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Bang-bang and Singular Controls in Optimal Control Problems with Partial Differential Equations

Bang-bang and Singular Controls in Optimal Control Problems with Partial Differential Equations Hans Josef Pesch, Simon Bechmann, Jan-Eric Wurst Chair of Mathematics in Engineering Sciences University of Bayreuth, Germany Hans-josef.pesch@uni-bayreuth.de. Outline. Intro: from ODE to PDE

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Bang-bang and Singular Controls in Optimal Control Problems with Partial Differential Equations

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  1. Bang-bang and Singular Controls in Optimal Control Problems with Partial Differential Equations Hans Josef Pesch, Simon Bechmann, Jan-Eric Wurst Chair of Mathematics in Engineering Sciences University of Bayreuth, Germany Hans-josef.pesch@uni-bayreuth.de

  2. Outline • Intro: from ODE to PDE • The elliptic van der Pol oscillator • A wave equations with a singular control • A direct postprocessing method • Outlock: An adjoint-based postprocessing method

  3. The van der Pol Oscillator (uncontrolled, limit cycle) Maurer: SADCO course 2011

  4. The van der Pol Oscillator (minimum time, minimum damping) Minimize subject to

  5. bang-bang singular The van der Pol Oscillator (minimum time, minimum damping) [Kaya, Noakes, Maurer, Vossen] bang-bang

  6. pseudo- PDE ellip. van der Pol The „damped“ „elliptic van der Pol Oscillator“

  7. The „damped“ „elliptic van der Pol Oscillator“: pseudo-PDE W S state computed by AMPL + IPOPT

  8. E bang - singular N E W S control + =0 zoom - W negative adjoint The „damped“ „elliptic van der Pol Oscillator“: pseudo-PDE feedback formula reduced regularity due to double initial conditions of state

  9. The „damped“ „elliptic van der Pol Oscillator“: E W S state

  10. The „damped“ „elliptic van der Pol Oscillator“: W E bang – bang - singular control with jumps as in ODE

  11. The „damped“ „elliptic van der Pol Oscillator“: singular region difference: zoom a posteriori verification of necessary conditions negative adjoint

  12. Wave equation with an unusual control constraint pointwise in time Kunisch, D. Wachsmuth

  13. Wave equation with a singular control (example 1) negative adjoint control state

  14. Wave equation with a singular control (example 2) negative adjoint state control

  15. Based on a partion of the domain with fixed toplogy and prescribed control laws on the interior of each subdomain Direct postprocessing step: definitions and assumptions feedback control

  16. matching of state variable Direct postprocessing step: idea optimization variable partition of fixed topology

  17. Semi-infinite shape optimization problem if the curve is parameterized appropriately Direct postprocessing step: Switching Curve Optimization Analogon to switching point optimization in ODE optimal control

  18. Direct postprocessing step: Switching Time Optimization

  19. Indirect postprocessing step: idea optimization variable partition of fixed topology

  20. Indirect postprocessing step: Multiple Domain Optimization shape optimization inner optimization Analogon to multipoint boundary value formulation in ODE optimal control

  21. Conclusion • A challenge in theory • First Discretize Then Optimize • Direct postprocessing possible • Indirect postprocessing: a challenge • Ref.: Karsten Theißen, PhD thesis, Maurer, 2006 • Our paper in the proceedings • Frederic Bonnans, Report, Oct. 2012 was done for state-constrained problems: Michael Frey, Diss. 2012

  22. Thank you very much for your attention

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