I discuss in this talk a framework, the framework of "tautological control systems" for addressing structural problems in geometric control theory. The tools presented in the talk are aimed at addressing such problems as controllability, stabilisability, and optimality, complete understandings of which have (somewhat strangely) eluded researchers for many years.
The talk will begin with an overview of why there is a need for an alternative framework for geometric control theory. The essential idea here is that of feedback-invariance, i.e., of a framework where the models do not depend on an explicit parameterisation of the control set, and where there is no dependence of dynamics on control in the usual manner. It will be seen that even the most elementary construction of nonlinear control theory - linearisation about an equilibrium point - is not feedback-invariant.
The remainder of the talk will be devoted to an outline of the main ideas of the framework, of which there are two. The first can be thought of as an abstraction of the "usual" notion of a control system, and relies on specifications of topologies for spaces of vector fields. These topologies were presented in this seminar last year by my student Saber Jafarpour. The other main idea of the tautological control system framework is more difficult to understand, and involves the use of sheaf theory. This will be sketched, and some results presented that, I hope, will illustrate the utility of the "sheaf" way of thinking about things.
The full presentation of these ideas tends to be quite technical. In this talk, I will focus the attention on the main ideas, and away from the (difficult but necessary) mathematical details.