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.
Analysis and Mathematical Physics – Springer Journals
Published: Dec 21, 2010
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