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Sensitivity and Optimal Control of Evolutions with Scalar Hysteresis

Sensitivity and Optimal Control of Evolutions with Scalar Hysteresis. Martin Brokate Department of Mathematics, TU München. INdAM Meeting Ocerto 2016, Cortona, 20.-24.6.2016. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A.

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Sensitivity and Optimal Control of Evolutions with Scalar Hysteresis

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  1. Sensitivity and Optimal Control of Evolutions with Scalar Hysteresis Martin Brokate Department of Mathematics, TU München INdAM Meeting Ocerto 2016, Cortona, 20.-24.6.2016 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

  2. Parabolic control problem with hysteresis Minimize

  3. Abstract optimization problem Reduced cost functional

  4. Parabolic control problem with hysteresis Minimize not monotone

  5. Abstract optimization problem Reduced cost functional Chain rule is needed!

  6. Parabolic equation with hysteresis

  7. Play w r v

  8. Play w r v

  9. Play: Derivatives ? w r v you expect:

  10. Play: Derivatives ? w r v Moreover: Does exist ?

  11. Play: Directional differentiability w w r v v (accumulated maximum) M. Brokate, P. Krejci, Weak differentiability of scalar hysteresis operators, Discrete Cont. Dyn. Syst. Ser. A 35 (2015), 2405-2421

  12. From maximum to play Maximum: Accumulated maximum: (gliding maximum) Play:

  13. Play: local description Locally, the play can be represented as a finite composition of variants of the accumulated maximum. w o r v o

  14. w r v either completely to the right of the red line, or completely to the left of the green line.

  15. w r v w.r.t the max norm

  16. The maximum functional (fixed interval) convex, Lipschitz Directional derivative: Directional derivative is a Hadamard derivative:

  17. Bouligand derivative is directionally differentiable Assume: The directional derivative is called a Bouligand derivative if Still nonlinear w.r.t the direction h. Better approximation property than the directional derivative: Limit process is uniform w.r.t. the direction h

  18. Maximum functional: Bouligand derivative is not a Bouligand derivative is a Bouligand derivative

  19. Newton derivative Semismooth Newton method:

  20. Newton derivative (Set valued) Semismooth Newton method:

  21. Maximum functional: Newton derivative Directional derivative:

  22. Maximum functional: Newton derivative is not a Newton derivative is a Newton derivative

  23. Proof

  24. The accumulated maximum Directional derivative exists ``pointwise in time‘‘: Denote

  25. The accumulated maximum The pointwise derivative is a regulated function, but discontinuous in general. Thus, is not directionally differentiable But is directionally differentiable.

  26. Accumulated maximum: Newton derivative weakly measurable. is a Newton derivative

  27. Play: Newton derivative Theorem. The play operator is Newton differentiable. The proof uses the chain rule for Newton derivatives, and yields a recursive formula based on the successive accumulated maxima.

  28. Play: Bouligand derivative Theorem. The play operator is Bouligand differentiable.

  29. Parabolic problem Solution operator M. Brokate, K. Fellner, M. Lang-Batsching, Weak differentiability of the control-to-state mapping in a parabolic control problem with hysteresis, Preprint IGDK 1754

  30. Parabolic problem Solution operator Assumptions: Then

  31. Parabolic problem Theorem: (Visintin) Solution operator

  32. First order problem Theorem: (variant of Visintin)

  33. Auxiliary estimates Assume Then

  34. First order problem Theorem:

  35. Sensitivity result Theorem: The control-to-state mapping has a Bouligand derivative when considered as a mapping Proof: Estimates for the remainder problem

  36. Parabolic control problem with hysteresis Minimize

  37. Optimality condition Reduced cost functional

  38. Optimality conditions, related results ODE control problem with scalar hysteresis: My habilitation thesis. ODE control problem with vector hysteresis: M. Brokate, P. Krejci, Optimal control of ODE systems involving a rate independent variational inequality, Discrete Cont. Dyn. Syst. Ser. B 18 (2013), 331–348. A complete set of 1st order optimality conditions is obtained, under certain regularity assumptions (otherwise, only weak stationarity conditions). Employs regularization, but not weak derivatives of the hysteresis operator (their existence is an open problem in the vector case).

  39. Optimality conditions, related results Parabolic control problem with Lipschitz nonlinearity: C. Meyer, L.M. Susu, Optimal control of nonsmooth semilinear parabolic equations, preprint 2015 local in time, possibly nonlocal in space and directionally differentiable

  40. Optimality conditions, related results Parabolic control problem with Lipschitz nonlinearity: C. Meyer, L.M. Susu, Optimal control of nonsmooth semilinear parabolic equations, preprint 2015 If moreover there exists a regularization

  41. Optimality conditions, related results Parabolic control problem with hysteresis operator: C. Münch, ongoing PhD project parabolic system (e.g. reaction-diffusion), a single hysteresis operator Employs regularization and directional differentiability, obtains optimality conditions

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