Peter Vassiliou (Australian National University) Symmetry and Geometric Control Theory Sponsored by the Meyer Fund
Thu, Jul. 19 10am (MATH 350)
Geometry/Analysis
Peter Vassiliou (Australian National University) Hyperbolic Reduction of Partial Differential Equations 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.
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In previous centuries mathematicians sought solutions of differential equations in formula as a matter of course. This approach to the study of differential systems has fallen out of fashion (and possibly favor) as a result of the vast revolutions in analysis and the types of problems requiring solution in the current epoch. Nevertheless a tremendous heritage in this area has been bequeathed to us by previous generations which we have begun to explore only in the last 4 decades or so. This resurgence has produced a fascinating body of mathematics and its applications, such solitons, Poisson geometry, infinite dimensional hamiltonian systems, Backlund transformations; to name a few. One such 19th century topic, Darboux integrability, has been the subject of a number of recent studies which reveal it to have strong geometric content and utility for constructing explicit solutions in formula of certain partial differential equations. However, because Darboux integrability is a comparatively rare phenomenon it is natural to ask if some weaker form of it may exist so as to be of wider applicability while retaining some useful aspects of the original. In this talk I will briefly recall the basic ideas of this form of integrability and then propose a weaked version which will be explored via a few elementary examples. Time permitting we will also explore its application to certain harmonic maps.