Ergodic Theory and Control Theory

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Ergodic Theory and Control Theory
Igor Mezic
Department of Mechanical and Environmental
Engineering, University of California, Santa
Barbara
Mohammed Dahleh Symposium, UCSB 2002
Sponsored by AFOSR, NSF, Sloan Foundation
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Introduction

• Control of Hamiltonian systems

Applications -molecular dynamics, vortex
dynamics, satellite
motion control, quantum control, power systems

• Volume-preserving systems on groups.

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Introduction
T
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Ergodicity and Controllability
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Ergodicity and Controllability
Observation 2 For Uw, w irrational, the
system is almost reachable. In
fact, for irrational w the system is
ergodic.
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Ergodicity and Controllability
Theorem Assume A is a compact abelian
topological group, U an input set of
positive Haar measure and g an element
of U such that gn is dense in G. Then
the system TU is controllable.
Remark The element g is an ergodic element. Its
presence in the input set is
necessary Let G-1,1 under multiplication.
Then U1 is an input set of measure ½ but 1 is
not an ergodic element, and the element -1 is NOT
reachable.
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Ergodicity and Controllability

• Stronger results are possible, where only
  dim(U)gt0 is necessary.

• Extensions for more general systems e.g. linear
  systems
• with input constraints continuous-time
  Hamiltonian systems
• vortex dynamics.

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Hamiltonian systems twist maps
Jim Swifts page http//odin.math.nau.edu/jws/st
d.map/
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Hamiltonian systems twist maps
By Zoran Levnajic (UCSB/Trieste).
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Hamiltonian systems twist maps
Kolmogorov-Arnold-Moser theorem implies
No controllability!
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Hamiltonian systems twist maps
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Discussion and Conclusions

• Controllability results for Hamiltonian and
• volume-preserving systems extensions for N DOF
• With D. Vainchtein control of (Hamiltonian)
• vortex dynamics using properties of nominal
  dynamics.
• Control of statistical properties conserved
  quantities.

• Linking ergodic theory and control theory
• 1. Control of mixing.
• 2. Nonlinear model validation.
• 3. Ergodicity and controllability.