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

Hans-josef.pesch@uni-bayreuth.de

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