Rory Biggs

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We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in th...

We consider the left-invariant sub-Riemannian and Riemannian structures on the Heisenberg groups. A classification of these structures was found previously. In the present paper, we find (for each normalized structure) the isometry group, the exponential map, the totally geodesic subgroups, and the conjugate locus. Finally, we determine the minimiz...

A classification of the real four-dimensional connected Lie groups is obtained. Those groups which are linearizable are identified; accompanying matrix Lie groups are exhibited.

Quadratic Hamilton-Poisson systems on three-dimensional Lie-Poisson spaces are considered. The homogeneous (positive) semidefinite systems are classified up to linear isomorphism; an exhaustive and nonredundant list of 23 normal forms is exhibited. For each normal form, the stability nature of the equilibria is determined. Each normal form is expli...

We define the extension of a left-invariant sub-Riemannian structure in terms of an extension of the underlying Lie group and compatibility of the respective distributions and metrics. We show that geodesics of a structure can be lifted to geodesics of any extension of the structure. In the case of central extensions, we show that the normal geodes...

This is the poster of the event.

We consider state space equivalence and (a specialization of) feedback equivalence in the context of left-invariant control affine systems. Simple algebraic characterizations of both local and global forms of these equivalence relations are obtained. Several illustrative examples regarding the classification of systems on low-dimensional Lie groups...

This is a short survey of our recent research on invariant control systems (and their associated optimal control problems). We are primarily concerned with equivalence and classification, especially in three dimensions.

We consider equivalence, stability and integration of quadratic Hamilton–Poisson systems on the semi-Euclidean Lie–Poisson space \(\mathfrak{se}(1,1)^{*}_{-}\). The inhomogeneous positive semidefinite systems are classified (up to affine isomorphism); there are 16 normal forms. For each normal form, we compute the symmetry group and determine the L...

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...

The realification of the -dimensional complex Heisenberg Lie algebra is a -dimensional real nilpotent Lie algebra with a 2-dimensional commutator ideal coinciding with the centre, and admitting the compact algebra of derivations. We investigate, in general, whether a real nilpotent Lie algebra with 2-dimensional commutator ideal coinciding with the...

Left-invariant control affine systems on simply connected nilpotent Lie groups of dimension ≤4 are considered. First, we classify these control systems under two natural equivalence relations. Second, the associated cost-extended systems (with invariant quadratic Lagrangian) are classified. The latter classification is reinterpreted in terms of inv...

Positive semidefinite quadratic Hamilton-Poisson systems on the three-dimensional Heisenberg Lie-Poisson space are classified. Stability and integration of each normal form are briefly covered. The relation of these systems to optimal control is also briefly discussed.

We consider left-invariant control affine systems on the matrix Lie group SO(2,1). 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...

We consider Riemannian manifolds endowed with a nonholonomic distribution. These structures model mechanical systems with a (positive definite) quadratic Lagrangian and nonholonomic constraints linear in velocities. We classify the left-invariant nonholonomic Riemannian structures on three-dimensional simply connected Lie groups, and describe the e...

The invariant affine distributions on a four-dimensional central extension of the semi-Euclidean group are classified (up to group automorphism). This classification is briefly discussed in the context of invariant control theory and sub-Riemannian geometry.

We consider inhomogeneous quadratic Hamilton-Poisson systems on the Lie-Poisson space so(3)*_. There are nine such systems up to affine equivalence. We investigate the stability nature of the equilibria for each of these systems. For a subclass of systems, we find explicit expressions for the integral curves in terms of Jacobi elliptic functions.

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 cost-extended systems (associated to left-invariant optimal control problems with quadratic cost) are also classified....

We consider state space equivalence and feedback equivalence in the context of (full-rank) left-invariant control systems on Lie groups. We prove that two systems are state space equivalent (resp. detached feedback equivalent) if and only if there exists a Lie group isomorphism relating their parametrization maps (resp. traces). Local analogues of...

We classify the subspaces of each real four-dimensional Lie algebra, up to automorphism. Enumerations of the subalgebras, ideals, and full-rank (or bracket generating) subspaces are obtained. Also, the interplay between quotients (resp. extensions) of algebras and such classifications is briefly considered.

In this paper we consider quadratic Hamilton–Poisson systems on the semi-Euclidean Lie–Poisson space se(1,1)*. 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...

We consider left-invariant control affine systems, evolving on three-dimensional matrix Lie groups. Equivalence and controllability are investigated. All full-rank systems are classified, under detached feedback equivalence. A representative is identified for each equivalence class. The controllability nature of these representatives is determined.

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.

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 optim...

We classify the full-rank affine subspaces (resp. parametrized affine subspaces) of the semi-Euclidean Lie algebra se(1,1). The equivalence relations under consideration (L-equivalence and P-equivalence) are motivated by the study of invariant control affine systems. Exhaustive lists of equivalence representatives are obtained, along with classifyi...

The structure of the four-dimensional oscillator Lie algebra is examined. The adjoint orbits are determined; these are linearly isomorphic to the coadjoint orbits. The linear subspaces are classified; as a by-product, we arrive at classifications of the full-rank subspaces, the subalgebras, and the ideals. The associated connected Lie groups are th...

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.

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.

We classify the left-invariant control affine systems evolving on the orthogonal group SO(4). The equivalence relation under consideration is detached feedback equivalence. Each possible number of inputs is considered; both the homogeneous and inhomogeneous systems are covered. A complete list of class representatives is identified and controllabil...

Any two-input left-invariant control affine system of full rank, evolving on the Euclidean group SE (2), is (detached) feedback equivalent to one of three typical cases. In each case, we consider an optimal control problem which is then lifted, via the Pontryagin Maximum Principle, to a Hamiltonian system on the dual space 𝔰𝔢 (2)*. These reduced Ha...

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 ident...

We apply Williamson's theorem for the diagonalization of quadratic forms by symplectic matrices to sub-Riemannian (and Riemannian) structures on the Heisenberg groups. A classification of these manifolds, under isometric Lie group automorphisms, is obtained. A (parametrized) list of equivalence class representatives is identified; a geometric chara...

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 II, IV , and V in the Bianchi-Behr classification.

We classify, under affine equivalence, the quadratic Hamilton-Poisson systems on the Lie-Poisson space so(3)*. For the simplest strictly inhomogeneous quadratic system, we find explicit expressions for the inte- gral curves in terms of Jacobi elliptic functions.

We classify homogeneous positive semidefinite quadratic Hamilton-Poisson systems on a certain subclass of three-dimensional Lie-Poisson spaces.

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).

We identify a class of quadratic Hamilton-Poisson systems on the three-dimensional Euclidean Lie-Poisson space. Specifically, we consider systems that are both homogeneous and for which the underlying quadratic form is positive semidefinite. Any such system is shown to be equivalent to one of four normal forms (of which two are parametrized familie...

We investigate a certain class of left-invariant control systems evolving on the Lie group SO(4). Two such systems are L-equivalent provided their traces are related by a Lie algebra automorphism. We produce structural results regarding L-equivalence of all homogeneous control affine systems on SO(4). An illustrative example is provided.

We consider the equivalence of cost-extended control systems corresponding to certain invariant optimal control problems. We prove that two equivalent cost-extended systems have the same optimal controlled trajectories and the same extremal curves (up to a diffeomorphism). We also prove that equivalent cost-extended systems are lifted (via the Pont...

We classify, under (local) state space equivalence, all full-rank left-invariant control affine systems evolving on the Euclidean group SE(2).

We consider a general single-input left-invariant control affine system, evolving on the Euclidean group SE(2). Any such controllable control system is (detached feedback) equivalent to one of two typical cases. In each case, we consider an optimal control problem (with quadratic cost) which is then lifted, via the Pontryagin Maximum Principle, to...

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.

A classification of full-rank affine subspaces of (real) three-dimensional Lie algebras is presented. In the context of invariant control affine systems, this is exactly a classification of all full-rank systems evolving on three-dimensional Lie groups.