Ju n 20 14 On Three Generalizations of Contraction

We introduce three forms of generalized contraction (GC). Roughly speaking, these are motivated by allowin g contraction to take place after small transients in time and/or amplitude. Indeed, contraction is usually used to prove asy mptotic properties, like convergence to an attractor or entrainment to a periodic excitation, and allowing initial transients does not affect this asymptotic behavior. We provide sufficient conditions for GC, and demonstrate their usefulness using examples of systems that are not contractive, with respect to any norm, yet are GC. I. I NTRODUCTION A dynamical system is called contractive if any two trajectories converge to one other at an exponential rate. T his implies many desirable properties including convergence to an attractor (if it exists), and entrainment to periodic excitations [1], [2]. Contraction theory is a powerful tool for analyzing nonlinear dynamical systems, with applications in control the ory [3], observer design [4], synchronization of coupled oscillators [5], and more. Recent extensions include: the notion of partial contraction [6], analyzing a network of interacting contractive elements [7], a Lyapunov-like cha racterization of incremental stability [8], and a LaSalle-t ype principle for contractive systems [9]. A contractive syste m with added diffusion terms or random noise still satisfies certain asymptotic properties [10], [11]. In this respect, contraction is a robust property. In this paper, we introduce three forms of generalized contraction (GC). These are motivated by requiring contractio n to take place only after arbitrarily small transients in tim e and/or amplitude. Indeed, contraction is usually used to pr ove asymptoticproperties, and thus allowing (arbitrarily small) transients seems reasonable. We demonstrate the usefulnes s of these generalizations by showing several examples of systems that arenot contractive with respect to any norm, yet are a GC. The remainder of this paper is organized as follows. Section II provides a brief overview of contraction theory. Section III describes our main results. The proofs of these results are detailed in Section VI. Section IV demonstrates the results using a simple model of a biochemical control system. EDS (sontag@math.rutgers.edu) is with the Dept. of Mathema tics and the Cancer Institute of New Jersey, Rutgers University, Pis cataway, NJ 08854, USA; MM (michaelm@eng.tau.ac.il) is with the School f Electrical Engineering-Systems and the Sagol School of Neuroscience, Tel Aviv University, Israel 69978; TT (tamirtul@post.tau.ac.il) i s with the Dept. of Biomedical Engineering and the Sagol School of Neuroscienc e, Tel-Aviv University, Tel-Aviv 69978, Israel. EDS’s work is supporte d in part by grants NIH 1R01GM086881 and 1R01GM100473, and AFOSR FA9550 11-1-0247. The research of MM is partly supported by the ISF. II. CONTRACTION THEORY We begin with a brief review of some ideas from contraction theory. For more details, including the historic development of contraction theory, and the relation to othe r notions, see e.g. [12], [13], [14]. Consider the time-varying system ẋ = f(t, x), (1) evolving on a convex setΩ ⊂ R. We assume that f(t, x) is differentiable with respect tox, and that bothf(t, x) andJ(t, x) := ∂f ∂x (t, x) are continuous in(t, x). Let x(t, t0, x0) denote the solution of (1) at time t ≥ t0 with x(t0) = x0 (for the sake of simplicity, we assume from here on that x(t, t0, x0) exists and is unique for all t ≥ t0 ≥ 0 and allx0 ∈ Ω). Recall that (1) is calledcontractive[1] on Ω with respect to a norm| · | : R → R+ if there existsc > 0 such that |x(t2, t1, a)− x(t2, t1, b)| ≤ exp(−(t2 − t1)c)|a− b| (2) for all t2 ≥ t1 ≥ 0 and alla, b ∈ Ω. In other words, any two trajectories contract to one another at an exponential rate. This implies in particular that the initial condition is “forgotten”. Recall that a vector norm| · | : R → R+ induces a matrix measureμ : Rn×n → R defined byμ(A) := limǫ↓0 1ǫ (||I + ǫA|| − 1), where || · || : Rn×n → R+ is the matrix norm induced by| · |. A standard approach for proving contraction is based on bounding some matrix measure of the Jacobian J . (This is in fact a particular case of using a Lyapunov-Finsle r function to prove contraction [9]). Theorem 1 [2] If there exists a vector norm| · | and c > 0 such that the induced matrix measure μ : Rn×n → R satisfies μ(J(t, x)) ≤ −c, (3) for all t2 ≥ t1 ≥ 0 and all x ∈ Ω then (2) holds. One important implication of contraction is entrainment to a periodic excitation. Recall that f : R+ × Ω → R is calledT -periodic if f(t, x) = f(t+ T, x) for all t ≥ 0 and allx ∈ Ω. Theorem 2 [2] If (1) is contractive andf is T -periodic then there exists a unique periodic solution α : [0,∞] → Ω of (1), of periodT , and lim t→∞ |x(t, 0, a)− α(t)| = 0, for all a ∈ Ω. In other words, every trajectory x(t, 0, a) converges to the unique periodic solution. Entrainment is important in vari ous applications ranging from biological systems [15], [2] to t he stability of the power grid [16]. The next section presents our main results. All the proofs are placed in Section VI.