On an Instationary Mixed ODE/PDE Optimal Control Problem with State-Constraints and Free End Time
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Transcript of On an Instationary Mixed ODE/PDE Optimal Control Problem with State-Constraints and Free End Time
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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
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Outline
• Motivation: Flight path optimization of hypersonic passenger jets
• The hypersonic rocket car problem
• Necessary conditions
• Numerical results
• Conclusion
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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
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The ODE-Part of the Model: The Rocket Car
minimum time control costs
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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
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The Optimal Trajectories (Regularized, Control Constrained)
distributed casestate unconstrained
space
time
space
time
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• 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
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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
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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
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Theoretical results (ODE formulations)
Integro-state constraint
Transformation
Integro-ODE
pointwise
corresponds toMaurer‘s intermediateadjoining approach
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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
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Theoretical results (PDE formulations)
non-standard
+ free terminal time
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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)
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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
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Numerical results: Direct Method (AMPL + IPOPT)
non-linearlinear
control is
(AD and a-posteriori verification of nec. cond.)
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Numerical results
touch point (TP) and boundary arc (BA)
time order 2
TP
TP
TP BA
BA
BA
TP
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Numerical results for boundary control problem
only boundary arc
BA
BA
BA
BA
BA
time order 1
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Numerical results: Verification
A posteriori verfication of optimality conditions:projection formula (ODE)
Method:Ampl + IPOPT
Ref.: IPOPTAndreas Wächter 2002
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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
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A posteriori verfication of optimality conditions:The PDE formulation: adjoint temperature
numericalartefacts
estimate from NLP solution by IPOPT
Numerical results: Verification
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is discontinous
A posteriori verfication of optimality conditions:comparison of adjoints (ODE + PDE)
Numerical results: Verification
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A posteriori verfication of optimality conditions:comparison of adjoints/jump conditions (ODE + PDE)
is discontinous
correct signsof jumps
Numerical results: Verification
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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
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Thank you for your attention!
Visit our homepage for further information
www.ingenieurmathematik.uni-bayreuth.de
Email: [email protected]
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