Physica D 189 (2004) 234–246 Controllability for a class of area-preserving twist maps Umesh Vaidya a,∗ , Igor Mezi´ c a,b a Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 93105-5070, USA b Department of Mathematics, University of California, Santa Barbara, CA 93105-5070, USA Received 21 November 2002; received in revised form 10 October 2003; accepted 10 October 2003 Communicated by C.K.R.T. Jones Abstract In this paper, we study controllability of two-dimensional integrable twist maps with bounded area-preserving time-dependent (control) perturbations. In contrast to the time-independent perturbation case of the Kolmogorov–Arnold–Moser theorem, there are no invariant sets other than the whole phase space if the perturbation is made a function of time. We give necessary and sufficient conditions for global controllability of these maps. © 2003 Elsevier B.V. All rights reserved. PACS: 07.05.Dz; 47.20.Ky; 47.52.+j Keywords: Control; Hamiltonian system; Twist map; KAM 1. Introduction Control of Hamiltonian systems is a topic that has received a lot of attention lately [1]. Besides the intrinsic beauty of the subject, this is due to a number of exciting applications such as satellite control [2], quantum control [3,4], and control of mixing [5,6]. In this paper, we combine the control-theoretical and dynamical systems point of view to study a class of sys- tems that are well understood from the dynamical systems perspective: perturbations of integrable planar twist maps [7]. These two-dimensional maps defined on an annulus can arise from discretization of continuous-time integrable Hamiltonian systems. Integrable twist maps on an annulus have very simple dynamics given by (x,y) → (x + G(y),y), with G ′ (y) > 0, where x and y are the usual Cartesian coordinates on the plane and x is con- sidered mod 1. Thus, all the initial conditions stay at the same y for all time and y = constant is an invariant manifold for the dynamics. The Kolmogorov–Arnold–Moser (KAM) theorem [8] (in Moser’s version [7]) con- siders a time-independent perturbation of an integrable twist map. Under the condition that the perturbed map is area-preserving (in fact that it possesses the so-called intersection property, that is implied by area-preservation), KAM theorem states that the majority of initial conditions stay on one-dimensional invariant curves close to the unperturbed invariant curves on which G(y) satisfies the Diophantine condition (strong irrationality). It is commonly ∗ Corresponding author. Tel.: +1-8058934711; fax: +1-8058938651. E-mail addresses: ugvaidya@engineering.ucsb.edu (U. Vaidya), mezic@engineering.ucsb.edu (I. Mezi´ c). 0167-2789/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2003.10.008