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Fixed points and solutions of boundary value problems at resonance

Alaa Almansour, Paul Eloe (2015)

Annales Polonici Mathematici

We consider a simple boundary value problem at resonance for an ordinary differential equation. We employ a shift argument and construct a regular fixed point operator. In contrast to current applications of coincidence degree, standard fixed point theorems are applied to give sufficient conditions for the existence of solutions. We provide three applications of fixed point theory. They are delicate and an application of the contraction mapping principle is notably missing. We give a partial explanation...

Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states

David Avanessoff, Jean-Baptiste Pomet (2007)

ESAIM: Control, Optimisation and Calculus of Variations

This paper studies Monge parameterization, or differential flatness, of control-affine systems with four states and two controls. Some of them are known to be flat, and this implies admitting a Monge parameterization. Focusing on systems outside this class, we describe the only possible structure of such a parameterization for these systems, and give a lower bound on the order of this parameterization, if it exists. This lower-bound is good enough to recover the known results about “(x,u)-flatness”...

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