Gunther Reißig.
Convexity of reachable sets of nonlinear ordinary differential
equations.
Automat. Remote Control, vol. 68, no. 9, Sep. 2007, pp. 1527-1543.
Russian translation published in Avtomatika i Telemekhanika, 2007, No. 9,
pp. 64-78.
Full text.
(Definitive publication; restricted access.)
Full text.
(Accepted version; free access.)
Abstract:
We present a necessary and sufficient condition for the reachable set,
i.e., the set of states reachable from a ball of initial states at
some time, of an ordinary differential equation to be convex.
In particular, convexity is guaranteed if the ball of initial states
is sufficiently small, and we provide an upper bound on the radius of
that ball, which can be directly obtained from the right hand side of
the differential equation. In finite dimensions, our results cover the
case of ellipsoids of initial states.
A potential application of our results is inner and outer polyhedral
approximation of reachable sets, which becomes extremely simple
and almost universally applicable if these sets are known to be
convex. We demonstrate by means of an example that the balls of
initial states for which the latter property follows from our results
are large enough to be used in actual computations.
BibTeX entry:
@article{Reissig07a,
author = {Rei{\ss}ig, Gunther},
title = {Convexity of reachable sets of nonlinear ordinary differential equations},
journal = {Automat. Remote Control},
year = 2007,
volume = 68,
number = 9,
pages = {1527-1543},
month = sep,
doi = {10.1134/S000511790709007X},
eprint = {1211.6080},
note = {Russian transl. in \emph{Avtomat. i Telemekh.}, 2007, no. 9, pp. 64-78.}
}
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