Publication

On two isomorphic Lie algebroids for Feedback Linearization

Philippe Müllhaupt
2019
Report or working paper
Abstract

Two Lie algebroids are presented that are linked to the construction of the linearizing output of an affine in the input nonlinear system.\ The algorithmic construction of the linearizing output proceeds inductively, and each stage has two structures, namely a codimension one foliation defined through an integrable 1-form ω\omega , and a transversal vectorfield gg\ to the foliation. Each integral manifold of the vectorfield gg defines an equivalence class of points. Due to transversality, a leaf of the foliation is chosen to represent these equivalence classes. A Lie groupoid is defined with its base given as the particular chosen leaf and with the product induced by the pseudogroup of diffeomorphisms that preserve equivalence classes generated by the integral manifolds of g. Two Lie algebroids associated with this groupoid are then defined. The theory is illustrated with an example using polynomial automorphisms as particular cases of diffeomorphisms and shows the relation with the Jacobian conjecture.

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Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.
Poisson manifold
In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics. A Poisson structure (or Poisson bracket) on a smooth manifold is a functionon the vector space of smooth functions on , making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra).
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In mathematics, especially in and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: Group with a partial function replacing the binary operation; in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called inverse by analogy with group theory. A groupoid where there is only one object is a usual group.
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