New Necessary Conditions for State-constrained Elliptic Optimal Control
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New Necessary Conditions for State-constrained Elliptic Optimal Control
Problems and Their Numerical Treatment
Simon Bechmann, Michael Frey, Armin Rund,and Hans Josef Pesch
Chair of Mathematics in Engineering SciencesUniversity of Bayreuth, Germany
The 2011 Annual Australian and New ZealandIndustrial and Applied Mathematics Conference
Glenelg, Australia, Jan. 30 - Feb. 3, 2011
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Outline
• Introduction
• New Necessary Conditions
• The Algorithm
• Numerical Results
• Conclusion
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Outline
• Introduction
• State constraints in ODE optimal control• Model problem: elliptic optimal control problem• Standard necessary conditions in PDE optimal control• Idea and Goals
• New Necessary Conditions
• The Algorithm
• Numerical Results
• Conclusion
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State constraints in optimal control of ODE (1)
Minimize
subject to
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State constraints in optimal control of ODE (2)
Order of the state constraint
Hamiltonian: Jacobson, Lele, Speyer, 1971 , via Maurer, 1976 , to Bryson, Denham, Dreyfus, 1963 :Maximum principle: stationarity condition adjoint equations, transversality conditions complementarity conditions jump conditions, sign condition
The higher q the higher the regularity
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Model Problem: elliptic, distributed control, state constraint
Minimize
with
subject to
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Standard necessary conditions
• BVP posseses a unique weak solution for all
• Since , we have an explicit Slater point
Theorem (Casas, 1986; analogon to JLS, 1971)
Let the pair be an optimal solution of the model problem.Then there exist
such that the following optimality system holds
• a real regular Borel measure • an associated adjoint state for all
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Standard necessary conditions: optimality system
adjoint equationwith measures
gradient equation
complementarity conditions
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Definition of active set and assumptions
Definition: active / inactive set / interface
Assumptions
no degeneracylike appendices
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Splitted optimality system
cf. Bergounioux, Kunisch, 2003
with
better regularity butnot numerically exploited
matchingconditions
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Idea and goals
• Apply the Bryson-Denham-Dreyfus approach • Lift the regularity of the multiplier component to
• Lift the regularity of the multiplier component to resp. exploit
• Obtain new necessary conditions without measures, but piecewise multipliers
• resulting in a more efficient numerical method
DirichletNeumann
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Outline
• Introduction
• New Necessary Conditions• Reformulation of the state constraint• Reformulation of the model problem• New necessary conditions• Regularity of multipliers
• The Algorithm
• Numerical Results
• Conclusion
Reformulation of the optimal control problem
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Reformulation of the state constraint
Splitting of the boudary value problem
with
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(Neumann variant)
Reformulation of the state constraint
Transfering the Bryson-Denham-Dreyfus approach
Using the state equation
(Dirichlet variant)
Optimal solution on given by data, but optimization variable
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Reformulation as topology-shape optimal control problem
Minimize
subject to
interface conditions
equality constrainton subdomain
non-standard
Problem is equivalent to original problem
No proof of Zowe-Kurcyuszpossible
of same class as and
Problem is a complicated differential game
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Reformulation as shape optimal control problem of bi-level type
Minimize
subject to
a posteriori check
Problem is not equivalent to original problem
Proof of Zowe-Kurcyuszpossible
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New necessary conditions
Theorem Let be an optimal solution of theshape optimal control problem.Then there exist
such that
• multipliers • and functions
jump condition
modified gradient
Proof by Zowe-Kurcyusz constraint qualification +derivatives of Lagrangian
hereneeded
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obtainable by shape derivativeof a bilevel optimization problem
results incontinuous control
Regularity of multipliers: comparision with Casas‘ multiplier
Proposition
Alternative BDD approach (using Neumann BDD ansatz)
with
jump condition
improved regularity
existence of multipliers!!!
Dirichlet BDD ansatz:continuous adjoint, jump in normal derivative
discontinuous adjoint, continuous normal derivative
improved regularityexploits splitting
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Outline
• Introduction
• New Necessary Conditions
• The Algorithm
• Numerical Results
• Conclusion
• The condensed optimality system• The trial algorithm
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The condensed optimality system
Free boundary value problem for a coupled system of two elliptic equations
control eliminated control eliminated
state matching
adjoint matching
boundary control eliminated
continuity of control
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Solving the optimality system
Different idea to solve the system
• Relax one condition and formulate a shape optimization problem (cf. Hintermüller, Ring, 2004)
• Derive a shape linearization and perform a Newton-type algorithm (similar as in Kärkkainen, 2005)
• Derive a „partial shape linearization“ of one equation while the others are kept (trial method)
needs shape adjoints
no shape adjoints, difficult implementation
no shape adjoints, easier implementation
However, no convergence analysis,but mesh independency observed;algorithm formulated in function space
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• initial guess for
• solve the optimality system without on
• get a displacement of the interface by solving
in the variable , which is a normal component of a displacement vector field
• update and
• if stop criterion is not fulfilled, go to
• otherwise check . If indicated adjust topology of active set .
The trial algorithm
The trial algorithm
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Outline
• Introduction
• New Necessary Conditions
• The Algorithm
• Numerical Results
• Conclusion
• test problems• comparison with PDAS
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Test problem „Dump-Bell“
Construction: Prescribe ,choose small,press down .
Initial guess: automatically from unconstrained problem
Iter No. 123456789
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Test problem „Dump-Bell“
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Test problem „Smiley“
Construction: Prescribe ,choose small,press down .
Initial guess: automatically from unconstrained problem
Iter No. 123456789I made it!
topology changes
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Test problem „Smiley“
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Comparison with PDAS
Trial method
locally convergent
formulated in function space
potentially mesh-independent
no regularization necessary
PDAS
globally convergent
not formulated in function space
not mesh-independent
regularization essential
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Outline
• Introduction
• New Necessary Conditions
• The Algorithm
• Numerical Results
• Conclusion
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Conclusion
• New necessary conditions
• Higher regulatity on multipliers, no measures
• Optimality system is a free boundary value problem
• Extentable to semilinear equations and more complex state constraints
• Trial algorithm formulated in function space
• Trial algorithm needs no regularization
• Trail algorithm exhibits mesh-independency
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Thank you