multiple-boundary-value-problem formulation for pde constrained optimal control problems
DESCRIPTION
Multiple-Boundary-Value-Problem Formulation for PDE constrained Optimal Control Problems with a Short History on Multiple Shooting for ODEs Hans Josef Pesch Chair of Mathematics in Engineering Sciences University of Bayreuth, Germany [email protected]. Outline. - PowerPoint PPT PresentationTRANSCRIPT
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Multiple-Boundary-Value-Problem Formulationfor PDE constrained Optimal Control Problems
with a Short History on Multiple Shooting for ODEs
Hans Josef PeschChair of Mathematics in Engineering Sciences
University of Bayreuth, Germany
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Outline
• A short history on multiple shooting
• Multipoint-boundary-value-problem formulation
• A state constrained elliptic problem
• A state constrained parabolic PDE-ODE problem • A singular hyperbolic optimal control problem
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Outline
• A short history on multiple shooting
• Multipoint-boundary-value-problem formulation
• A state constrained elliptic problem
• A state constrained parabolic PDE-ODE problem • A singular hyperbolic optimal control problem
![Page 4: Multiple-Boundary-Value-Problem Formulation for PDE constrained Optimal Control Problems](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814ab1550346895db7c431/html5/thumbnails/4.jpg)
Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The not Well-known Stone Age of Multiple Shooting
Engineers: Morrison, Riley, Zancanaro (1962)
Multiple Shooting Method for Two-Point Boundary Value Problems,Communications of the ACM, 1962, pp. 613 - 614.
One serious shortcoming of shooting becomes apparent when, as happens altogether too often, the differential equations are so unstablethat they „blow up“ before the initial value problem can be completely integrated.This can occur even in the face of extremely accurate guesses for the initial values. Hence, shooting seems to offer no hope for some problems. A finite difference method does have a chance for it tends to keep a firm hold on the entire solutionat once. The purpose of this note is to point out a compromising procedurewhich endows shooting-type methods with this particular advantage of finite difference methods. For such problems, then, all hope need not be abandoned for shooting methods. This is desirable because shooting methods are generally faster than finite difference methods.
Parallel shooting on equidistant intervals
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The Pioneers
Keller, Osborne (1968,69): first analysis
Bulirsch, Stoer (1971,73): first algorithmic realisation
Concept and first analysis of multiple shooting and parallel shooting
First code (1968): BOUNDSOL: nonlinear boundary value problemsSecond code (1970): OPTSOL: optimal control problems with inequality constraints
Bulirsch coined the term Mehrzielmethode
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The Followers
Deuflhard (1974,75): improved Newton method (DLOPTR)
Oberle (1977,83): multipoint bvps (BOUNDSCO)
Bock (1984): direct multiple shooting (MUSCOD)
Various error normsAlmost singular coefficient matrixImproved relaxation strategy
Improved robustness due to multipoint boundary value formulationReduced condition number by eliminating condensation
First-discretize-then-optimize code with multiple shooting
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Outline
• A short history on multiple shooting
• Multipoint-boundary-value-problem formulation
• A state constrained elliptic problem
• A state constrained parabolic PDE-ODE problem • A singular hyperbolic optimal control problem
![Page 8: Multiple-Boundary-Value-Problem Formulation for PDE constrained Optimal Control Problems](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814ab1550346895db7c431/html5/thumbnails/8.jpg)
Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Abort Landing in a Wind Shear
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Maximal Minimum Altitude Optimal Solutionfor Different Wind Profiles
Montrone, P. 1991, Berkmann, P. 1995
max!
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Maximal Minimum Altitude Optimal Solution
bangsingular3rd order state constr
1st order state constr
first optimizethen discretize
byindirect
multiple shooting
control versus time: rate of angle of attack
altitude versus rangeof abort landing
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
A Very Complicated Switching Structure
3 bang-bang subarcs
2 singular subarcs
1 boundary subarc of a 1st order state constraint
1 boundary subarc of a 3rd order state constraint
1 touch point of a 3rd order state constraint
switching structure (7 pts, 12 add. var.):number of interiorboundary conditions:
4
2
6
plus 5 additional interior boundary conditions due to modellingplus 11 usual boundary conditions given or by optimality conditions
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Outline
• A short history on multiple shooting
• Multipoint-boundary-value-problem formulation
• A state constrained elliptic problem jointly with Michael Frey, Simon Bechmann & Armin Rund
• A state constrained parabolic PDE-ODE problem • A singular hyperbolic optimal control problem
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Model Problem: elliptic, distributed control, state constraint
Minimize
subject to
with
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Definition of active set and assumptions
Definition: active / inactive set / interface
Assumptionon addmissbleactive sets
No degeneracy.No active set
of zero measure.No common points
with boundary
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Reformulation of the state constraint
Transfering the Bryson-Denham-Dreyfus approach
Using the state equation
Optimal solution on given by data, but optimization variable
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Reformulation as set optimal control problem
Minimize
subject to
a posteriori check
inner outer
topology is assumed to be known
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Theorem:
For each admissible the objective is shape differentiable. The semi-derivative in the direction
is
Optimality system in the inner optimization of a bilevel problem
subject to the optimality system of the inner optimization problem
determines the interface
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The Smiley example: rational initial guess
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The Smiley example: bad initial guess
Algorithm can cope with topology changes to some extent
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Outline
• A short history on multiple shooting
• Multipoint-boundary-value-problem formulation
• A state constrained elliptic problem
• A state constrained parabolic PDE-ODE problem jointly with Armin Rund
• A singular hyperbolic optimal control problem
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Motivation: Super-Concorde - Hypersonic Passenger Jet
Project LAPCATReading Engines, UK
ODE
PDE
2 box constraints1 control-state constraint1 state constraint
quasilinear PDEnon-linear boundary conditionscoupled with ODE
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The Hypersonic Rocket Car Problem: The ODE Part
minimum time control costs
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The PDE-Part of the Model: The PDE Part
friction term
instationary heating of the entire vehicle
control via ODE state
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The State Constraint
ODE
PDE
The state constraintregenerates
the PDE with the ODE
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Numerical results
non-linearlinear
control is
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Numerical results (topology and order of state constraint)
touch point (TP) and boundary arc (BA)
time order 2
TP
TP
BA BA
BA
BA
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Numerical results (indirect boundary control)
only boundary arc
BA
BA
BA
BA
BA
time order 1
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Numerical results (adjoint temperature)
active set
jump innormal derivative
essential singularitiesat junction points:
Dirac impulses
non-local jump cond. in the energy
non-local jump cond. in the energy
except on the set of active constraintand on the junction lines
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Outline
• A short history on multiple shooting
• Multipoint-boundary-value-problem formulation
• A state constrained elliptic problem
• A state constrained parabolic PDE-ODE problem • A singular hyperbolic optimal control problem jointly with Simon Bechmann & Jan-Eric Wurst
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The „damped“ „elliptic van der Pol Oscillator“
ellip.van der Pol
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
WE
state
The „damped“ „elliptic van der Pol Oscillator“:
S
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The „damped“ „elliptic van der Pol Oscillator“:
W E
controlwith jumps as in ODE
bang – bang - singular
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
The „damped“ „elliptic van der Pol Oscillator“:
difference:
negative
adjoint
zoom
singular region
a posteriori verificationof necessary conditions
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Kunisch, D. Wachsmuth
Wave equation with an unusual control constraint pointwise in time
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
negative
adjoint controlstate
Wave equation with a singular control (example 1)
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
negative
adjoint controlstate
Wave equation with a singular control (example 2)
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Direct postprocessing step: definitions and assumptions
and prescribed control laws on the interior of each subdomain
Based on a partion of the domain with fixed toplogy
feedbackcontrol
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Direct postprocessing step: idea
optimization variable
partition of fixed topology
matching of state variable
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Direct postprocessing step: Switching Curve Optimization
Analogon to switching point optimization in ODE optimal control
Semi-infinite shape optimization problemif the curve is parameterized appropriately
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Direct postprocessing step: Switching Time Optimization
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Indirect postprocessing step: idea
optimization variable
partition of fixed topology
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Indirect postprocessing step: Multiple Domain Optimization
Analogon to multipoint boundary value formulation in ODE optimal control
inner optimization
shape optimization
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Conclusion
In state constrained or bang-singular optimal control problems
there is a natural domain decomposition
with matching conditions
along spatial and/or temporal interior boundaries
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
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Multiple Shooting and Time Domain DecompositionMay. 6-8, 2013, IWR, Heidelberg, Germany
Thank you for your attention