Optimal Control of Coupled Systems of ODEs and PDEs with Applications to Hypersonic Flight

1. Optimal Control of Coupled Systems of ODEs and PDEs with Applications to Hypersonic Flight Part 2 Hans Josef Pesch University of Bayreuth, Germany Part 2: Armin Rund, Wolf von Wahl & Stefan Wendl The 8th International Conference on Optimization: Techniques and Applications (ICOTA 8), Shanghai, China, Dec. 10-14, 2010

2. Outline • Introduction/Motivation • The hypersonic trajectory optimization problem • The instationary heat constraint • Numerical results • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

3. have nothing to do with the supersonic we are going with hypersonic speed The Hypersonic Rocket Car Problems Bloodhound-SSC-Projekt 1997, Oct. 15: Thrust SSC officially 1.228 km/h less than Ma 1=1.234,8 km/h at 20°C Aim: 1.000 m/h, hence 1.609 km/h, faster than a speeding bullet. http://www.bloodhoundssc.com/

4. The ODE-Part of the Model: The Rocket Car minimum time control costs

5. Problem 1 The PDE-Part of the Model: The Distributed Control Case friction term control via ODE state instationary heating of the entire vehicle

6. Problem 2 The PDE-Part of the Model: The Boundary Control Case Transformation to homogeneous Robin type b.c. control via ODE state friction term instationary heating at the stagnation point

7. The State Constraint PDE The state constraint regenerates the PDE with the ODE ODE

8. Problem 1 switching curve space time The Optimal Trajectories (Non-regularized, Minimum Time) distributed case state unconstrained

9. Problem 1 space space time time The Optimal Trajectories (Regularized, Control Constrained) distributed case state unconstrained

10. Problem 2 Boundary Control Case 1.4 3.0 7.0 space space space time time time Coulomb Stokes Newton

11. Outline • Introduction/Motivation • The hypersonic trajectory optimization problem • The instationary heat constraint • Numerical results • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

12. Existence, uniqueness, and continuous dependence on data Problem 1 • Non-negativity of • Symmetry • Strong maximum in • Classical solution time space • Maximum regularity Theoretical results for Problem 1

13. Existence, uniqueness, and continuous dependence on data Problem 2 • Non-negativity of • Continuity of • Global maximum on time space Theoretical results for Problem 2

14. loss of convergence if differentiated non-standard 2) as PDE optimal control problem plus two isoperimetric constraints on due two ODE boundary conds. Theoretical results (two formulations) Problem 1: Two equivalent formulations 1) as ODE optimal control problem non-local, resp. integro-state constraint

15. Problem 1 Transformation Integro-ODE pointwise Theoretical results (ODE formulations, distributed control) Integro-state constraint corresponds to Maurer‘s intermediate adjoining approach

16. Theoretical results (ODE formulations, distributed control) Necessary conditions: optimal control law

17. Usual jump conditions for adjoint auxiliary state discontinouities complementarity condition Theoretical results (ODE formulations, distributed control) Necessary conditions: adjoint equations Retrograde integro-ODE for the adjoint velocity difficult to solve no standard software

18. Problem 1 Theoretical results (PDE formulations, distributed control)

19. one obtains By the continuously differentiable solution operator subject to with the convex cone Theoretical results (optimization problem in Banach space)

20. Theoretical results (existence of Lagrange multiplier)

21. Necessary condition: integro optimal control law Theoretical results (PDE formulations, distributed control) Necessary conditions: adjoint equations , but so far all seems to be standard extremely difficult to solve no standard software

22. discont. deriv. / jump jump Theoretical results (necessary conditions: ODE vs. PDE) By comparing the two optimal control laws

23. Theoretical results (necessary conditions) The two formulations allow the comparison of the necessary conditions: ODE version • known theory • local jump conditions for ODE formulation • multipoint boundary value problem for integro-ODEs • existence of Lagrange parameter • non-local jump conditions for PDE formulation • projection formula for control with integro-terms PDE version

24. Outline • Introduction/Motivation • The hypersonic trajectory optimization problem • The instationary heat constraint • Numerical results • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

25. Jump condition (PDE formulations, distributed control) Jump condition in the direction of (equivalent to ODE case): Jump condition in the direction of (no counterpart in ODE case):

26. Outline • Introduction/Motivation • The hypersonic trajectory optimization problem • The instationary heat constraint • Numerical results • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

27. Numerical results (Type: Problem 1) control is non-linear linear

28. Numerical results (Type: Problem 1) bang bang

29. Numerical results (Type: Problem 1) time order 2 BA BA TP BA TP BA touch point (TP) and boundary arc (BA)

30. Numerical results (Type: Problem 2)

31. Numerical results (Type: Problem 2) time order 1 BA BA BA BA BA only boundary arc

32. Numerical results (Type: Problem 2) time order 1 two boundary arcs – typical for order 1

33. Numerical results (FOTD vs. FDTO) A posteriori verfication of optimality conditions: projection formula (ODE) Method: Ampl + IPOPT Ref.: IPOPT Andreas Wächter 2002

34. essential singularities non-local jump cond. in the energy non-local jump cond. in the energy jump in except on the set of active constraint Numerical results (FOTD vs. FDTO) A posteriori verfication of optimality conditions: The PDE formulation: adjoint temperature solution by method of lines

35. Numerical results (FOTD vs. FDTO) A posteriori verfication of optimality conditions: The PDE formulation: adjoint temperature numerical artefacts estimate by IPOPT

36. is discontinous Numerical results (FOTD vs. FDTO) A posteriori verfication of optimality conditions: comparison of adjoints (ODE + PDE)

37. correct signs of jumps is discontinous Numerical results (FOTD vs. FDTO) A posteriori verfication of optimality conditions: comparison of adjoints/jump conditions (ODE + PDE)

38. Outline • Introduction/Motivation • The hypersonic trajectory optimization problem • The instationary heat constraint • Numerical results • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

39. Conclusions (hypersonic aircraft problem) Pros: • Detailed model of high complexity for instationary heating • Reduction of temperature of TPS due to optimal control • Challenging problem: ODE - PDE state - constrained optimal control Cons: • Lack of theory: nonlinear, state - constrained, ODE-PDE coupling • Method of lines and SQP methods seem to be at their limits • Adjoint based methods desirable, but almost impossible to handle

40. Conclusions (hypersonic rocket car problems) • Staggered optimal control problems with state constraints • Structural analysis w.r.t. switching structure • Unexpectedly complicated necessary conditions • Problems with free terminal time for PDE problems • Jump conditions in Integro-ODE and PDE optimal control • First optimize, then discretize hardly applicable • First discretize, then optimize with reliable verification of necessary conditions, but with limitations in time and storage • Motivation from hypersonic flight path optimization