# Some properties of reachable sets for control affine systems

Some properties of reachable sets for control affine systems Let $${\dot{q}=X_{0}+\sum_{j=1}^{k}u_{j}X_{j}}$$ be a control affine system on a manifold M, let C be a convex compact subset of $${\mathbb{R}^{k}}$$ , dim C > 0, let q 0 be a fixed point of M, and let U be a neighbourhood of q 0. We consider three reachable sets from q 0 for our system which are generated by square integrable controls with values in C, riC—the relative interior of C, and rbC—the relative boundary of C, respectively, with contraints on a state variable q of the form $${q\in U}$$ . Among other things, we investigate the relation between closures, interiors and boundaries of the three reachable sets. We also show how methods of the sub-Lorentzian geometry can serve as an auxiliary tool in the study of control affine systems. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Some properties of reachable sets for control affine systems

, Volume 1 (1) – Dec 21, 2010
11 pages

Loading next page...

/lp/springer-journals/some-properties-of-reachable-sets-for-control-affine-systems-YfPa3Fe2Bu
Publisher
Springer Journals
Copyright
Copyright © 2010 by The Author(s)
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-010-0001-y
Publisher site
See Article on Publisher Site

### Abstract

Let $${\dot{q}=X_{0}+\sum_{j=1}^{k}u_{j}X_{j}}$$ be a control affine system on a manifold M, let C be a convex compact subset of $${\mathbb{R}^{k}}$$ , dim C > 0, let q 0 be a fixed point of M, and let U be a neighbourhood of q 0. We consider three reachable sets from q 0 for our system which are generated by square integrable controls with values in C, riC—the relative interior of C, and rbC—the relative boundary of C, respectively, with contraints on a state variable q of the form $${q\in U}$$ . Among other things, we investigate the relation between closures, interiors and boundaries of the three reachable sets. We also show how methods of the sub-Lorentzian geometry can serve as an auxiliary tool in the study of control affine systems.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Dec 21, 2010