Locally convex topologies and control theory

Using recent characterisations of topologies of spaces of vector fields for general regularity classes—e.g., Lipschitz, finitely differentiable, smooth, and real analytic—characterisations are provided of geometric control systems that utilise these topologies. These characterisations can be expressed as joint regularity properties of the system as a function of state and control. It is shown that the common characterisations of control systems in terms of their joint dependence on state and control are, in fact, representations of the fact that the natural mapping from the control set to the space of vector fields is continuous. The classes of control systems defined are new, even in the smooth category. However, in the real analytic category, the class of systems defined is new and deep. What are called “real analytic control systems” in this article incorporate the real analytic topology in a way that has hitherto been unexplored. Using this structure, it is proved, for example, that the trajectories of a real analytic control system corresponding to a fixed open-loop control depend on initial condition in a real analytic manner. It is also proved that control-affine systems always have the appropriate joint dependence on state and control. This shows, for example, that the trajectories of a control-affine system corresponding to a fixed open-loop control depend on initial condition in the manner prescribed by the regularity of the vector fields.