A note on norm-based Lyapunov functions via contraction analysis

It is well know that for globally contractive autonomous systems, there exists a unique equilibrium and the distance to the equilibrium evaluated along any trajectory decreases exponentially with time. We show that, additionally, the magnitude of the velocity evaluated along any trajectory decreases exponentially, thus giving an alternative choice of Lyapunov function. 1 Main Result Consider the nonlinear system ẋ = f(x) (1) for x ∈ R and continuously differentiable f(·). Denote the Jacobian as J(x) , ∂f ∂x (x). (2) Let | · | be a vector norm on R, ‖ · ‖ its induced matrix norm, and μ(A) , limh→0+ 1 h (‖I + hA‖ − 1) the associated matrix measure. Theorem 1. If there exists c > 0 such that μ(J(x)) ≤ −c for all x ∈ R, then |f(x(t))| ≤ |f(x(0))|e. (3) Proof. Let V (x) , |f(x)|. V (x(t)) is then absolutely continuous as a function of t and therefore V̇ (x(t)) , lim h→0+ V (x(t+ h))− V (x(t)) h (4) The authors are with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA. {scoogan,arcak}@eecs.berkeley.edu