New Necessary Conditions for State-constrained Elliptic Optimal Control

1. 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 Sciences University of Bayreuth, Germany The 2011 Annual Australian and New Zealand Industrial and Applied Mathematics Conference Glenelg, Australia, Jan. 30 - Feb. 3, 2011

2. Outline • Introduction • New Necessary Conditions • The Algorithm • Numerical Results • Conclusion

3. 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

4. State constraints in optimal control of ODE (1) Minimize subject to

5. Hamiltonian: Jacobson, Lele, Speyer, 1971 , via Maurer, 1976 , to Bryson, Denham, Dreyfus, 1963 : State constraints in optimal control of ODE (2) Order of the state constraint Maximum principle: stationarity condition adjoint equations, transversality conditions complementarity conditions jump conditions, sign condition The higher q the higher the regularity

6. Minimize subject to with Model Problem: elliptic, distributed control, state constraint

7. 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

8. adjoint equation with measures gradient equation Standard necessary conditions: optimality system complementarity conditions

9. no degeneracy like appendices Definition of active setand assumptions Definition: active / inactive set / interface Assumptions

10. Splitted optimality system cf. Bergounioux, Kunisch, 2003 matching conditions with better regularity but not numerically exploited

11. Dirichlet • 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 Neumann Idea and goals

12. Outline Reformulation of the optimal control problem • 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

13. Reformulation of the state constraint Splitting of the boudary value problem with

14. Optimal solution on given by data, but optimization variable Reformulation of the state constraint (Dirichlet variant) Transfering the Bryson-Denham-Dreyfus approach (Neumann variant) Using the state equation

15. of same class as and non-standard Reformulation as topology-shape optimal control problem Problem is a complicated differential game Problem is equivalent to original problem Minimize subject to interface conditions equality constraint on subdomain No proof of Zowe-Kurcyusz possible

16. Reformulation as shape optimal control problem of bi-level type Problem is not equivalent to original problem Minimize subject to Proof of Zowe-Kurcyusz possible a posteriori check

17. here needed • multipliers • and functions New necessary conditions Theorem Let be an optimal solution of the shape optimal control problem. Then there exist such that jump condition modified gradient Proof by Zowe-Kurcyusz constraint qualification + derivatives of Lagrangian

18. improved regularity exploits splitting Alternative BDD approach (using Neumann BDD ansatz) with Regularity of multipliers: comparision with Casas‘ multiplier Dirichlet BDD ansatz: continuous adjoint, jump in normal derivative Proposition results in continuous control obtainable by shape derivative of a bilevel optimization problem discontinuous adjoint, continuous normal derivative jump condition improved regularity existence of multipliers!!!

19. Introduction • New Necessary Conditions • The Algorithm • Numerical Results • Conclusion • The condensed optimality system • The trial algorithm Outline

20. The condensed optimality system Free boundary value problem for a coupled system of two elliptic equations control eliminated control eliminated state matching adjoint matching continuity of control boundary control eliminated

21. 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

22. The trial algorithm The trial algorithm • 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 .

23. Outline • Introduction • New Necessary Conditions • The Algorithm • Numerical Results • Conclusion • test problems • comparison with PDAS

24. Test problem „Dump-Bell“ Construction: Prescribe , choose small, press down . Initial guess: automatically from unconstrained problem Iter No. 1 3 2 4 5 6 7 8 9

25. Test problem „Dump-Bell“

26. Test problem „Smiley“ topology changes Construction: Prescribe , choose small, press down . Initial guess: automatically from unconstrained problem I made it! Iter No. 1 3 2 4 5 6 7 8 9

27. Test problem „Smiley“

28. Comparison with PDAS Trial method locally convergent formulated in function space potentially mesh-independentno regularization necessary PDAS globally convergentnot formulated in function space not mesh-independent regularization essential

29. Outline • Introduction • New Necessary Conditions • The Algorithm • Numerical Results • Conclusion

30. 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

31. Thank you