Nice reachability for planar bilinear control systems with applications to planar linear switched systems [corrected]

We consider planar bilinear control systems with measurable controls. We show that any point in the reachable set can be reached by a ldquonicerdquo control, specifically, a control that is a concatenation of a bang arc with either 1) a bang-bang control that is periodic after the third switch; or 2) a piecewise constant control with no more than two discontinuities. Under the additional assumption that the bilinear system is positive (or invariant for any proper cone), we show that the reachable set is spanned by a concatenation of a bang arc with either 1) a bang-bang control with no more than two discontinuities; or 2) a piecewise constant control with no more than two discontinuities. In particular, any point in the reachable set can be reached using a piecewise-constant control with no more than three discontinuities. Several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result. We demonstrate this with an example.

I. Introduction

Consider the planar bilinear control system $$\mathdot{\mmb x}(t)=A{\mmb x}(t)+u(t)B{\mmb x}(t),\qquad u\in{\cal U}\eqno{\hbox{(1)}}$$ where , , and is the set of measurable functions taking values in [0,1]. For , and , we use to denote the solution at time of (1) for the initial condition .