Rory Biggs

A Category of Control Systems

Abstract We construct the concrete category LiCS of left-invariant control systems (on Lie groups) and point out some very basic properties. Morphisms in this category are examined briefly. Also, covering control systems are introduced and organized into a (comma) category associated with LiCS

Control affine systems on solvable three-dimensional Lie groups, I

We seek to classify the full-rank left-invariant control affine systems evolving on solvable three-dimensional Lie groups. In this paper we consider only the cases corresponding to the solvable Lie algebras of types III, V I, and V II in the Bianchi-Behr classification.

CONTROL AFFINE SYSTEMS ON SEMISIMPLE THREE-DIMENSIONAL LIE GROUPS

Abstract We classify the full-rank left-invariant control affine systems evolving on (real) semisimple three-dimensional Lie groups. This is accomplished by reducing the problem to that of classifying the affine subspaces of the Lie algebras so (2; 1) and so (3).

Control systems on the Heisenberg group: equivalence and classification

Left-invariant control affine systems on the three-dimensional Heisenberg group are classified under detached feedback equivalence, strongly detached feedback equivalence, and state space equivalence. The corresponding controllable costextended systems (associated to left-invariant optimal control problems with quadratic cost) are also classified. As a corollary, a classification of the left-invariant metric pointaffine structures is obtained.

Equivalence of Control Systems on the Pseudo-Orthogonal Group SO (2, 1)0

Abstract We consider left-invariant control affine systems on the matrix Lie group SO (2, 1)0. A classification, under state space equivalence, of all such full-rank control systems is obtained. First, we identify certain subsets on which the group of Lie algebra automorphisms act transitively. We then systematically identify equivalence class representatives (for single-input, two-input and three-input control systems). A brief comparison of these classification results with existing results concludes the paper.

Optimal control of drift-free invariant control systems on the group of motions of the minkowski plane

We consider a class of invariant optimal control problems on the three-dimensional semi-Euclidean group. Specifically, we consider only drift-free left-invariant control affine systems and positive definite quadratic costs. The associated cost-extended systems are classified. Explicit expressions for the extremal controls of the corresponding optimal control problems are obtained.

Quadratic Hamilton–Poisson systems on

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.

\mathfrak{s}\mathfrak{e}(1, 1)^{*}_{-}

We consider left-invariant control affine systems evolving on three-dimensional matrix Lie groups. Equivalence and controllability are addressed. The full-rank systems are classified under detached feedback equivalence; a representative is identified for each equivalence class. A characterization of controllability on each group is then determined.

: The homogeneous case

Abstract We consider left-invariant control affine systems evolving on Lie groups. In this context, feedback equivalence specializes to detached feedback equivalence. We characterize (local) detached feedback equivalence in a simple algebraic manner. We then classify all (full-rank) systems evolving on three-dimensional Lie groups. A representative is identified for each equivalence class. Systems on the Heisenberg group, the Euclidean group, and the orthogonal group are treated in full, as typical examples. In these three cases, simple algebraic characterizations of the equivalence classes are also exhibited. A few remarks conclude the paper.

Control systems on three-dimensional lie groups

This is a survey of our research (conducted over the last few years) on invariant control systems, the associated optimal control problems, and the associated Hamilton–Poisson systems. The focus is on equivalence and classification. State space and detached feedback equivalence of control systems are characterized in simple algebraic terms; several classes of systems (in three dimensions, on the Heisenberg groups, and on the six-dimensional orthogonal group) are classified. Equivalence of cost-extended systems is shown to imply equivalence of the associated Hamilton–Poisson systems. Cost-extended systems of a certain kind are reinterpreted as invariant sub-Riemannian structures. A classification of quadratic Hamilton–Poisson systems in three dimensions is presented. As an illustrative example, the stability and integration of a typical system is investigated.