Marta Peña

BIMODAL PIECEWISE LINEAR DYNAMICAL SYSTEMS: REDUCED FORMS

Piecewise linear systems constitute a class of nonlinear systems which have recently attracted the interest of researchers because of their interesting properties and the wide range of applications from which they arise. Different authors have used reduced forms when studying these systems, mostly in the case where they are observable. In this work, we focus on bimodal continuous dynamical systems (those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane) depending on two or three state variables, which are the most common piecewise linear systems found in practice. Reduced forms are obtained for general systems, not necessarily observable. As an application, we calculate the dimension of the equivalence classes.

Dimension of the orbit of marked subspaces

Given a nilpotent endomorphism, we consider the manifold of invariant subspaces having a fixed Segre characteristic. In [Linear Algebra Appl., 332–334 (2001) 569], the implicit form of a miniversal deformation of an invariant subspace with respect to the usual equivalence relation between subspaces is obtained. Here we obtain the explicit form of this deformation when the invariant subspace is marked, and we use it to calculate the dimension of the orbit and in particular to characterize the stable marked subspaces (those with open orbit). Moreover, we study the rank of the endomorphisms in the quotient space by the subspaces in the miniversal deformation of the giving subspace.

Controllability of Continuous Bimodal Linear Systems

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.

The change of feedback invariants under column perturbations: particular cases

We study the variation of the feedback invariants of a complex rectangular n × (n + m) matrix when we make small additive perturbations to the elements of the last m columns. First of all, we obtain necessary conditions for the feedback invariants of all the matrices obtained by means of sufficiently small perturbations. Conversely, we prove that these conditions are also sufficient to find a matrix, as close as desired to the fixed matrix, with prescribed feedback invariants, in some particular cases: when the rectangular matrix is completely controllable, when the rectangular matrix is completely uncontrollable and when m = 1.

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.

Local differentiable pole assignment

Given a general local differentiable family of pairs of matrices, we obtain a local differentiable family of feedbacks solving the pole assignment problem, that is to say, shifting the spectrum into a prefixed one. We point out that no additional hypothesis is needed. In fact, simple approaches work in particular cases (controllable pairs, constancy of the dimension of the controllable subspace, and so on). Here the general case is proved by means of Arnolds techniques: the key point is to reduce the construction to a versal deformation of the central pair; in fact to a quite singular miniversal one for which the family of feedbacks can be explicitly constructed. As a direct application, a differentiable family of stabilizing feedbacks is obtained, provided that the central pair is stabilizable.

Bifurcation Diagram of Saddle/Spiral Bimodal Linear Systems

We complete the study of the bifurcations of saddle/spiral bimodal linear systems, depending on the respective traces T and τ: one 2-codimensional bifurcation; four kinds of 1-codimensional bifurcations. We stratify the bifurcation set in the (T,τ)-plane and we describe the qualitative changes of the dynamical behavior at each bifurcation point.

Structural stability of planar bimodal linear systems

Structural stability ensures that the qualitative behavior of a system is preserved under small perturbations. We study it for planar bimodal linear dynamical systems, that is, systems consisting of two linear dynamics acting on each side of a given hyperplane and assuming continuity along the separating hyperplane. We describe which one of these systems is structurally stable when (real) spiral does not appear and when it does we give necessary and sufficient conditions concerning finite periodic orbits and saddle connections. In particular, we study the finite periodic orbits and the homoclinic orbits in the saddle/spiral case.

Miniversal deformations of observable marked matrices

Given the set of vertical pairs of matrices

Differentiable families of stabilizers for planar bimodal linear control systems

{\cal M}\subset M_{m,n}(\mathbb C)\times M_n(\mathbb C)