On an Instationary Mixed ODE/PDE Optimal Control Problem with State-Constraints and Free End Time

1. On an Instationary Mixed ODE/PDE Optimal Control Problem with State-Constraints and Free End Time Armin Rund University of Bayreuth, Germany jointly with Hans Josef Pesch & Stefan Wendl Workshop on PDE Constrained Optimization Trier, June 3-5, 2009

2. Outline • Motivation: Flight path optimization of hypersonic • passenger jets • The hypersonic rocket car problem • Necessary conditions • Numerical results • Conclusion

3. PDE ODE Motivation: Hypersonic Passenger Jets quasilinear PDE non-linear boundary conditions both coupled with ODE 2 box constraints 1 control-state constraint 1 state constraint Project LAPCAT Reaction Engines, UK

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

5. The PDE-Part of the Model: Heating of the Entire Vehicle friction term control via ODE state The state constraint regenerates the PDE with the ODE for boundary control cf. [Pesch, R., v. Wahl, Wendl]

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

7. Existence, uniqueness, and continuous dependence on data • Non-negativity of • Symmetry • Classical solution time space • Maximum regularity Theoretical results: jointly with Wolf von Wahl • Strong maximum in

8. Only if regular Hamiltonian Theoretical results (order concept w.r.t. the ODE/PDE) touch points boundary arcs yields feedback laws for optimal controls on subarcs [boundary control: order 1, only boundary arcs] space order with respect to the PDE touch points boundary arcs

9. non-standard 2) as PDE optimal control problem plus two isoperimetric constraints on due two ODE boundary conds. Theoretical results (two formulations) Solution formula for T by separation of variables and series expansion Two equivalent formulations 1) as ODE optimal control problem non-local, resp. integro-state constraint

10. Transformation Integro-ODE pointwise Theoretical results (ODE formulations) Integro-state constraint corresponds to Maurer‘s intermediate adjoining approach

11. Theoretical results (ODE formulations) Lagrangian and necessary conditions → Standard adjoint ODEs, projection formula, jump conditions and complementarity conditions, but: Retrograde integro-ODE for the adjoint velocity difficult to solve no standard software

12. Theoretical results (PDE formulations) non-standard + free terminal time

13. Theoretical results (PDE formulations, distributed control) • We follow the well-known proceeding: • Frechet-differentiability of the solution operator • Formulation of optimization problem in Banach Space • Existence of Lagrange multiplier for the state constraint • → Lagrange-Formalism

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

15. control is non-linear linear Numerical results: Direct Method (AMPL + IPOPT) (AD and a-posteriori verification of nec. cond.)

16. Numerical results time order 2 TP BA TP TP BA TP BA touch point (TP) and boundary arc (BA)

17. Numerical results for boundary control problem time order 1 BA BA BA BA BA only boundary arc

18. Numerical results: Verification A posteriori verfication of optimality conditions: projection formula (ODE) Method: Ampl + IPOPT Ref.: IPOPT Andreas Wächter 2002

19. essential singularities: jump in jump in except on the set of active constraint Ansatz for Lagrange multiplier and jump conditions Construction of Lagrange multiplier (justified by analysis): solution of IBVP by method of lines

20. Numerical results: Verification A posteriori verfication of optimality conditions: The PDE formulation: adjoint temperature numerical artefacts estimate from NLP solution by IPOPT

21. is discontinous Numerical results: Verification A posteriori verfication of optimality conditions: comparison of adjoints (ODE + PDE)

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

23. Conclusions • Staggered optimal control problems with state constraints motivated from hypersonic flight path optimization • Prototype problem with unexpectedly complicated necessary conditions • Discussion from ODE or PDE point of view possible • → Comparison and transfer of concepts possible. • Structural analysis w.r.t. switching structure • Jump conditions in Integro-ODE and PDE optimal control, • free terminal time • First discretize, then optimize with reliable verification of necessary conditions, but with limitations in time and storage

24. Thank you for your attention! Visit our homepage for further information www.ingenieurmathematik.uni-bayreuth.de Email: armin.rund@uni-bayreuth.de