Maria Patrizia Pera

Topological methods for the global controllability of nonlinear systems

Sufficient conditions for the local and global controllability of general nonlinear systems, by means of controls belonging to a fixed finite-dimensional subspace of the space of all admissible controls, are established with the aid of topological methods, such as homotopy invariance principles. Some applications to certain classes of nonlinear control processes are given, and various known results on the controllability of perturbed linear systems are also derived as particular cases.

On the uniqueness of the fixed point index on differentiable manifolds

It is well known that some of the properties enjoyed by the fixed point index can be chosen as axioms, the choice depending on the class of maps and spaces considered. In the context of finite-dimensional real differentiable manifolds, we will provide a simple proof that the fixed point index is uniquely determined by the properties of normalization, additivity, and homotopy invariance.

Global bifurcation of fixed points and the Poincaré translation operator on manifolds

Let M be a differentiable manifold and ϕ∶ [0, ∞)×M→M be a C1 map satisfying the condition ϕ(0, p)=p for all p∈M. Among other results, we prove that when the degree (also called Hopf index or Euler characteristic) of the tangent vector field w∶M→TM, given by w(p)=(∂ϕ/∂λ)(0, p), is well defined and nonzero, then the set (of nontrivial pairs) admits a connected subset whose closure is not compact and meets the slice {0}×M of [0, ∞)×M. This extends known results regarding the existence of harmonic solutions of periodic ordinary differential equations on manifolds.

Delay Differential Equations on Manifolds and Applications to Motion Problems for Forced Constrained Systems

We prove a global bifurcation result for T -periodic solutions of the delay T -periodic differential equation x′(t) = λf(t, x(t), x(t − 1)) on a smooth manifold (λ is a nonnegative parameter). The approach is based on the asymptotic fixed point index theory for C1 maps due to Eells–Fournier and Nussbaum. As an application, we prove the existence of forced oscillations of motion problems on topologically nontrivial compact constraints. The result is obtained under the assumption that the frictional coefficient is nonzero, and we conjecture that it is still true in the frictionless case.

Global continuation of periodic solutions for retarded functional differential equations on manifolds

We consider T-periodic parametrized retarded functional differential equations, with infinite delay, on (possibly) noncompact manifolds. Using a topological approach, based on the notions of degree of a tangent vector field and of the fixed point index, we prove a global continuation result for T-periodic solutions of such equations.Our main theorem is a generalization to the case of retarded equations of a global continuation result obtained by the last two authors for ordinary differential equations on manifolds. As corollaries we obtain a Rabinowitz-type global bifurcation result and a continuation principle of Mawhin type.MSC:34K13, 34C40, 37C25, 70K42.

A Continuation Result for Forced Oscillations of Constrained Motion Problems with Infinite Delay

Abstract We prove a global continuation result for T-periodic solutions of a T-periodic parametrized second order retarded functional differential equation on a boundaryless compact manifold with nonzero Euler-Poincaré characteristic. The approach is based on the fixed point index theory for locally compact maps on ANRs. As an application, we prove the existence of forced oscillations of retarded functional motion equations defined on topologically nontrivial compact constraints. This existence result is obtained under the assumption that the frictional coefficient is nonzero, and we conjecture that it is still true in the frictionless case.

The Fixed Point Index of the Poincaré Translation Operator on Differentiable Manifolds

The fixed point index of the Poincare translation operator associated to an ordinary differential equation is a very useful tool for establishing the existence of periodic solutions. In this chapter we focus on ODEs on differentiable manifolds embedded in Euclidean spaces. Our purpose is twofold: on the one hand we aim to provide a short and accessible introduction to some topological tools (such as the Topological Degree, the Degree of a tangent vector field and the Fixed Point Index) that are useful in Nonlinear Analysis; on the other hand we offer a unifying approach to several results about the fixed point index of the Poincare translation operator that were previously scattered among a number of publications. Our main concern will be a formula for the computation of the fixed point index of the flow operator induced on a manifold by a first order autonomous ordinary differential equation. Other formulas for the fixed point index of the translation operator associated with non-autonomous equations will be deduced as consequences. We emphasize that other results, unrelated to our approach, but still involving the fixed point index of the Poincare translation operator have been successfully exploited, for instance, by Srzednicki (see e.g. [Srz2, Srz3]).

NORMALIZED EIGENVECTORS OF A PERTURBED LINEAR OPERATOR VIA GENERAL BIFURCATION

Let X be a real Banach space, A : X → X a bounded linear operator, and B : X → X a (possibly nonlinear) continuous operator. Assume that λ = 0 is an eigenvalue of A and consider the family of perturbed operators A + eB, where e is a real parameter. Denote by S the unit sphere of X and let SA = S ∩ KerA be the set of unit 0-eigenvectors of A. We say that a vector x0 ∈ SA is a bifurcation point for the unit eigenvectors of A+ eB if any neighborhood of (0, 0, x0) ∈ R × R × X contains a triple (e, λ, x) with e 6= 0 and x a unit λ-eigenvector of A+ eB, i.e. x ∈ S and (A+ eB)x = λx. We give necessary as well as sufficient conditions for a unit 0-eigenvector of A to be a bifurcation point for the unit eigenvectors of A + eB. These conditions turn out to be particularly meaningful when the perturbing operator B is linear. Moreover, since our sufficient condition is trivially satisfied when KerA is one-dimensional, we extend a result of the first author, under the additional assumption that B is of class C2.

A Set of Axioms for the Degree of a Tangent Vector Field on Differentiable Manifolds

Given a tangent vector field on a finite-dimensional real smooth manifold, its degree (also known as characteristic or rotation) is, in some sense, an algebraic count of its zeros and gives useful information for its associated ordinary differential equation. When, in particular, the ambient manifold is an open subset of , a tangent vector field on can be identified with a map , and its degree, when defined, coincides with the Brouwer degree with respect to zero of the corresponding map . As is well known, the Brouwer degree in is uniquely determined by three axioms called Normalization, Additivity, and Homotopy Invariance. Here we shall provide a simple proof that in the context of differentiable manifolds the degree of a tangent vector field is uniquely determined by suitably adapted versions of the above three axioms.

Topological persistence of the normalized eigenvectors of a perturbed self-adjoint operator

Abstract Let T be a self-adjoint bounded operator acting in a real Hilbert space H , and denote by S the unit sphere of H . Assume that λ 0 is an isolated eigenvalue of T of odd multiplicity greater than 1 . Given an arbitrary operator B : H → H of class C 1 , we prove that for any e ≠ 0 sufficiently small there exists x e ∈ S and λ e near λ 0 , such that T x e + e B ( x e ) = λ e x e . This result was conjectured, but not proved, in a previous article by the authors. We provide an example showing that the assumption that the multiplicity of λ 0 is odd cannot be removed.