Saber Jafarpour

Time-Varying Vector Fields and Their Flows

Introduction.- Fibre Metrics for Jet Bundles.- Finitely Differentiable, Lipschitz, and Smooth Topologies.- The COhol-topology for the Space of Holomorphic Vector Fields.- The Cw-topology for the Space of Real Analytic Vector Fields.- Time-Varying Vector Fields.- References.

Real analytic control systems

Using a suitable locally convex topology for the space of real analytic vector fields, we give a characterization of real analytic control systems. Among other things, this class of real analytic control systems has the property that, upon substitution of an open-loop control, the resulting time-varying vector field has a flow depending on initial condition in a real analytic manner. To give context to the real analytic case, we also consider the cases of finitely differentiable and smooth control systems.

Fibre Metrics for Jet Bundles

One of the principal devices we use in the monograph is convenient seminorms for the various topologies we use for spaces of sections of vector bundles. Since such topologies rely on placing suitable norms on derivatives of sections, i.e., on jet bundles of vector bundles, in this chapter we present a means for defining such norms, using as our starting point a pair of connections, one for the base manifold, and one for the vector bundle. These allow us to provide a direct sum decomposition of the jet bundle into its component “derivatives”, and so then a natural means of defining a fibre metric for jet bundles using metrics on the tangent bundle of the base manifold and the fibres of the vector bundle. As we shall see, in the smooth case these constructions are a convenience, whereas in the real analytic case they provide a crucial ingredient in our global, coordinate-free description of seminorms for the topology of the space of real analytic sections of a vector bundle. For this reason, in this chapter we shall also consider the existence of, and some properties of, real analytic connections in vector bundles.

The C ω -Topology for the Space of Real Analytic Vector Fields

In this chapter we examine a topology on the set of real analytic vector fields. As we shall see, this requires some considerable effort. Agrachev and Gamkrelidze (Mathematics of the USSR-Sbornik 107(4):467–532, 1978) consider the real analytic case by considering bounded holomorphic extensions to neighbourhoods of real Euclidean space in complex Euclidean space. Our approach is more general, more geometric, and global, using a natural real analytic topology described, for example, in the work of Martineau (Mathematische Annalen 163:62–88, 1966). This allows us to dramatically broaden the class of real analytic vector fields that we can handle to include “all” analytic vector fields.

Time-Varying Vector Fields

In this chapter we consider time-varying vector fields. The ideas in this chapter originate (for us) with the paper of Agrachev and Gamkrelidze (Mathematics of the USSR-Sbornik 107(4):467–532, 1978), and are nicely summarised in the more recent book by Agrachev and Sachkov (Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences, vol. 87. Springer-Verlag, New York/Heidelberg/Berlin, 2004), at least in the smooth case. A geometric presentation of some of the constructions can be found in the paper of Sussmann (Geometry of Feedback and Optimal Control, pp. 463–557. Dekker Marcel Dekker, New York, 1997), again in the smooth case, and Sussmann also considers regularity less than smooth, e.g., finitely differentiable or Lipschitz. There is some consideration of the real analytic case in Agrachev and Gamkrelidze (Mathematics of the USSR-Sbornik 107(4):467–532, 1978), but this consideration is restricted to real analytic vector fields admitting a bounded holomorphic extension to a fixed-width neighbourhood of real Euclidean space in complex Euclidean space. One of our results, the rather nontrivial Theorem 6.25, is that this framework of Agrachev and Gamkrelidze (Mathematics of the USSR-Sbornik 107(4):467–532, 1978) is sufficient for the purposes of local analysis. However, our treatment of the real analytic case is global, general, and comprehensive. To provide some context for our novel treatment of the real analytic case, we treat the smooth case in some detail, even though the results are probably mostly known. (However, we should say that, even in the smooth case, we could not find precise statements with proofs of some of the results we give.) We also treat the finitely differentiable and Lipschitz cases, so our theory also covers the “standard” Caratheodory existence and uniqueness theorem for time-varying ordinary differential equations, e.g., Sontag (Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. No. 6 in Texts in Applied Mathematics. Springer-Verlag, New York/Heidelberg/Berlin, 1998), Theorem 54. We also consider holomorphic time-varying vector fields, as these have a relationship to real analytic time-varying vector fields that is sometimes useful to exploit.One of the unique facets of our presentation is that we fully explain the role of the topologies developed in Chaps. 3, 4, and 5. Indeed, one way to understand the principal results of this chapter is that they show that the usual pointwise—in state and time—conditions placed on vector fields to regulate the character of their flows can be profitably phrased in terms of topologies for spaces of vector fields. While this idea is not entirely new—it is implicit in the approach of Agrachev and Gamkrelidze (Mathematics of the USSR-Sbornik 107(4):467–532, 1978)—we do develop it comprehensively and in new directions.

The COhol-Topology for the Space of Holomorphic Vector Fields

Even if one has no per se interest in holomorphic vector fields, it is the case that an understanding of certain constructions for real analytic vector fields relies in an essential way on their holomorphic extensions. Also, as we shall see, we will arrive at a description of the real analytic topology that, while often easy to use in general arguments, is not well suited for verifying hypotheses in examples. In these cases, it is often most convenient to extend from real analytic to holomorphic, where things are easier to verify. Thus in this section we overview the holomorphic case. We begin with vector bundles, as in the smooth case.

The Compact-Open Topologies for the Spaces of Finitely Differentiable, Lipschitz, and Smooth Vector Fields

As mentioned in Chap. 1, one of the themes of this work is the connection between topologies for spaces of vector fields and regularity of their flows. In this and the subsequent two chapters we describe appropriate topologies for finitely differentiable, Lipschitz, smooth, holomorphic, and real analytic vector fields. The topology we use in this chapter in the smooth case (and the easily deduced finitely differentiable case) is classical, and is described, for example, in Agrachev and Sachkov (Encyclopedia of Mathematical Sciences, vol. 87, Springer-Verlag, New York/Heidelberg/Berlin (2004), §2.2) see also Michor (Manifolds of Differentiable Mappings, No. 3 in Shiva Mathematics Series, Shiva Publishing Limited, Orpington, UK (1980), Chapter 4). What we do that is original is provide a characterisation of the seminorms for this topology using the jet bundle fibre metrics from Sect. 2.2.. The fruits of the effort expended in the next three chapters is harvested when our concrete definitions of seminorms permit a relatively unified analysis in Chap. 6 of time-varying vector fields.