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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. Outline. - PowerPoint PPT Presentation
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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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