Bang-bang and Singular Controls in Optimal Control Problems with Partial Differential Equations
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Transcript of Bang-bang and Singular Controls in Optimal Control Problems with Partial Differential Equations
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
Bang-bang and Singular Controlsin Optimal Control Problems
with Partial Differential Equations
Hans Josef Pesch, Simon Bechmann, Jan-Eric Wurst
Chair of Mathematics in Engineering SciencesUniversity of Bayreuth, Germany
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
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
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
The van der Pol Oscillator (uncontrolled, limit cycle)
Maurer: SADCO course 2011
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
The van der Pol Oscillator (minimum time, minimum damping)
Minimize
subject to
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
The van der Pol Oscillator (minimum time, minimum damping)
bang-bang
bang-bangsingular
[Kaya, Noakes, Maurer, Vossen]
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
The „damped“ „elliptic van der Pol Oscillator“
pseudo-PDE
ellip.van der Pol
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
The „damped“ „elliptic van der Pol Oscillator“: pseudo-PDE
WS
state
computed by AMPL + IPOPT
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
The „damped“ „elliptic van der Pol Oscillator“: pseudo-PDE
control
W
S
Nbang - singular
E
E
W
negative
adjoint
zoom-
+=0
reduced regularitydue to double initialconditions of state
feedback formula
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
WE
state
The „damped“ „elliptic van der Pol Oscillator“:
S
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
The „damped“ „elliptic van der Pol Oscillator“:
W E
controlwith jumps as in ODE
bang – bang - singular
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
The „damped“ „elliptic van der Pol Oscillator“:
difference:
negative
adjoint
zoom
singular region
a posteriori verificationof necessary conditions
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
Kunisch, D. Wachsmuth
Wave equation with an unusual control constraint pointwise in time
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
negative
adjoint controlstate
Wave equation with a singular control (example 1)
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
negative
adjoint controlstate
Wave equation with a singular control (example 2)
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
Direct postprocessing step: definitions and assumptions
and prescribed control laws on the interior of each subdomain
Based on a partion of the domain with fixed toplogy
feedbackcontrol
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
Direct postprocessing step: idea
optimization variable
partition of fixed topology
matching of state variable
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
Direct postprocessing step: Switching Curve Optimization
Analogon to switching point optimization in ODE optimal control
Semi-infinite shape optimization problemif the curve is parameterized appropriately
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
Direct postprocessing step: Switching Time Optimization
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
Indirect postprocessing step: idea
optimization variable
partition of fixed topology
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
Indirect postprocessing step: Multiple Domain Optimization
Analogon to multipoint boundary value formulation in ODE optimal control
inner optimization
shape optimization
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.
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
51st IEEE Conference on Decision and Control,Dec. 10-13, 2012, Grand Wailea, Maui, Hawaii, USA.Thank you very much for your attention