18,016 research outputs found
Stabilizing Randomly Switched Systems
This article is concerned with stability analysis and stabilization of
randomly switched systems under a class of switching signals. The switching
signal is modeled as a jump stochastic (not necessarily Markovian) process
independent of the system state; it selects, at each instant of time, the
active subsystem from a family of systems. Sufficient conditions for stochastic
stability (almost sure, in the mean, and in probability) of the switched system
are established when the subsystems do not possess control inputs, and not
every subsystem is required to be stable. These conditions are employed to
design stabilizing feedback controllers when the subsystems are affine in
control. The analysis is carried out with the aid of multiple Lyapunov-like
functions, and the analysis results together with universal formulae for
feedback stabilization of nonlinear systems constitute our primary tools for
control designComment: 22 pages. Submitte
On the monopole Lefschetz number of finite order diffeomorphisms
Let be a knot in an integral homology 3-sphere , and the
corresponding -fold cyclic branched cover. Assuming that is a
rational homology sphere (which is always the case when is a prime power),
we give a formula for the Lefschetz number of the action that the covering
translation induces on the reduced monopole homology of . The proof
relies on a careful analysis of the Seiberg--Witten equations on 3-orbifolds
and of various -invariants. We give several applications of our formula:
(1) we calculate the Seiberg--Witten and Furuta--Ohta invariants for the
mapping tori of all semi-free actions of on integral homology 3-spheres;
(2) we give a novel obstruction (in terms of the Jones polynomial) for the
branched cover of a knot in being an -space; (3) we give a new set of
knot concordance invariants in terms of the monopole Lefschetz numbers of
covering translations on the branched covers.Comment: 39 page, 2 figures. Added a reference to Langte Ma's paper
arXiv:1909.01533, which contains an independent proof of our Theorem B. Final
version, to appear in Geometry and Topolog
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