Further Remarks on Strict Input-to-State Stable Lyapunov Functions for Time-Varying Systems

We study the stability properties of a class of time-varying nonlinear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given Lyapunov functions which guarantee the existence of strict ISS Lyapunov functions for our systems. Next, we provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body.Comment: 6 pages, submitted for publication in June 200

For differential equations with r parameters, 2r+1 experiments are enough for identification

Given a set of differential equations whose description involves unknown parameters, such as reaction constants in chemical kinetics, and supposing that one may at any time measure the values of some of the variables and possibly apply external inputs to help excite the system, how many experiments are sufficient in order to obtain all the information that is potentially available about the parameters? This paper shows that the best possible answer (assuming exact measurements) is 2r+1 experiments, where r is the number of parameters.Comment: This is a minor revision of the previously submitted report; a couple of typos have been fixed, and some comments and two new references have been added. Please see http://www.math.rutgers.edu/~sontag for related wor

Asymptotic amplitudes and cauchy gains: A small-gain principle and an application to inhibitory biological feedback

The notions of asymptotic amplitude for signals, and Cauchy gain for input/output systems, and an associated small-gain principle, are introduced. These concepts allow the consideration of systems with multiple, and possibly feedback-dependent, steady states. A Lyapunov-like characterization allows the computation of gains for state-space systems, and the formulation of sufficient conditions insuring the lack of oscillations and chaotic behaviors in a wide variety of cascades and feedback loops. An application in biology (MAPK signaling) is worked out in detail.Comment: Updates and replaces math.OC/0112021 See http://www.math.rutgers.edu/~sontag/ for related wor