Contemporary Trends in Nonlinear Geometric Control Theory and Its Applications

The following sections are included:

• Contents

• Foreword

We consider n-dimensional extensions of some classical problems on curves which satisfy certain curvature constraints. The problems of elastica, Delaunay, and Dubins are treated under a unifying point of view provided by the framework of frame curves and the frame bundles. We shall treat all these problems on space forms of dimension n. All of these problems have natural control theoretic formulations on the underlying isometry group leading up to the appropriate Hamiltonian obtained through the Maximum Principle. We show that these Hamiltonians are completely integrable and we also show that their solutions reveal remarkable connections with the geometry of the top.

The purpose of this paper is to present conditions that allow one to determine the controllability of a Lie system on a connected Lie group G by pushing the controllability question forward to smaller dimensional coset spaces. Since non-controllable systems may become controllable on smaller dimensional coset spaces, the problem considered is the determination of closed subgroups H such that a system is controllable on G/H if and only if it is controllable on G.

We start with a brief overview of results characterizing distributions (equivalently Pfaffian systems) that are locally equivalent to a canonical contact system. Then we concentrate on distributions equivalent to the canonical contact system on Jⁿ(ℝ, ℝ^m), that is the canonical contact system for curves. We give various necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact systems for curves. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistributions contained in the successive elements of the derived flag. We give a new characterization of distributions locally equivalent to the canonical contact systems for curves that uses only one involutive subdistribution in which all information about the geometry and about the regularity is encoded. We also distinguish regular points, at which the system is equivalent to the canonical contact system, and singular points, at which we propose a new normal form that generalizes the canonical contact system on Jⁿ(ℝ, ℝ^m) in a way analogous to that how Kumpera-Ruiz normal form generalizes Goursat normal form, that is, the canonical contact system on Jⁿ(ℝ, ℝ). Finally, we briefly discuss how the geometry of distributions equivalent to the canonical contact systems for curves is reflected by their flatness and illustrate some of our results by a model of a simple nonholonomic control system.

The following sections are included:

• Introduction

• Johann Bernoulli and the brachistochrone problem
  • Johann Bernoulli's challenge
  • The cycloid's double role as the brachistochrone and the tautochrone
  • Why was the brachistochrone so important in 1696–97?

• The standard formulation and Johann Bernoulli's solution
  • Formulation of the brachistochrone problem
  • Johann Bernoulli's solution

• Spurious solutions and the calculus of variations approach
  • Remarks on Johann Bernoulli's argument
  • The calculus of variations approach

• The optimal control approach
  • The brachistochrone as an optimal control problem
  • Optimality proofs and the existence problem

• The differential-geometric connection
  • Is the standard formulation of the brachistochrone problem satisfactory?
  • Hamilton's principal function
  • What is Hamilton's principal function for the brachistochrone?
  • Lie brackets, integrals of motion, and the differential-geometric perspective
  • The diamond Lie algebra
  • Energy conservation
  • Naïve application of the Maximum Principle
  • Controllability
  • The existence problem
  • Application of the Maximum Principle on manifolds
  • The complete synthesis and Hamilton's principal function for the 4-dimensional brachistochrone problem

• Five modern variations on the theme of the brachistochrone
  • Hestenes' “variation of the brachistochrone problem”
  • The reflected brachistochrone problem
  • A modest claim
  • The brachistochrone with friction
  • The classical brachistochrone as a degenerate problem
  • Deriving Snell's law from the Maximum Principle
  • Was Johann Bernoulli close to discovering non-Euclidean geometry?

• Acknowledgements

• References

We announce a new sufficient condition for a bang-bang extremal to be a strong local optimum for a control problem in the Mayer form. The controls appear linearly and take values in a polyhedron and the state space and the constraints are smooth, finite dimensional manifolds.

In this paper we study the action of certain nilpotent Lie groups under which a differential system given in terms of a rank-two distribution remains invariant. Such systems describe the dynamics of electric charges in static inhomogeneous magnetic fields. The action of the Lie group associated to the sub-Riemannian structures in the simplest non-uniform cases, contains naturally the Heisenberg group which corresponds to a uniform field. A linear magnetic field yields the nilpotent group associated with the so-called Martinet case and the group corresponding to systems of Engel type. The Cartan five dimensional system is also the natural setting for the study of linear magnetic fields acting on classical electric charges. Our results generalize previous works on this class of non-holonomic systems.

This note deals with “Grobman-Hartman like” theorems for control systems (or in other words under-determined systems of ordinary differential equations). The main results (proved elsewhere) is that when a control system is topologically conjugate to a linear controllable one, then it is also “almost” differentiably conjugate. We focus on the meaning of this result, and on an open question resulting from it.

This paper concerns the variational approach to the study of controls problems. We construct local approximations of the reachable set from which sufficient conditions for a trajectory to be locally controllable and necessary conditions for a reference pair to be a weak solution of a Mayer control problem with end constraints can be derived.

We give preliminary remarks concerning the optimal control of the atmospheric arc for a space shuttle (earth re-entry or Mars sample return project). The system governing the trajectories is 6–dimensional, the control is the bank angle, the cost–integrand is the thermal flux and we have state constraints on the thermal flux and the normal acceleration. Our study is geometric and founded on the analysis of the solutions of a minimum principle and direct evaluation of the small–time reachable set for the problem taking into account the state constraints.

In this paper, following ideas already developped in ¹⁰, we construct an observer for nonlinear systems that looks like the extended Kalman filter. In fact, it is asymptotically (in time) exactly the deterministic version of the extended Kalman filter, and when the “innovation” is large, it is an high gain observer. In the context of the theory developped in ¹⁰, we show that it works for “all observable systems”. In the paper, we prove convergence of the estimation error, we give several estimates of this error, and we show a convincing illustrative application (a distillation column)

We show that the theory of systems of differential equations allowing a rule for expressing the general solution in terms of some particular ones, the so-called Lie systems, are relevant in Control Theory. These systems are related to some particular cases of control systems on Lie groups, whose exact solution is usually found by means of a method generalizing the one proposed by Wei and Norman. We establish some results concerning these systems. The general theory is illustrated with some examples.

This contribution goes along the lines of the belief that nonlinear observability has a deep geometric content. We try to set up that geometry, its algebraic translation all the way down to the algorithms needed to do some of the computations. The basic geometric picture of observability is that of a projection map, that of the system trajectories onto the data trajectories. Observability is then seen as the condition for this map to be finite in the sense that its fibres are generically finite. It is this definition which easily passes to differential algebraic geometric characterizations. Next, most of the picture is kept by defining the notion of singular observation data, that is, the special data for which the generic observability property is lost. The work is incomplete at least in that we hardly touch the so-called real case where we have to leave the abstract differential algebraic geometry to enter the differential real algebraic geometry. This is under investigation.

The following sections are included:

• Introduction

• Existence of analytic realizations

• Convergence along vector fields and their commutators

• Existence of analytic solutions of PDE's

• Acknowledgments

• References

The following sections are included:

• Introduction

• Background

• Normality implies seminormality

• Hamiltonian normality implies seminormality

• Normality implies seminormality for systems of Mayer type

• NOC for multiobjective optimal control problems

• References

Since the articles on nonlinear controllability of the early 1970s by Jurdjevic and Sussmann, Lobry, Brockett, Hermes and Haynes and others, one common thread of the differential geometric approach has been the desired conceptual separation of the time-varying features from the fixed geometrical structures.

Integral manifolds are intrinsically related to Lie-brackets of vector fields, and they determine accessibility. Iterated integral functional of the controls carry a natural chronological algebra structure, and they carry the information that distinguishes controllability from accessibility – in the case of nonlinear systems.

This article surveys some of the progress made in the study of nonlinear controllability since the 1970s, leading to a modern reformulation of the concept of obstructions to controllabilty (traditionally misnamed bad brackets) which takes advantage of newly found formulas for the logarithm of the Chen-Fliess series, i.e. using coordinates of the first kind, and comparing them to the established formulas using coordinates of the second kind.

We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new Lagrangian in an extended configuration space —which we call D'Alambert's—comprising both the original coordinates and the compatible “virtual displacements” joining two solutions of the original system. The variational principle is Hamilton's with the new Lagrangian. We use this formulation to obtain constants of motion in the Jacobi equations of any Lagrangian system with symmetries. These constants are related to constants in the original system and so with symmetries of the original Lagrangian. We cast our approach in an intrinsic coordinate free formulation. Our results can be of interest for reducing the dimensions of the equations that characterize perturbations in a Lagrangian control system.

A suitable combination of the differential flatness property and the second order sliding mode controller design technique is proposed for the specification of a robust dynamic feedback multivariable controller accomplishing prescribed trajectory tacking tasks for the earth coordinate position variables of a hovercraft vessel model.

We investigate minimization problems along a singular trajectory of a single-input affine control system with constraint on the control, and then as an application of a sub-Riemannian system of rank 2. Under generic assumptions we get necessary and sufficient conditions for optimality of such a singular trajectory. Moreover we describe precisely the contact of the accessibility sets at time T with the singular direction. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer γ into two sectors, bordered by the first Pontryagin's cone along γ, called the L^∞-sector and the L²-sector.

The following sections are included:

• Introduction

• Complex analytic considerations

• Holomorphic vector fields
  • Approximating solutions of ODE
  • The density property

• Jets
  • The jet theorems
  • Applications
  • The Fatou-Bieberbach Phenomenon
  • Completeness of vector fields
  • Interpolation results

• References

The following sections are included:

• Index