Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 342548, 14 pages http://dx.doi.org/10.1155/2013/342548 Research Article Controllability of Continuous Bimodal Linear Systems Josep Ferrer, Juan R. Pacha, and Marta Peña Deptartament de Matem´ atica Aplicada I, Universitat Polit` ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain Correspondence should be addressed to Juan R. Pacha; juan.ramon.pacha@upc.edu Received 25 January 2013; Revised 12 April 2013; Accepted 17 April 2013 Academic Editor: Rongni Yang Copyright © 2013 Josep Ferrer et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced to the unobservable case, and for these ones we obtain a simple explicit characterization of controllability for dimensions 2 and 3, as well as some partial criteria for higher dimensions. 1. Introduction A switched linear system (see, e.g., [1]) is a hybrid system (see, e.g., [2]) which consists of several linear subsystems and a rule that orchestrates the switching among them. Te classical piecewise linear systems (see [3]) occur when the switching law depends only on the state. In recent years, piecewise linear dynamical/control systems have again attracted the attention of the researchers, indeed, because they are the most natural extension to linear systems in order to capture nonlinear phenomena as, for instance, limit cycles, heteroclinic and homoclinic orbits, and strange attractors. Also, their application has been especially successful in many engineering problems, such as the analysis and design of electronic oscillators (see [4]) or control systems. Tey consist of two or more linear subsystems, each one acting in a diferent region—separated from the others by hypersurfaces—of the entire state space. In the basic case of bimodal systems, a couple of linear subsystems act at each side of a hyperplane. One says that the piecewise linear system is continuous if both adjacent subsystems coincide on the separating hypersurface. For elementary circuits (see [4]), the number of state variables is typically two or three. Moreover, most of the nonlinear behavior appears already in low dimensions, even in the bimodal case. For example, the study in [5] gives a complete characterization of the focus-center limit cycle bifurcation for planar bimodal linear systems. Later works (see [6–8]) extended this analysis to the 3D case. Contrarily to what happens in 2D, in 3D, a remarkable phenomenon occurs, involving bimodal linear systems. Tus, whereas in 2D, a point placed on the separating hyperplane is asymptot- ically stable when both subsystems are stable; in 3D, unstable global dynamics might arise under the same hypotheses. Concerning structural stability (i.e., with regard to small perturbations), we point the reader to [9] for an account of generic properties and to [10] for a full characterization (both studies on the 2D case). About control systems see, for example, [11] (again for the planar case). To investigate continuous bimodal systems, several authors have used reduced forms of the matrices involved, both for dynamical systems (e.g., in [4]) and control systems (e.g., in [12, 13]). Here, we will use the reduced forms in [14, 15]. For a MAPLE program useful to compute them in 2D and 3D, see http://www.ma1.upc.edu/joanr/html/cfpwls.html. Te goal of this paper is to fnd explicit and efective criteria for controllability of continuous bimodal linear con- trol systems (for a study of the discrete-time planar case, see [16], where explicit necessary and sufcient conditions for controllability and reachability are given). Other problems concerning discrete-time systems are considered in [17], where a controller is designed. In other cases as in [18] or in [19], the system is continuous and the controller is digital. A state feedback controller for piecewise linear systems is designed in [20]. Problems as stability and stabilization of switched systems are considered in [18, 21–23] or in [24]. In [25], the problem of controllability is addressed for linear time-invariant dynamical multiagent systems. Fur- ther contributions include switched systems with stochastic perturbations (see [26] and the references therein). In our