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IMA Journal of Mathematical Control and Information 1997 14(4):353-369; doi:10.1093/imamci/14.4.353
© 1997 by Institute of Mathematics and its Applications
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Closed trajectories and global controllability in the plane

C. YALÇIN KAYA and J. LYLE NOAKES

Department of Mathematics, University of Western Australia Nedlands, W.A., 6907 Australia

In the plane, results are given about the structure of closed trajectories which may occur as simple closed curves or general closed curves with self-intersections. Necessary conditions for the global controllability of nonlinear systems that are in the so-called linear-analytic form = f(x)+ug(x), where x {sum} R2 and ¦u¦ ≤ 1, are given. It is proved that, if there exists a closed trajectory {Gamma} of the system, then {Gamma} either contains a point where f and g are linearly dependent, or encloses some zeros of f + ug for all u {sum} [-1,1]. Then this result is used to prove that, if the linear-analytic system is controllable and the vector field g is never zero in W {subseteq} R2, then W contains some zeros of f + ug for some u {sum} [-1,1]. A topological approach is taken. Remarks are made about the size of the region where a closed trajectory can lie, and about the shape of the closed trajectories. Further implications are discussed.


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