Tautological Control Systems

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.

Lecture 1
Introduction; structure of jet bundles.


Lecture 2
Structure of jet bundles (cont'd); the Ck-compact-open topology.


Lecture 3
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.


Lecture 4
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.


Lecture 5
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.


Lecture 6
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.


Lecture 7
Cν-control systems; holomorphic extension of real analytic control systems; presheaves and sheaves of sets; étalé spaces and the étalé topology.


Lecture 8
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.


Lecture 9
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 flows.


Lecture 10
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.


Lecture 11
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 control systems.


Lecture 12
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).


Lecture 13
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 controllability.