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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
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