Transcript Multiple Shooting Method
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]
Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Outline
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A short history on multiple shooting
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Multipoint-boundary-value-problem formulation
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A state constrained elliptic problem
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A state constrained parabolic PDE-ODE problem
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A singular hyperbolic optimal control problem
Multiple Shooting and Time Domain Decomposition May. 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 Multiple Shooting and Time Domain Decomposition May. 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 unstable that 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 solution at once. The purpose of this note is to point out a compromising procedure which 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 Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
The Pioneers Keller, Osborne (1968,69): first analysis Concept and first analysis of multiple shooting and parallel shooting Bulirsch, Stoer (1971,73): first algorithmic realisation First code (1968): BOUNDSOL: nonlinear boundary value problems Second code (1970): OPTSOL: optimal control problems with inequality constraints Bulirsch coined the term Mehrzielmethode Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
The Followers Deuflhard (1974,75): improved Newton method (DLOPTR) Various error norms Almost singular coefficient matrix Improved relaxation strategy Oberle (1977,83): multipoint bvps (BOUNDSCO) Improved robustness due to multipoint boundary value formulation Reduced condition number by eliminating condensation Bock (1984): direct multiple shooting (MUSCOD) First-discretize-then-optimize code with multiple shooting Multiple Shooting and Time Domain Decomposition May. 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 Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Abort Landing in a Wind Shear Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Maximal Minimum Altitude Optimal Solution
for Different Wind Profiles
max!
Montrone, P. 1991, Berkmann, P. 1995 Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Maximal Minimum Altitude Optimal Solution bang singular 3rd order state constr 1st order state constr first optimize then discretize by indirect multiple shooting control versus time: rate of angle of attack altitude versus range of abort landing Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
A Very Complicated Switching Structure switching structure (7 pts, 12 add. var.): number of interior boundary conditions: 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 4 2 6 plus 5 additional interior boundary conditions due to modelling plus 11 usual boundary conditions given or by optimality conditions Multiple Shooting and Time Domain Decomposition May. 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
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A state constrained parabolic PDE-ODE problem
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A singular hyperbolic optimal control problem Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Model Problem: elliptic, distributed control, state constraint Minimize subject to with Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Definition of active set and assumptions Definition: active / inactive set / interface Assumption on addmissble active sets Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany No degeneracy.
No active set of zero measure.
No common points with boundary
Reformulation of the state constraint Transfering the Bryson-Denham-Dreyfus approach Using the state equation Optimal solution on given by data, but optimization variable Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Reformulation as set optimal control problem topology is assumed to be known Minimize
inner
subject to
outer
May. 6-8, 2013, IWR, Heidelberg, Germany
Optimality system in the inner optimization of a bilevel problem subject to the optimality system of the inner optimization problem Theorem: For each admissible the objective is shape differentiable. The semi-derivative in the direction is Multiple Shooting and Time Domain Decomposition
The Smiley example: rational initial guess Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
The Smiley example: bad initial guess Algorithm can cope with topology changes to some extent Multiple Shooting and Time Domain Decomposition May. 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 Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Motivation: Super-Concorde - Hypersonic Passenger Jet PDE ODE Project LAPCAT Reading Engines, UK May. 6-8, 2013, IWR, Heidelberg, Germany
The Hypersonic Rocket Car Problem: The ODE Part minimum time control costs Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
The PDE-Part of the Model: The PDE Part friction term control via ODE state instationary heating of the entire vehicle Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
The State Constraint PDE The state constraint regenerates the PDE with the ODE ODE Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Numerical results control is non-linear linear Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Numerical results (topology and order of state constraint) BA time order 2 BA TP BA TP BA Multiple Shooting and Time Domain Decomposition touch point (TP) and boundary arc (BA)
Numerical results (indirect boundary control) time order 1 BA BA BA BA BA May. 6-8, 2013, IWR, Heidelberg, Germany
Numerical results (adjoint temperature) active set essential singularities at junction points: Dirac impulses non-local jump cond. in the energy non-local jump cond. in the energy jump in normal derivative except on the set of active constraint and on the junction lines May. 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 Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
The „damped“ „elliptic van der Pol Oscillator“ Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
ellip.
van der Pol
The „damped“ „elliptic van der Pol Oscillator“: W
state
S Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany E
The „damped“ „elliptic van der Pol Oscillator“: W bang – bang - singular E
control with jumps as in ODE
Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
The „damped“ „elliptic van der Pol Oscillator“:
singular region
difference:
a posteriori verification of necessary conditions negative
adjoint
Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
zoom
Wave equation with an unusual control constraint pointwise in time Kunisch, D. Wachsmuth Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Wave equation with a singular control (example 1) negative
adjoint state
Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
control
Wave equation with a singular control (example 2) negative
adjoint state
Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
control
Direct postprocessing step: definitions and assumptions Based on a partion of the domain with fixed toplogy and prescribed control laws on the interior of each subdomain feedback control Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Direct postprocessing step: idea optimization variable matching of state variable partition of fixed topology Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Direct postprocessing step: Switching Curve Optimization Semi-infinite shape optimization problem if the curve is parameterized appropriately Analogon to switching point optimization in ODE optimal control Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Direct postprocessing step: Switching Time Optimization Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Indirect postprocessing step: idea optimization variable partition of fixed topology Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Indirect postprocessing step: Multiple Domain Optimization shape optimization inner optimization Analogon to multipoint boundary value formulation in ODE optimal control Multiple Shooting and Time Domain Decomposition May. 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 Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany
Thank you for your attention
Multiple Shooting and Time Domain Decomposition May. 6-8, 2013, IWR, Heidelberg, Germany