A Glimpse on Optimal Control of Partial Differential Equations: Theory, Numerics, and Applications

1. A Glimpse on Optimal Control of Partial Differential Equations: Theory, Numerics, and Applications Hans Josef Pesch Chair of Mathematics in Engineering Sciences University of Bayreuth, Bayreuth, Germany hans-josef.pesch@uni-bayreuth.de

2. multi-beam welding additional beams weld seam solidification hot crack mushy zone weld pool main laser beam compression Motivation: Optimal placement of laser beams to avoid hot cracking [Karkin, Ploshikin] Semi-infinite optimization problem for an elliptic PDE with state constraints

3. Motivation: Optimal placement of laser beams to avoid hot cracking [Petzet] isotherms surrounding the mushy zone limit hot crack criterium opening displacement zoom weld pool region

4. Motivation: Optimal load changes for fuel cell systems Molten Carbonate Fuel Cell cell stack

5. 1D counter-flow design Anode gas channel Air inlet CH4 H2O CH4 + H2O CO + 3H2 CO + H2O CO2 + H2 O2 N2 H2 +CO32-H2O+CO2+2e- CO+CO32-2CO2+2e- e- Catalytic burner Anode only ions can move throughelectrolyte Elektrolyte U CO32- Cathode Mixer Exhaust ½O2 + CO2 + 2e- CO32- Cathode gas channel Recirculation Motivation: Optimal load changes for fuel cell systems German Federal Pollution Control Act: Air

6. Motivation: Optimal load changes for fuel cell systems [Sundmacher] [Heidebrecht] 2D cross-flow design exhaust air inlet recirculation cathode exhaust anode exhaust catalytic burner CO32- mixer anode solid anode inlet cathode cathode inlet 28 semi-linear partial integro-differential equations with non-standard non-linear boundary conditions

7. 0.7 0.6 for a load change cell voltage using controls optimal control simulation 0.4 sec 0 0.1 1.1 11.1 111.1 1111.1 scaled time using control 0.8 sec optimal control simulation 0 0.1 1.1 11.1 111.1 1111.1 scaled time Motivation: Optimal placement of laser beams to avoid hot cracking [Sternberg]

8. PDE ODE Motivation: Minimum fuel transcontinental flights at hypersonic speeds Europe - USA in 2 hrs / Europe - Australia in 4.5 hrs quasilinear heat equation non-linear boundary conditions coupled with ODE 2 box constraints 1 control-state constraint 1 state constraint

9. Motivation: Minimum fuel transcontinental flights at hypersonic speeds [Wächter, Chudej, LeBras] altitude [10,000 m] flight path angle [deg] velocity [m/s] [s] temperature [K] temperature [K] temperature [K] limit temperature 1000 K on a boundary arc 2nd layer 3rd layer 1st layer [s]

10. Outline A glimpse on the theory A glimpse on the numerics An application Conclusions

11. Outline A glimpse on the theory A glimpse on the numerics An application Conclusions

12. set of admissible controls tracking functional Tikhonov regularization Elliptic optimal control problem with distributed control A simple elliptic optimal control problems Lions (since 1970s), Casas (1987-), Tröltzsch (1980-) An example: optimal stationary temperature distribution subject to

13. set of admissible controls tracking functional Tikhonov regularization Elliptic optimal control problem with boundary control A simple elliptic optimal control problems An example: optimal stationary temperature distribution subject to

14. An example: Optimal stationary temperature distribution Optimization problem in Hilbert space Necessary condition: variational inequality Elliptic optimal control problem with distributed control Necessary conditions subject to with linear and continuous solution operator

15. Optimization problem in Hilbert space Necessary condition: variational inequality Necessary conditions Descripton with the adjoint solution operator

16. Description with the adjoint state Necessary conditions Description with the adjoint solution operator

17. Optimality system: semi-linear elliptic, distributed + boundary control

18. Functional Analysis Partial Differential Equations Optimization in Banach spaces Numerical Methods of Linear Algebra Parallel Numerical Methods Numerics of PDE Numerical Methods of Optimization High Performance Scientific Computing Challenges in PDE constrained optimization Optimal Control of PDE

19. Outline A glimpse on the theory A glimpse on the numerics An application Conclusions

20. effort of optimization small constant effort of simulation Methods for PDE constrained optimization The general problem The aims concepts for real-life application

21. appropriate choice of and ansatz for ? appropriate choice of and ansatz for ? First Discretize then Optimize vs. First Optimize then Discretize First discretize then optimze (fDtO) Solve large scale NLP capture as much structure of ( P) as possible on discrete level ( Ph ) First optimze then discretize(fOtD) Solve coupled PDE system Questions appropriate ansatz for adjoint variables and multipliers?

22. First Discretize then Optimize vs. First Optimize then Discretize First discretize then optimze (fDtO): replace all quantities of the infinite dimensional optimization problem by finite dimensional substitutes and solve an NLP First optimze then discretize(fOtD): Derive optimality conditions of the infinite dimensional system, discretize the optimality system and find solution of the discretized optimality system In general Ideal: discrete concept for which both approaches commute Discontinuous Galerkin methods

23. Mathematical Toolbox (incomplete list) • Structure of optimality system allows one-shot-iterations • Griewank, Schulz • Constraints require non-smooth solution techniquesIto, Hintermüller, Kunisch, M. Ulbrich • Structure of optimality system allows multigrid methods • Borzi, Schulz • Structure of optimality system allows taylored discrete concepts • Hinze, Meyer, Rösch • Relaxation of constraints by penalty or barrier methods • Hintermüller, Kunisch, Schiela • State constraints: set optimal control problem with shape calculus • Frey, Bechmann, Pesch, Rund • Adaptive algorithms • Becker, Rannacher; Vexler; Hintermüller, Hoppe; Hinze, Günther; et.al. • Surrogate models for the PDE system in the optimality system • Hinze et.al., Sachs et.al., Kunisch, Tröltzsch, S. Ulbrich, Volkwein • Shape calculus for shape optimization • Sokolowski, Zolesio; Gauger, Schulz; Hintermüller, Ring; M. Ulbrich, S. Ulbrich • Automatic differentiation provides adjoints • Griewank, Walther

24. Outline A glimpse on the theory A glimpse on the numerics An application optimal control of a molten carbonate fuel cell process control via model reduction techniques Conclusions

25. Configuration and function ofMCFC 2D cross-flow design [Heidebrecht] [Sundmacher] controllable exhaust air inlet slow state variable recirculation fast very fast cathode exhaust anode exhaust algebraic controllable catalytic burner load changes input mixer anode solid anode inlet cathode boundary conditions by ODAE cathode inlet controllable

26. Numerical results: simulation of load change [Chudej, Sternberg] cathode gas temperature anode gas temperature [2.8 ≈ 560 °C] flow directions [3.2 ≈ 680 °C] reforming reactions are endothermic oxidation reaction is exothermic reduction reaction is exothermic

27. Numerical results: simulation of load change [Chudej, Sternberg] solid temperature [3.2 ≈ 680 °C] [2.8 ≈ 560 °C] flow directions in anode and cathode

28. state constraint would be desirable Numerical results: simulation of load change solid temperature [3.2 ≈ 680 °C] [2.8 ≈ 560 °C]

29. fast 0.7 0.6 slow on on Numerical results: optimal control of fast load changewhile temperature gradients stay small Pareto performance index: instead of state constraint with

30. Aim for process control How to apply optimal solutions in practise?

31. measurable: cell voltage, gas temperatures and concentrations at anode and cathode outlet diserable for process control: information on spatial temperatur and concentration profiles solution ansatz: observer / state estimator Problem: complexity of model Aim for process control ? ?? Remedy: model reduction technique

32. 2002-2005 Model reduction by POD (proper orthogonal decomposition) or Karhunen-Loève decomposition (K.: 1946, L: 1955, Lumley: 1967,…) Demands on model reduction techniques • good accuracy for a wide range of operation conditions • suitable for describing the nonlinear behavior of the cell • ability for extrapolation in case of varying parameter German Industrial Partners: CFC Solutions GmbH, offspring of MTU, Munich; IPF Berndt KG, Reilingen, constructor and operator of power plants

33. Ansatz (separation of variables): orthogonal snapshots Method of weighted residuals: low order model: ODAE of index 1 Reduced model: Model reduction by POD: idea Complete model:

34. test signal 1. temperature basis function 2. temperature basis function Model reduction by POD: computation of snapshots by the complete model orthogonalization by singular value decomposition

35. Model reduction by POD: comparison of reduced vs. complete model [Mangold, Sheng] random variation of cell current #eqs. 4759 vs. 131 3200 sec vs. 82 sec 2 < N < 10 perfect coincidence with reference model appropriate for process control

36. complete temperature reduced voltage Model reduction by POD: comparison of reduced vs. complete model example: response to changes of the steam-to-carbon ratio in the feed steam-to-carbon ratio

37. MCFC sensors ? input state measurement process observer correction + - Simulator observer sensor models MCFC model Scheme for state estimator for discrete measurements

38. Temperature control at Hotmodule: nonlinear feed forward controller + PID controller [Sheng et al] - PID controller 1 feed forward controller MCFC system PID controller 2 - state estimator state estimator

39. Temperature control at Hotmodule: nonlinear feed forward controller + PID controller feed forward controller only feed forward controller + PID controller significantly better process behaviour

40. References Focus on Theory: Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications AMS, Graduate Studies in Mathematics, Vol. 112, 2010. Focus on Methods: Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints Mathematical Modelling: Theorie and Applications, Vol. 23, 2008. Focus on Applications: See my homepage: google: Hans Josef Pesch

41. Conclusions Concerning theory: already well developed Concerning numerics: still improving Concerning applications: has to be intensified one always abuts against limits

42. Thank you for your attention