Mathematical models for geometric control theory

TL;DR

This paper introduces a geometric framework for control systems called tautological control systems that avoids explicit control parameterisation, providing a more intrinsic and regularity-aware approach to modeling in geometric control theory.

Contribution

It develops a novel, parameterisation-independent framework for control systems using topologies on vector fields, enabling better understanding of regularity and system equivalences.

Findings

Established topologies for vector fields across various regularity classes.

Proved regular dependence of flows on initial conditions for time-varying vector fields.

Demonstrated that control-affine systems are special cases within the new framework.

Abstract

Just as an explicit parameterisation of system dynamics by state, i.e., a choice of coordinates, can impede the identification of general structure, so it is too with an explicit parameterisation of system dynamics by control. However, such explicit and fixed parameterisation by control is commonplace in control theory, leading to definitions, methodologies, and results that depend in unexpected ways on control parameterisation. In this paper a framework is presented for modelling systems in geometric control theory in a manner that does not make any choice of parameterisation by control; the systems are called "tautological control systems." For the framework to be coherent, it relies in a fundamental way on topologies for spaces of vector fields. As such, classes of systems are considered possessing a variety of degrees of regularity: finitely differentiable; Lipschitz; smooth; real analytic. In each case, explicit geometric seminorms are provided for the topologies of spaces of vector fields that enable straightforward descriptions of time-varying vector fields and control systems. As part of the development, theorems are proved for regular (including real analytic) dependence on initial conditions of flows of vector fields depending measurably on time. Classes of "ordinary" control systems are characterised that interact with the regularity under consideration in a comprehensive way. In this framework, for example, the statement that "a smooth or real analytic control-affine system is a smooth or real analytic control system" becomes a theorem. Correspondences between ordinary control systems and tautological control systems are carefully examined, and trajectory correspondence between the two classes is proved for control-affine systems and for systems with general control dependence when the control set is compact.