 1,w by minimizing along all such path Because of 19 and the fact that P is C 3, Hopf-Rinow Theorem implies the metric space we obtain this way is complete, and, given any w 1 in R n and w in R n, there exists a C 3 normalized minimal geodesic γ, solution of 58, such that w 1 = γ, w = γ s s dw 1,w =Lγ = s 64 Following 7], for each s in,s ] consider the C 1 function t Γs, t solution of Γ s, t =f Γs, t t + l W w, t g Γs, t α Γs, t U W j w, t U Γs, t] with initial condition j=1 Γs, = γ s 65 With 39, we have Γ,t=W 1 w, t, Γs,t=W w, t and so, for each t, s,s ] Γs, t is a C path between W 1 w, t and W w, t From the first variation formula see 31, Theorem 614] for instance, 3 we have where d dt L Γs, t s t= = Aw+Cw Aw =lwuw 1 Uw ] dγ ds s P w gw αw lwuw Uw 1 ] dγ ds P w 1 gw 1 αw 1 Cw = dγ ds s P w fw dγ ds P w 1 fw 1 But, with 54, we obtain: Aw = lwuw Uw 1 ] x w dγ ds s + ] x w 1 dγ ds This means that γ satisfies dγ /dss P γ sdγ /dss =1 3 In 31, Theorem 614], the result is stated with γ C note however that C is enough Also, with the Euler-Lagrange form of the geodesics58, we get: Cw = = s s d dγ ds ds s P γ s f γ s + dγ ds s P γ s d ] ds f γ s ds 1 s dγ x ds s P x dγ f γ ds x=γ s s + dγ ds s P γ s f ] x γ s dγ ds s ds Here the integrand is nothing but the left hand side of 59 With a compactness argument 4 we can show that condition 59 in Proposition 7 is equivalent to the existence of a smooth function ν : R n R + such that, for all x, v, 1 v d f P xv + v P x f x xv λv P xv + νx x xv Hence, the geodesic being normalized, we have: Cw+λ s with the notation: Bw = dγ ds s P γ s dγ ds sds s = Cw+λdw 1,w Bw ν γ s x γ s dγ ds s ds From A and B we define two C functions a and b by dividing by dw 1,w =s Namely, we define: a γ w, r = U γ r + Uw 1 r x γ r+ dγ ds γ r+ + ] x w 1 dγ ds 66 b γ w, r = 1 r+s 1 ν γ s r x γ s dγ ds s ds 67 4 The following two properties are equivalent a v fxv < for all v with v =1and all x satisfying gxv =b there exists ν such that v fxv νx gxv for all v with v =1and all x Proof b a is trivial For the converse, let C be an arbitrary compact set, if b does not hold for some η C and all x in C, thereexistx i and v i with v i =1satisfying v i fx iv i i gx i v i With compactness this implies the existence of x ω and v ω with v ω =1satisfying gx ωv ω =and v ω fxωvω This contradicts a

14 ANDRIEU et al: TRANSVERSE EXPONENTIAL STABILITY AND APPLICATIONS 349 They are defined on R n \D ],dw 1,w ] and depend apriorion the particular minimizing geodesic γ we consider We extend by continuity in r their definition to R n \D,dw 1,w ] by letting a γ w, = x γ dγ ds s 1 b γ w, = ν γ x γ dγ ds s 1 In this way, for any pair w 1,w in R n \Dand any minimizing geodesic γ between w 1 and w, the function r a γ w, r,b γ w, r is defined and C 15 on,dw 1,w ], wehave: 1 d s L Γs, t + λ dw 1,w dt t= b γ w, dw 1,w lwa γ w, dw 1,w Also, Lemma 1: For any pair w 1,w in R n \D and any minimizing geodesic γ between w 1 and w, a γ w, dw 1,w is non-negative, and if it is zero, the same holds for b γ w, dw 1,w Proof: For any pair w in R n \D, the function r a γ w, r is defined and continuous on,dw 1,w ] If the real number a γ w, = x γ dγ ds s 1 is zero, the level sets of U being totally geodesic, we have x γ r + dγ ds r + = U γ r + = Uw 1, r,dw 1,w ] and so Uw =Uw 1 and a γ w, dw 1,w and b γ w, dw 1,w are zero 5 This comes from this general result Let f be a C function defined on a neighborhood of in R, where it is The function ϕ defined as ϕr = fr/r if r and ϕ = f is C 1 Indeed it is clearly C everywhere except may be at Its first derivative is ϕ r =fr rf /r Itis also continuous at since lim r ϕr =f = ϕ Its first derivative at exists if lim r ϕr ϕ/r = lim r fr rf /r exists But this is the case, since f being C,wehave fr rf r = 1 r = 1 r = 1 r r f s f ] ds r s r f tdt ds f tr t]dt which leads to ϕ = 1/f Wehavealsofr rf r/r = 1/r r sf sds This implies lim r ϕ r =ϕ and therefore ϕ is continuous If instead that real number is positive and a γ w, dw 1,w = U γ s Uw 1 dw 1,w x γ s dγ ds γ s + ] x w 1 dγ ds is non-positive, then there exists r in ],dw 1,w ] such that either Uγ r + =Uw 1 But,thelevelsetsofU being totally geodesic and γ being a minimizing geodesic between w 1 and w, and therefore between w 1 and γ r +, it follows from the proof of 7, Proposition A3] that γ taes its values in the level set {x : Ux =Uw 1 }, at least on,r + ] This implies x γ s dγ s = ds for all s in,r + ] and, consequently, in,s ] This yields Uw =Uw 1 and a γ w, dw 1,w and b γ w, dw 1,w are zero; or we have x γ r + dγ ds γ r + + x γ dγ ds = This implies that / xγ dγ /ds and / xγ r + dγ /dsr + have opposite signs and so the function r / xγ r + dγ /dsr + must vanish on ],r Again, this implies Uw =Uw 1 and and a γ w, dw 1,w and b γ w, dw 1,w are zero Now, to each pair of integers i 1,i, we associate the compact set C i1 i = { w 1,w R n : i 1 dw 1, i 1 +1, i dw, i +1} Lemma : For any pair i 1,i, there exists a real number l i1 i such that, for all w 1,w in C i1 i \Dand all l larger or equal to l i1 i,wehave: b w, dw 1,w la w, dw 1,w λ 68 Proof: For the sae of getting a contradiction, assume there exist a pair i 1,i and a sequence w 1,w,γ N of points and minimizing geodesic satisfying dw 1,w w 1 = γ w = γ dw 1,w = w 1 + dw 1,w dγ sds 69 ds λ + b γ w 1,w,dw 1,w a γ w 1,w,dw 1,w 7

15 341 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 61, NO 11, NOVEMBER 16 Because the sequence b γ w 1,w,dw 1,w N bounded, we have is we get, for all s in,dw 1,w ], lim a γ w 1,w,dw 1,w = 71 The sequence w 1,w N has a cluster point w 1ω,w ω To eep the notations simple, we still denote by, the index of the subsequence for which we have convergence to this point Case 1 w 1ω = w ω = w ω : Assume we have lim dw 1,w = lim dw 1,w ω = lim dw,w ω = By compactness, C 1 property, and boundedness, there exists a real number M and an integer such that, for all larger than,wehave a γ w 1,w,dw 1,w a γ w 1,w, Mdw 1,w x w 1 dγ ds Mdw 1,w b γ w 1,w,dw 1,w b γ w 1,w, + Mdw 1,w νw 1 x w 1 dγ ds + Mdw 1,w This implies λ + νw1 and, therefore, ] x w 1 dγ ds ] 1 M +1 lim x w 1 dγ ds dw 1,w = 7 Also, by compactness, w w 1 /dw 1,w has a cluster point we denote as v ω With, again, as index for the subsequence of the subsequence!, we have v ω = lim w w 1 /dw 1,w But, with 69, this gives also v ω = lim dγ /ds which, with 7, gives / xw ω v ω = and implies lim b γ w 1,w, dw 1,w = Sincea γ w 1,w,dw 1,w is nonnegative, this contradicts 7 Case w 1ω w ω : Assume now we have lim dw 1,w 1ω = lim dw,w ω = dw 1ω,w ω w 1,w is in the compact set C i1 i and γ is a minimal geodesic at least on,dw 1p,w p ] So, from: p w1 w dw 1,w p w 1 w w 1,w C C p γ s w 1 d γ s,w 1 dw 1,w D i1 i where D i1 i =sup w1,w C i1 i dw 1,w Wehavealso w 1 w w 1 + w dw,w 1 +dw, p D i 1 i +i +1 p With the completeness of the metric, this implies that γ :,D i1 i ] R n taes its values in a compact set independent of the index and is a solution of the geodesic equation It follows, for instance, from 1, Theorem 5, Section 1], that there exist a subsequence of the subsequence! again with as index and a solution γω of the geodesic equation satisfying, γ ω = w 1ω, γωdw 1ω,w ω = w ω and lim γ s =γ ωs uniformly in s,d i1 i ] Also, each γ being minimizing between w 1 and w, γω is minimizing between w 1ω and w ω see 6, Lemma II 4] With the definitions 66 and 67 of a γ w 1,w,dw 1,w and b γ w 1,w,dw 1,w and with 71 we obtain a γ ω w 1ω,w ω,dw 1ω,w ω = lim a γ w 1,w,dw 1,w = b γ ω w 1ω,w ω,dw 1ω,w ω = lim b γ w 1,w,dw 1,w With Lemma 1, we get b γ ω w 1ω,w ω,dw 1ω,w ω = So, as in the previous case, we get a contradiction of 7 To complete the proof of Proposition 7, it is sufficient to observe that for any pair w 1,w in R n, we can find i 1,i such that C i1 i contains it It is then sufficient to pic lw 1,w larger or equal to l i1 i to obtain, with 68, D + dw 1,w d s L Γs, t dt λ dw 1,w t= This gives: p W1 w 1,w,t W w 1,w,t d W 1 w 1,w,t W w 1,w,t exp λ t dw 1,w exp λ p w1 t w ACKNOWLEDGMENT The authors would lie to than the anonymous reviewers for their helpful comments

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Japan, 1966, vol 9 36] T Yoshizawa, Extreme stability and almost periodic solutions of functional-differential equations, Archive for Rational Mech Anal, vol 17, no, pp , 1964 Vincent Andrieu graduated in applied mathematics from INSA de Rouen, France, in 1 After woring for ONERA a French aerospace research company, he obtained a PhD degree from Ecole des Mines de Paris in 5 In 6, he had a research appointment at the Control and Power Group, Dept EEE, Imperial College London In 8, he joined the CNRS-LAAS lab in Toulouse, France, as a CNRS-charg de recherche Since 1, he has been woring in LAGEP-CNRS, Universit de Lyon 1, France In 14, he joined the functional analysis group from Bergische Universitt Wuppertal in Germany, for two years His main research interests are in the feedbac stabilization of controlled dynamical nonlinear systems and state estimation problems He is also interested in practical application of these theoretical problems, and especially in the field of aeronautics and chemical engineering Bayu Jayawardhana SM 13 received the BSc degree in electrical and electronics engineering from the Institut Tenologi Bandung, Bandung, Indonesia, in, the MEng degree in electrical and electronics engineering from the Nanyang Technological University, Singapore, in 3, and the PhD degree in electrical and electronics engineering from Imperial College London, London, UK, in 6 Currently, he is an associate professor in the Faculty of Mathematics and Natural Sciences, University of Groningen, Groningen, The Netherlands He was with Bath University, Bath, UK, and with Manchester Interdisciplinary Biocentre, University of Manchester, Manchester, UK His research interests are on the analysis of nonlinear systems, systems with hysteresis, mechatronics, systems and synthetic biology He is a Subject Editor of the International Journal of Robust and Nonlinear Control, an associate editor of the European Journal of Control, and a member of the Conference Editorial Board of the IEEE Control Systems Society Laurent Praly graduated as an engineer from École Nationale Supèrieure des Mines de Paris in 1976 and got his PhD in Automatic Control and Mathematics in 1988 from Université Paris IX Dauphine After woring in industry for three years, in 198 he joined the Centre Automatique et Systèmes at École des Mines de Paris From July 1984 to June 1985, he spent a sabbatical year as a visiting assistant professor in the Department of Electrical and Computer Engineering at the University of Illinois at Urbana- Champaign Since 1985 he has continued at the Centre Automatique et Systèmes where he served as director for two years He has made several long term visits to various institutions Institute for Mathematics and its Applications at the University of Minnesota, University of Sydney, University of Melbourne, Institut Mittag-Leffler, and University of Bologna His main interest is in feedbac stabilization of controlled dynamical systems under various aspects linear and nonlinear, dynamic, output, under constraints, with parametric or dynamic uncertainty, disturbance attenuation or rejection On these topics he is contributing both on the theoretical aspect with many academic publications and the practical aspect with applications in power systems, mechanical systems, aerodynamical and space vehicles