We consider cost-extended control systems corresponding to a certain class of left-invariant optimal control problems. These systems are organized as a category; some basic properties are investigated. Cost-equivalence is characterized and the Pontryagin lift is realized as a contravariant functor. A number of illustrative examples are provided.

https://www.ru.ac.za/media/rhodesuniversity/content/mathematics/images/ggc/Talk-2012-03-Biggs-RU-ceLiCAS.pdf

Cost-Extended Control Systems on Lie Groups

In this paper we analyze the quadratic and homogeneous Hamiltonian systems on (&#x1d530;&#x1d52c;(3))* from the Poisson dynamics and geometry point of view.

New Aspects on the Geometry and Dynamics of Quadratic Hamiltonian Systems on ((3))

We identify a class of quadratic Hamilton-Poisson systems on the three-dimensional Euclidean Lie-Poisson space. Speciflcally, we consider systems that are both homogeneous and for which the underlying quadratic form is positive semideflnite. Any such system is shown to be equivalent to one of four normal forms (of which two are parametrized families of systems). For the cases with non-trivial dynamics, the stability nature of the equilibrium states is fully investigated. Furthermore, we flnd explicit expressions for the integral curves (in terms of Jacobi elliptic functions).

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On some quadratic Hamilton-Poisson systems

Asymptotic stabilization with phase of periodic orbits of three-dimensional Hamiltonian systems

In this paper we consider quadratic Hamilton–Poisson systems on the semi-Euclidean Lie–Poisson space . The homogeneous positive semidefinite systems are classified; there are exactly six equivalence classes. In each case, the stability nature of the equilibrium states is determined. Explicit expressions for the integral curves are found. A characterization of the equivalence classes, in terms of the equilibria, is identified. Finally, the relation of this work to optimal control is briefly discussed.

Quadratic Hamilton–Poisson systems on $\mathfrak{s}\mathfrak{e}(1, 1)^{*}_{-}$: The homogeneous case

We study some geometrical and dynamical properties of a quadratic and homogeneous Hamilton–Poisson system defined on the dual of the Lie algebra so(3) with its minus-Lie-Poisson structure.

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Papers

New Aspects on the Geometry and Dynamics of Quadratic Hamiltonian Systems on ((3))

QUADRATIC AND HOMOGENEOUS HAMILTON-POISSON SYSTEM ON (so(3))