Singularities of optimal control problems on some 6-D lie groups

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a 3-D space. A Lie group formulation arises naturally and the vehicles are modeled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically, the 3-D space forms Euclidean space E-3, the sphere S-3, and the hyperboloid H'. The corresponding frame bundles are equal to the Euclidean group of motions SE(3), the rotation group SO(4), and the Lorentz group SO (1, 3). The maximum principle of optimal control shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case, the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It. is then shown that the projections, of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.

Controllability of control systems on complex simple lie groups and the topology of flag manifolds

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Fundamental domains for proper affine actions of Coxeter groups in three dimensions

We study proper actions of groups $G \cong \Z/2\Z \ast \Z/2\Z \ast \Z/2\Z$ on affine space of three real dimensions. Since $G$ is nonsolvable, work of Fried and Goldman implies that it preserves a Lorentzian metric. A subgroup $\Gamma < G$ of index two acts freely, and $\R^3/\Gamma$ is a Margulis spacetime associated to a hyperbolic surface $\Sigma$. When $\Sigma$ is convex cocompact, work of Danciger, Gu{\'e}ritaud, and Kassel shows that the action of $\Gamma$ admits a polyhedral fundamental domain bounded by crooked planes. We consider under what circumstances the action of $G$ also admits a crooked fundamental domain. We show that it is possible to construct actions of $G$ that fail to admit crooked fundamental domains exactly when the extended mapping class group of $\Sigma$ fails to act transitively on the top-dimensional simplices of the arc complex of $\Sigma$. We also provide explicit descriptions of the moduli space of $G$ actions that admit crooked fundamental domains.

Justifications and limits in the politically motivated lawbreaking of environmental activist groups

In recent events, notions of political protest, civil disobedience, extremism, and criminal action have become increasingly blurred. The London Riots, the Occupy movement, and the actions of hacking group Anonymous have all sparked heated debate about the limits of legitimate protest, and the distinction between an acceptable action and a criminal offence. Long before these events, environmental activists were challenging convention in protest actions, with several groups engaging in politically motivated law-breaking. The emergence of the term ‘eco-tage’ (the sabotage of equipment in order to protect the environment) signifies the important place environmental activists hold in challenging the traditional boundaries between illegal action and legitimate protest. Many of these groups establish their own boundaries of legitimacy, with some justifying their actions on the basis of civil disobedience or extensional self-defence. This paper examines the statements of environmental activist organisations that have engaged in politically motivated law breaking. It identifies the parameters that these groups set on their illegal actions, as well as the justifications that they provide, with a view to determining where these actions fit in the vast grey area between legal protest and violent extremism.

Domains in complex surfaces with non-compact automorphism groups

Let X be an arbitrary complex surface and D a domain in X that has a non-compact group of holomorphic automorphisms. A characterization of those domains D that admit a smooth, weakly pseudoconvex, finite type boundary orbit accumulation point is obtained.

On the unreasonable effectiveness of Gutzmer's formula

In this article we plan to demonstrate the usefulness of `Gutzmer's formula' in the study of various problems related to the Segal-Bargmann transform. Gutzmer's formula is known in several contexts: compact Lie groups, symmetric spaces of compact and noncompact type, Heisenberg groups and Hermite expansions. We apply Gutzmer's formula to study holomorphic Sobolev spaces, local Peter-Weyl theorems, Paley-Wiener theorems and Poisson semigroups.

Helping ourselves: the role of self-help groups in poverty alleviation through aquaculture

Self-help groups (SHGs) are ways for farmers and fishers, especially those who are poor, to come together and work together. They can be a useful entry point for outsiders, promote a supportive local environment, strengthen voices in decision-making and in negotiations with more powerful forces, increase the effectiveness of local actions, and provide easier access to micro-credit and other resources and services. This case study describes a rural aquaculture development context, in India, the development of SHGs and the concept of a ‘one-stop aqua shop’, set up and run by a federation of self-help groups in Kaipara village, West Bengal (a pilot state along with Jharkhand and Orissa). It outlines testing new ways to share information, as part of a series of revised procedures and institutional arrangements for service delivery recommended by farmers and fishers and prioritized by government, with support from the Department of International Development, London (DFID) Natural Resources Support Programme (NRSP) and the Network of Aquaculture Centres in Asia-Pacific (NACA) to the Support to Regional Aquatic Resources Management (STREAM) Initiative (10 p.)

Coordinated motion design on lie groups

The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics. It thereby shows that coordination can be studied in a systematic way once the Lie group geometry of the configuration space is well characterized. Applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds. A link with Brockett's double bracket flows is also made. The concepts are illustrated on SO(3), SE(2) and SE(3). © 2010 IEEE.

Coordination on Lie groups

This paper studies the coordinated motion of a group of agents evolving on a Lie group. Left-or rightinvariance with respect to the absolute position on the group lead to two different characterizations of relative positions and two associated definitions of coordination (fixed relative positions). Conditions for each type of coordination are derived in the associated Lie algebra. This allows to formulate the coordination problem on Lie groups as consensus in a vector space. Total coordination occurs when both types of coordination hold simultaneously. The discussion in this paper provides a common geometric framework for previously published coordination control laws on SO(3), SE(2) and SE(3). The theory is illustrated on the group of planar rigid motion SE(2). © 2008 IEEE.

Antagonistic effects of two novel GIP analogs, (Hyp(3))GIP and (Hyp(3)) GIPLys(16)PAL, on the biological actions of GIP and longer-term effects in diabetic ob/ob mice

This study examines the actions of the novel enzyme- resistant, NH2- terminally modified GIP analog ( Hyp(3)) GIP and its fatty acid- derivatized analog ( Hyp(3)) GIPLys(16)PAL. Acute effects are compared with the established GIP receptor antagonist ( Pro(3)) GIP. All three peptides exhibited DPP IV resistance, and significantly inhibited GIP stimulated cAMP formation and insulin secretion in GIP receptor- transfected fibroblasts and in clonal pancreatic BRIN- BD11 cells, respectively. Likewise, in obese diabetic ob/ob mice, intraperitoneal administration of GIP analogs significantly inhibited the acute antihyperglycemic and insulinreleasing effects of native GIP. Administration of once daily injections of ( Hyp(3)) GIP or ( Hyp(3)) GIPLys(16)PAL for 14 days resulted in significantly lower plasma glucose levels ( P <0.05) after ( Hyp3) GIP on days 12 and 14 and enhanced glucose tolerance ( P <0.05) and insulin sensitivity ( P <0.05 to P <0.001) in both groups by day 14. Both ( Hyp(3)) GIP and ( Hyp(3)) GIPLys(16)PAL treatment also reduced pancreatic insulin ( P <0.05 to P <0.01) without affecting islet number. These data indicate that ( Hyp3) GIP and ( Hyp(3)) GIPLys(16)PAL function as GIP receptor antagonists with potential for ameliorating obesity- related diabetes. Acylation of ( Hyp(3)) GIP to extend bioactivity does not appear to be of any additional benefit.

Integrating Hamiltonian systems defined on the Lie groups SO(4) and SO(1,3)

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E³, the spheres S³ and the hyperboloids H³ with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions is illustrated.