Connecting nonlinear incremental Lyapunov stability with the linearizations Lyapunov stability Vincent Fromion and G´ erard Scorletti Abstract—In this paper, we reveal new connections between the incremental Lyapunov properties of a nonlinear system and the Lyapunov properties of its linearizations. We focus on (incremental) asymptotic and exponential stability. In contrast with other works on the incremental Lyapunov properties of nonlinear systems, our approach is based on extended spaces, Gˆ ateaux derivative and the mean value theorem in norm. I. I NTRODUCTION Two different approaches, based on incremental stability, have recently emerged for the nonlinear system analysis. The first one, the incremental Lyapunov stability approach focuses on the analysis of the nonlinear system trajectories associated to a given set of initial conditions, that is, quali- tative properties. The second one, the (weighted) incremen- tal norm approach, focuses on the input-output properties of Lipschitz continuous nonlinear systems. As in the first approach, the second approach allows qualitative property analysis. In contrast with, it allows to analyze desensitivity, robustness and more generally many quantitative properties. Note that nonlinear control specifications include both qual- itative and quantitative properties. It emphasizes the strong advantage of the weighted incremental norm approach, which encompasses in a single mathematical framework both kind of properties (see [18, 15] for illustrative examples). In this paper, we first prove that the incremental Lyapunov asymptotic stability of a nonlinear system for a convex set U of initial conditions is implied by the Lyapunov asymptotic stability of all its linearizations associated to each trajectory with an initial condition in U . A stronger result is then obtained when Lyapunov exponential stability is considered. The incremental Lyapunov exponential stability of a nonlinear system for a convex set U of initial conditions is proved to be equivalent to the Lyapunov exponential stability of all its linearizations associated to each trajectory with an initial condition in U . We then prove that the incremental Lyapunov exponential stability on a convex set U is equivalent to the Lyapunov exponential stability on U at any initial condition in U . In other words, the exponential stability of the trajectories associated to a convex set U of initial conditions is equivalent to the exponential stability of the equilibrium trajectories with respect to U . This result is close to a recent one in [2] derived with a different proof. Our paper is a contribution in the scope of the incremental Lyapunov stability approach. In this approach, many contri- V. Fromion is with MIG-INRA, Domaine de Vilvert, 78350 Jouy-en- Josas, France, e-mail: vincent.fromion@jouy.inra.fr. G. Scorletti is with GREYC Equipe Auto, 6 Bd. du Marchal Juin, 14050 Caen cedex, France, e-mail: scorletti@greyc.ensicaen.fr. butions focus on the analysis of trajectories associated to different systems or associated to the same system but with two different initial conditions. Two arguments are usually applied. The first one is based on “incremental” Lyapunov functions, see e.g. [36, 3, 30]. The second one focuses on the analysis of the (time dependent) distance between two trajectories (see [24] for a survey and also [23]). The obtained conditions involve the nonlinear system linearizations (see [29, 25, 21] and more recently the contraction analysis [26, 27]). Note that related problems were considered for the error analysis of numerical integration schemes (see [4, 5] and for related problems [7]). In contrast with these results, our proof is based on the machinery of (weighted) incremental norm approach. The weighted incremental approach focuses on the properties of incrementally stable systems, i.e. Lipschitz continuous sys- tems, defined as causal operators from a normed functional space to another one. This approach clearly roots in the input- output approach, more precisely in the Zames’ and Sand- berg’s pioneering works. The (Lipschitz) continuity was early pointed out as a natural extension to the nonlinear systems of the linear bounded input bounded output stability (see [37] and also [35]). Nevertheless, in the input-output approach of nonlinear system analysis, most of the results focus on ensuring only the input-output boundedness e.g., the L 2 gain stability, see [6]. Few works investigated the properties of incrementally stable systems, see the book [35], chapter 7 on linearizations. Some results were obtained in the sixties for a restricted class of nonlinear systems (interconnections between an LTI system and a memoryless nonlinearity), see [32, 33]. Recently, the (weighted) incremental norm approach emerged as a fruitful extension of the linear H ∞ approach to nonlinear systems, see [9, 12] and [19]. The major interest is to propose a quantitative evaluation of the robustness and the performance of closed-loop nonlinear systems. This evaluation reduces to the computation of the weighted incremental norm of some closed-loop nonlinear functions [12]. In addition, in [10, 11, 8, 13], it was also proved that the (weighted) incremental norm approach en- sures many interesting incremental Lyapunov properties as the uniqueness of the steady-state, the uniform Lyapunov stability of all the unperturbed trajectories and many other ones, see [2] in the ISS context. Based on this approach, in [16, 14], we reveal the connection between the incremental norm of a nonlinear system with the induced norm of its linearizations with an emphasis on the input-output properties. In this paper, we complete the picture by focusing on the Lyapunov properties. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005 WeA17.4 0-7803-9568-9/05/$20.00 ©2005 IEEE 4736