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

Outline

• Introduction

• New Necessary Conditions

• The Algorithm

• Numerical Results

• Conclusion

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

State constraints in optimal control of ODE (1)

Minimize

subject to

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

Model Problem: elliptic, distributed control, state constraint

Minimize

with

subject to

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

Standard necessary conditions: optimality system

adjoint equationwith measures

gradient equation

complementarity conditions

Definition of active set and assumptions

Definition: active / inactive set / interface

Assumptions

no degeneracylike appendices

Splitted optimality system

cf. Bergounioux, Kunisch, 2003

with

better regularity butnot numerically exploited

matchingconditions

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

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

Reformulation of the state constraint

Splitting of the boudary value problem

with

(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

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

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

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

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

Outline

• Introduction

• New Necessary Conditions

• The Algorithm

• Numerical Results

• Conclusion

• The condensed optimality system• The trial algorithm

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

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

• 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

Outline

• Introduction

• New Necessary Conditions

• The Algorithm

• Numerical Results

• Conclusion

• test problems• comparison with PDAS

Test problem „Dump-Bell“

Construction: Prescribe ,choose small,press down .

Initial guess: automatically from unconstrained problem

Iter No. 123456789

Test problem „Dump-Bell“

Test problem „Smiley“

Construction: Prescribe ,choose small,press down .

Initial guess: automatically from unconstrained problem

Iter No. 123456789I made it!

topology changes

Test problem „Smiley“

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

Outline

• Introduction

• New Necessary Conditions

• The Algorithm

• Numerical Results

• Conclusion

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

Thank you