Differentiable Families of Planar Bimodal Linear Control Systems

We consider bimodal linear control systems consisting of two subsystems acting on each side of a given hyperplane, assuming continuity along it. For a differentiable family of planar bimodal linear control systems, we obtain its stratification diagram and, if controllability holds for each value of the parameters, we construct a differentiable family of feedbacks which stabilizes both subsystems for each value of the parameters. 1. Introduction Piecewise linear control systems (in particular, the bimodal ones: see, for example, [1–3]) have attracted the interest of the researchers in recent years, as a special class of switched systems (see, e.g., [4–6]), by their wide range of applications, as well as by the possible theoretical approaches, even in the planar case (see, e.g., [7]). Bimodal linear control systems (BLCS) consist of two subsystems acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. These systems present a complex dynamical behaviour, even for low dimensions, as has been shown in several works. For example, in [8], it is proved that a planar bimodal linear system is stable if each subsystem is stable, but this does not hold for a bimodal linear system with three-state variables. On the other hand, since typically the number of state variables of systems describing elementary circuits is two or three (see [9]), we devote a special attention to the planar case. Here we tackle two problems concerning parameterized families of planar BLCS. Firstly, obtaining of its stratification diagram with regard to the natural equivalence relation is defined by change of basis in the state space (which preserve the hyperplanes parallel to the separating one). Previously, it is necessary to list the possible equivalence classes and to obtain a complete set of classifying invariant parameters (Theorem 6). By the way, Arnold's theory allows us to restrict this study to the so-called “miniversal” deformation families. Indeed, the equivalence classes are just the orbits of a certain group action, so that they are differentiable manifolds and Arnold's machinery is applicable. Moreover we remark that by joining the orbits according to the discrete classifying invariants one obtains differentiable “strata” (each one formed by the union of classes differing only on continuous classifying invariants). We list the dimension of each orbit and the corresponding strata (Proposition 7). As an application of the previous results, we present the unobservable bifurcation diagram of a miniversal deformation (Example 10). Secondly