Multiple-Boundary-Value-Problem Formulation for PDE constrained Optimal Control Problems

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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

hans-josef.pesch@uni-bayreuth.de

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

Outline

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

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

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

Outline

Abort Landing in a Wind Shear

Maximal Minimum Altitude Optimal Solutionfor Different Wind Profiles

Montrone, P. 1991, Berkmann, P. 1995

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

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:

plus 5 additional interior boundary conditions due to modellingplus 11 usual boundary conditions given or by optimality conditions

Outline

• A state constrained elliptic problem jointly with Michael Frey, Simon Bechmann & Armin Rund

Model Problem: elliptic, distributed control, state constraint

Minimize

subject to

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

Reformulation of the state constraint

Transfering the Bryson-Denham-Dreyfus approach

Using the state equation

Optimal solution on given by data, but optimization variable

Reformulation as set optimal control problem

Minimize

subject to

a posteriori check

inner outer

topology is assumed to be known

Theorem:

For each admissible the objective is shape differentiable. The semi-derivative in the direction

Optimality system in the inner optimization of a bilevel problem

subject to the optimality system of the inner optimization problem

determines the interface

The Smiley example: rational initial guess

The Smiley example: bad initial guess

Algorithm can cope with topology changes to some extent

Outline

• A state constrained parabolic PDE-ODE problem jointly with Armin Rund

• A singular hyperbolic optimal control problem

Motivation: Super-Concorde - Hypersonic Passenger Jet

Project LAPCATReading Engines, UK

2 box constraints1 control-state constraint1 state constraint

quasilinear PDEnon-linear boundary conditionscoupled with ODE

The Hypersonic Rocket Car Problem: The ODE Part

minimum time control costs

The PDE-Part of the Model: The PDE Part

friction term

instationary heating of the entire vehicle

control via ODE state

The State Constraint

The state constraintregenerates

the PDE with the ODE

Numerical results

non-linearlinear

control is

Numerical results (topology and order of state constraint)

touch point (TP) and boundary arc (BA)

time order 2

Numerical results (indirect boundary control)

only boundary arc

time order 1

Numerical results (adjoint temperature)

active set

jump innormal derivative

essential singularitiesat junction points:

Dirac impulses

non-local jump cond. in the energy

except on the set of active constraintand on the junction lines

Outline

• A state constrained parabolic PDE-ODE problem • A singular hyperbolic optimal control problem jointly with Simon Bechmann & Jan-Eric Wurst

The „damped“ „elliptic van der Pol Oscillator“

ellip.van der Pol

The „damped“ „elliptic van der Pol Oscillator“:

controlwith jumps as in ODE

bang – bang - singular

difference:

negative

adjoint

singular region

a posteriori verificationof necessary conditions

Kunisch, D. Wachsmuth

Wave equation with an unusual control constraint pointwise in time

negative

adjoint controlstate

Wave equation with a singular control (example 1)

negative

adjoint controlstate

Wave equation with a singular control (example 2)

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

Direct postprocessing step: idea

optimization variable

partition of fixed topology

matching of state variable

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

Direct postprocessing step: Switching Time Optimization

Indirect postprocessing step: idea

optimization variable

partition of fixed topology

Indirect postprocessing step: Multiple Domain Optimization

Analogon to multipoint boundary value formulation in ODE optimal control

inner optimization

shape optimization

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

Thank you for your attention

Multiple-Boundary-Value-Problem Formulation for PDE constrained Optimal Control Problems

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