Some properties of reachable sets for control affine systems

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 q0 be a fixed point of M, and let U be a neighbourhood of q0. We consider three reachable sets from q0 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.