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New necessary conditions for a type of ODE-PDE constrained optimal control problems

New necessary conditions for a type of ODE-PDE constrained optimal control problems Hans Josef Pesch University of Bayreuth, Germany Joint work with: Armin Rund, Wolf von Wahl & Stefan Wendl 1st Workshop of the ESF-Network OPTPDE Warsaw, Dec. 11-12, 2008. Outline.

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New necessary conditions for a type of ODE-PDE constrained optimal control problems

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  1. New necessary conditions for a type of ODE-PDE constrained optimal control problems Hans Josef Pesch University of Bayreuth, Germany Joint work with: Armin Rund, Wolf von Wahl & Stefan Wendl 1st Workshop of the ESF-Network OPTPDE Warsaw, Dec. 11-12, 2008

  2. Outline • Motivation by a hypersonic trajectory optimization problem • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

  3. Introduction: Hypersonic Passenger Jets http://www.reactionengines.co.uk/lapcat_anim.html Project LAPCAT Reading Engines, UK

  4. homotopy parameter Model: Space Vehicle: Equations of Motion Two-dimensional flight over a great circle of a rotational Earth multiply control and state constrained ODE optimal control problem

  5. Model: Instationary Heat Constraint: Equations quasi-linear parabolic initial-boundary value problem with nonlinear boundary conditions

  6. Model: Instationary Heat Constraint: State Constraint State-constraint for the temperature: ODE-PDE state-constrained optimal control problem PDE: quasilinear parabolic with nonlinear bound. conds. CONTROL: distributed and boundary controls indirectly via ODE states and controls CONSTRAINT: state constraint

  7. Numerical Results: Stagnation Point: States, Heat Loads altitude [10,000 km] flight path angle [deg] velocity [m/s] [s] temperature [K] temperature [K] temperature [K] limit temperature 1000 K on a boundary arc order concept? 2nd layer 3rd layer 1st layer [s] jointly with K. Chudej, M. Wächter, G. Sachs, F. le Bras

  8. Outline • Motivation • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

  9. have nothing to do with the supersonic we are going with hypersonic speed The Hypersonic Rocket Car Problems Bloodhound-SSC-Projekt http://www.bloodhoundssc.com/

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

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

  12. 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

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

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

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

  16. Outline • Motivation • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

  17. Theoretical results • Thm. 1 (Problem 1 and 2): Existence, uniqueness for all , • and continuous dependence on data: Regularity (first result): • Thm. 2 (Problem 1): • Thm. 3 (Problem 1): • Thm. 4 (Problem 1): takes its strong maximum in for each increases / decreases strictly monotonic in

  18. Theoretical results (joined work with Wolf von Wahl) • Thm. 1 (Problem 1 and 2): Existence, uniqueness for all , • and continuous dependence on data: Regularity (first result): • Thm. 2 (Problem 1): • Thm. 3 (Problem 1): • Thm. 4 (Problem 1): takes its strong maximum in for each increases / decreases strictly monotonic in

  19. Theoretical results (joined work with Wolf von Wahl) • Thm. 1 (Problem 1 and 2): Existence, uniqueness for all , • and continuous dependence on data: Regularity (first result): • Thm. 2 (Problem 1): • Thm. 3 (Problem 1): • Thm. 4 (Problem 1): takes its strong maximum in for each increases / decreases strictly monotonic in

  20. Theoretical results (joined work with Wolf von Wahl) • Thm. 1 (Problem 1 and 2): Existence, uniqueness for all , • and continuous dependence on data: Regularity (first result): • Thm. 2 (Problem 1): • Thm. 3 (Problem 1): space time • Thm. 4 (Problem 1): takes its strong maximum in for each increases / decreases strictly monotonic in

  21. Theoretical results (joined work with Wolf von Wahl) • Thm. 5 (Problem 1): Problem 1 possesses a classical solution in satisfies in the strong sense in • Thm. 6 (Problem 1, maximum regularity)): For any and any space time

  22. Theoretical results (joined work with Wolf von Wahl) • Thm. 5 (Problem 1): Problem 1 possesses a classical solution in satisfies in the strong sense in • Thm. 6 (Problem 1, maximum regularity)): For any and any space time

  23. Only if Problem 2: After transformation to homogeneous boundary conditions boundary arcs space order Theoretical results (order concept for optimal control of PDEs) Problem 1: regular Hamiltonian touch points boundary arcs yields feedback laws for optimal controls on subarcs

  24. Outline • Motivation • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

  25. 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

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

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

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

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

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

  31. Theoretical results (existence of Lagrange multiplier)

  32. Necessary condition: integro optimal control law Theoretical results (PDE formulations, distributed control) Necessary conditions: adjoint equations extremely difficult to solve no standard software

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

  34. Ansatz for Lagrange multiplier (PDE formulations, distributed control) Construction of Lagrange multiplier (justified by analysis): with Dirac unit measure absolutely continuous w.r.t. the Lebesque measure with density function fit parameters, non-negative

  35. Derivation of jump condition: Weak adjoint equation: Derivation of jump condition: Going to the limit : Jump condition (PDE formulations, distributed control)

  36. 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):

  37. Outline • Motivation • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

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

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

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

  41. Numerical results (Type: Problem 2)

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

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

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

  45. 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

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

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

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

  49. numerical artefacts correct sign of jump numerical artefacts not yet obtained: quantitative verfication of jumps by the Ansatz for Numerical results (FOTD vs. FDTO) A posteriori verfication of optimality conditions: jump condition (PDE)

  50. Outline • Motivation • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

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