Integrability, Ergodicity, Controllability

Prof. Igor Mezic, University of California, Santa Barbara, Department of Mechanical and Environmental Engineering and Mathematics

In this talk I will present links between the control-theoretic concept of controllability and dynamical systems concepts of integrability and ergodicity. The links comes from studying some elements of the theory of control of mixing. This is, in fact, the control theory of measure-preserving systems with specific objectives, typically coming from ergodic theoretic considerations. The mixing objectives such as partition and Kolmogorov-Sinai entropy will be considered and a simple prototypical example of optimal control solved. Some results on controllability of vortex systems will be shown using flat coordinates, showing the use of the concept of integrability in achieving controllability. Finally, a contrast will be made between these results on controllability and a new concept of controllability via ergodicity. A result questioning robustness of the KAM-type results in the context of perturbations with arbitrary time-dependence will be presented using such ergodicity-controllability link.