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Mathematical models for geometric control theory

Jafarpour, Saber; Lewis, Andrew D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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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...