Peter Vassiliou (Australian National University) Symmetry and Geometric Control Theory Sponsored by the Meyer Fund
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Control systems, typically nonlinear, abound in our daily experience of the world. Mathematically, they can be described by differential equations endowed with functional parameters. The prototypical example is balancing a broomstick on the tip of your finger. A basic problem in control theory is the following. Suppose a control system $\bsy{\o}$ is in state $A$ and you want to ``steer" it to state $B$. How do you choose the (functional) parameters in the system -- the controls -- in order to achieve this? One oft-used technique is to find a special type of transformation, $T$, that maps $\bsy{\o}$ to a certain type of universal linear control system called a Brunovsky normal form, $\bsy{\b}$. The trajectories of $\bsy{\b}$ can easily be written down in terms of arbitrary functions of time and their time derivatives after which $T}^{-1$ maps the trajectories of $\bsy{\b}$ to those of $\bsy{\o}$, from which the steering problem can be solved. If we can construct such a solution, $s$ of $\o$, we will say that it is {\em explicitly integrable}. This settles the steering problem for linearizable control systems which form a small subset of control systems of interest. But while every linearizable system is explicitly integrable, the converse is false and for the past few decades there has been a research program underway to characterize the phenomenon of explicit integrability beyond the linearizable case. However it turns out that the Brunovsky normal forms are familiar objects in differential geometry. They correspond to the contact systems on a special class of jet spaces over the real line. In this talk we will explore this geometry to show how to establish explicit integrability for control systems which are intrinsically nonlinear - those that have no linearizing diffeomorphisms. A key ingredient is the invariance of $\bsy{\o}$ under the free and regular action of a Lie group.