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

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 OptimizationTrier, June 3-5, 2009

Outline

• Motivation: Flight path optimization of hypersonic passenger jets

• The hypersonic rocket car problem

• Necessary conditions

• Numerical results

• Conclusion

Motivation: Hypersonic Passenger Jets

Project LAPCATReaction Engines, UK

ODE

PDE

2 box constraints1 control-state constraint1 state constraint

quasilinear PDEnon-linear boundary conditionsboth coupled with ODE

The ODE-Part of the Model: The Rocket Car

minimum time control costs

The PDE-Part of the Model: Heating of the Entire Vehicle

for boundary control cf. [Pesch, R., v. Wahl, Wendl]

control via ODE state

friction term

The state constraintregenerates

the PDE with the ODE

The Optimal Trajectories (Regularized, Control Constrained)

distributed casestate unconstrained

space

time

space

time

• Existence, uniqueness, and continuous dependence on data

• Symmetry

• Strong maximum in

spacetime

Theoretical results: jointly with Wolf von Wahl

• Classical solution

• Non-negativity of

• Maximum regularity

Theoretical results (order concept w.r.t. the ODE/PDE)

yields feedback laws for optimal controls on subarcs

[boundary control: order 1, only boundary arcs]

touch pointsboundary arcs

Only if

regular Hamiltonian

space order with respect to the PDE touch pointsboundary arcs

Theoretical results (two formulations)

Two equivalent formulations

1) as ODE optimal control problem

non-local, resp. integro-state constraint

2) as PDE optimal control problem

plus two isoperimetric constraints on due two ODE boundary conds.

Solution formula for T byseparation of variablesand series expansion

non-standard

Theoretical results (ODE formulations)

Integro-state constraint

Transformation

Integro-ODE

pointwise

corresponds toMaurer‘s intermediateadjoining approach

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 solveno standard software

Theoretical results (PDE formulations)

non-standard

+ free terminal time

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

Theoretical results (PDE formulations, distributed control)

Necessary conditions: adjoint equations

Necessary condition: integro optimal control law

extremely difficult to solveno standard software

Theoretical results (PDE formulations, distributed control)

so far all seems to be standard , but

Numerical results: Direct Method (AMPL + IPOPT)

non-linearlinear

control is

(AD and a-posteriori verification of nec. cond.)

Numerical results

touch point (TP) and boundary arc (BA)

time order 2

TP

TP

TP BA

BA

BA

TP

Numerical results for boundary control problem

only boundary arc

BA

BA

BA

BA

BA

time order 1

Numerical results: Verification

A posteriori verfication of optimality conditions:projection formula (ODE)

Method:Ampl + IPOPT

Ref.: IPOPTAndreas Wächter 2002

solution of IBVP by method of lines

essential singularities: jump in

except on the set of active constraint

Ansatz for Lagrange multiplier and jump conditions

Construction of Lagrange multiplier (justified by analysis):

jump in

A posteriori verfication of optimality conditions:The PDE formulation: adjoint temperature

numericalartefacts

estimate from NLP solution by IPOPT

Numerical results: Verification

is discontinous

A posteriori verfication of optimality conditions:comparison of adjoints (ODE + PDE)

Numerical results: Verification

A posteriori verfication of optimality conditions:comparison of adjoints/jump conditions (ODE + PDE)

is discontinous

correct signsof jumps

Numerical results: Verification

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

Thank you for your attention!

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Email: armin.rund@uni-bayreuth.de