50 research outputs found
Hamiltonian and self-adjoint control systems
This paper outlines results recently obtained in the problem of determining when an input-output map has a Hamiltonian realization. The results are obtained in terms of variations of the system trajectories, as in the solution of the Inverse Problem in Classical Mechanics. The variational and adjoint systems are introduced for any given nonlinear system, and self-adjointness defined. Under appropriate conditions self-adjointness characterizes Hamiltonian systems. A further characterization is given directly in terms of variations in the input and output trajectories, proving an earlier conjecture by the first author
Invariant higher-order variational problems II
Motivated by applications in computational anatomy, we consider a
second-order problem in the calculus of variations on object manifolds that are
acted upon by Lie groups of smooth invertible transformations. This problem
leads to solution curves known as Riemannian cubics on object manifolds that
are endowed with normal metrics. The prime examples of such object manifolds
are the symmetric spaces. We characterize the class of cubics on object
manifolds that can be lifted horizontally to cubics on the group of
transformations. Conversely, we show that certain types of non-horizontal
geodesics on the group of transformations project to cubics. Finally, we apply
second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics
on the group of transformations. This leads to a reduced form of the equations
that reveals the obstruction for the projection of a cubic on a transformation
group to again be a cubic on its object manifold.Comment: 40 pages, 1 figure. First version -- comments welcome
A Brauer's theorem and related results
Given a square matrix A, a Brauer's theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75-91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt's and Hotelling's deflations. An extension of the aforementioned Brauer's result, Rado's theorem, shows how to modify r eigenvalues of A at the same time via a rank-r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem. © 2012 Versita Warsaw and Springer-Verlag Wien.This work is supported by Fondecyt 1085125, Chile, the Spanish grant DGI MTM2010-18228 and the Programa de Apoyo a la Investigacion y Desarrollo (PAID-06-10) of the UPV.Bru GarcĂa, R.; CantĂł Colomina, R.; Soto, RL.; Urbano Salvador, AM. (2012). A Brauer's theorem and related results. Central European Journal of Mathematics. 10(1):312-321. https://doi.org/10.2478/s11533-011-0113-0S312321101Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75–91Crouch P.E., Introduction to Mathematical Systems Theory, Mathematik-Arbeitspapiere, Bremen, 1988Delchamps D.F., State-Space and Input-Output Linear Systems, Springer, New York, 1988Hautus M.L.J., Controllability and observability condition of linear autonomous systems, Nederl. Akad. Wetensch. Indag. Math., 1969, 72, 443–448Kailath T., Linear Systems, Prentice Hall Inform. System Sci. Ser., Prentice Hall, Englewood Cliffs, 1980Langville A.N., Meyer C.D., Deeper inside PageRank, Internet Math., 2004, 1(3), 335–380Perfect H., Methods of constructing certain stochastic matrices. II, Duke Math. J., 1955, 22(2), 305–311Saad Y., Numerical Methods for Large Eigenvalue Problems, Classics Appl. Math., 66, SIAM, Philadelphia, 2011Soto R.L., Rojo O., Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem, Linear Algebra Appl., 2006, 416(2–3), 844–856Wilkinson J.H., The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 196
On post-Lie algebras, Lie--Butcher series and moving frames
Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on
differential manifolds. They have been studied extensively in recent years,
both from algebraic operadic points of view and through numerous applications
in numerical analysis, control theory, stochastic differential equations and
renormalization. Butcher series are formal power series founded on pre-Lie
algebras, used in numerical analysis to study geometric properties of flows on
euclidean spaces. Motivated by the analysis of flows on manifolds and
homogeneous spaces, we investigate algebras arising from flat connections with
constant torsion, leading to the definition of post-Lie algebras, a
generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately
associated with euclidean geometry, post-Lie algebras occur naturally in the
differential geometry of homogeneous spaces, and are also closely related to
Cartan's method of moving frames. Lie--Butcher series combine Butcher series
with Lie series and are used to analyze flows on manifolds. In this paper we
show that Lie--Butcher series are founded on post-Lie algebras. The functorial
relations between post-Lie algebras and their enveloping algebras, called
D-algebras, are explored. Furthermore, we develop new formulas for computations
in free post-Lie algebras and D-algebras, based on recursions in a magma, and
we show that Lie--Butcher series are related to invariants of curves described
by moving frames.Comment: added discussion of post-Lie algebroid