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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.
Note: This paper first appeared in IEEE Trans. Automat. Control, vol. 54, no. 4, pp. 900-905, Apr. 2009. Due to a production error by the publisher, the figure that appeared was not correct. The paper has been reprinted in full with a corrected figure.