Marco Spadini

Strong Local Optimality for a Bang-Bang Trajectory in a Mayer Problem

This paper gives sufficient conditions for a class of bang-bang extremals with multiple switches to be locally optimal in the strong topology. The conditions are the natural generalizations of those considered in [A. A. Agrachev, G. Stefani, and P. Zezza, SIAM J. Control Optim., 41 (2002), pp. 991-1014], [L. Poggiolini, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), pp. 1-23], and [L. Poggiolini and G. Stefani, Systems Control Lett., 53 (2004), pp. 269-279]. We require both the strict bang-bang Legendre condition and the second order conditions for the finite dimensional problem obtained by moving the switching times of the reference trajectory.

On the set of harmonic solutions of periodically perturbed autonomous differential equations on manifolds

where g : M → R and f : R×M → R are continuous vector fields, tangent to a (not necessarily closed) boundaryless differentiable manifold M ⊂ R, with f T -periodic with respect to the first variable. We investigate the structure of the set of solution pairs of (1); i.e. of those pairs (λ, x) ∈ [0,∞)× CT (M) such that x is a (necessarily T -periodic) solution of (1). We give conditions ensuring the existence of a non-compact connected component of solution pairs (λ, x) of (1) which emanates from the set of constant solutions of the unperturbed equation

Uniqueness of Local Control Sets

The local controllability behavior near an equilibrium is discussed. If the Jacobian of the linearized system is hyperbolic, uniqueness of local control sets is established.

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.

Periodic Perturbations With Delay of Autonomous Differential Equations on Manifolds

Abstract We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincaré T-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.

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]).

Harmonic perturbations with delay of periodic separated variables differential equations

We study the structure of the set of harmonic solutions to perturbed, nonautonomous,

About the notion of non‐T‐resonance and applications to topological multiplicity results for ODEs on differentiable manifolds

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

-periodic, separated variables ODEs on manifolds. The perturbing term, supposed to be

About the Sign of Oriented Fredholm Operators between Banach Spaces

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