I have recently published a book Tautological Control
Systems with Springer-Verlag on a framework for understanding structural
problems in geometric control theory. In an attempt to make this theory
accessible, here I will publish videotaped lectures I am giving on the
subject. Perhaps two or three people will find this interesting.
Introduction; structure of jet bundles.
Structure of jet bundles (cont'd); the
The C∞-compact-open topology;
Lipschitz and locally Lipschitz mappings; the
Clip-compact-open topology for spaces of locally Lipschitz
mappings between metric spaces; the
Ck+lip-compact-open topology for spaces of
Ck-mappings with locally Lipschitz kth derivative;
the Chol-compact-open topology; germs of real holomorphic
mappings about a subset of a real analytic manifold.
Final and initial topologies for spaces of
Cω-mappings; uniform structure for spaces of
mappings between manifolds using explicit semimetrics; topologies for spaces
of sections of vector bundles using explicit seminorms; weak-PB descriptions
for topologies for spaces of mappings between manifolds.
Local, i.e., coordinate, characterisations of topologies for
spaces of mappings; background on locally convex topologies in preparation
for talking about time-varying vector fields, mainly continuity,
boundedness, measurability, and integrability of mappings taking values in
locally convex spaces, with an emphasis on the rôle of the Suslin
property for measurability.
The weak-L topology for spaces of vector fields;
Cν-Carathéodory vector fields; locally integrally
and locally essentially bounded Cν-vector fields; flows
of locally integrally bounded Cν-vector fields;
holomorphic extension of locally integrally bounded real analytic vector fields.
Cν-control systems; holomorphic extension of
real analytic control systems; presheaves and sheaves of sets;
étalé spaces and the étalé topology.
Morphisms of presheaves; correspondence between sheaves and spaces of
local sections of the étalé space; the stalk topology; sheaf of
time-varying vector fields; local description of sections of sheaf of
time-varying vector fields; bare bones introduction to groupoids; the
groupoid of Cν-local diffeomorphisms.
More about the sheaf of time-varying vector fields; elementary
description of the flow of a time-varying vector field; absolute continuity
of curves of local diffeomorphisms; groupoid characterisation of flow of a
time-varying vector field; the stalk topology and the exponential map for
Definition of tautological control system; examples of tautological
control systems: (1) tautological control systems from control systems and
vice versa, (2) tautological control systems from differential inclusions and
vice versa, (3) differential inclusions from control systems and vice versa,
(4) tautological control systems from piecewise defined families of vector
fields, (5) tautological control systems from distributions; sheafification
of presheaves of sets of vector fields; implications of sheafification for
(control system)-(differential inclusion) equivalence and for (tautological
control system)-(differential inclusion) equivalence.
Differential inclusions of class Cν; global
generators for distributions; open-loop systems; open-loop subfamilies;
trajectories; linear control systems (these are not what you think);
trajectory correspondence between control systems and their tautological
Direct images of tautological control systems; morphisms of tautological
control systems; natural morphisms; isomorphisms and natural isomorphisms;
étalé open-loop systems; étalé open-loop subfamilies;
étalé trajectories; orbits (briefly).
Tangent and vertical lifts of vector fields; linearisation of
tautological control systems as tautological control systems on the tangent
bundle; trajectories of linearisation about a reference trajectory;
trajectories of linearisation about a reference flow; linearisation about an
equilibrium point; example from Lecture 1 revisited; linear