1. Sensitivity and Optimal Control of Evolutions with Scalar Hysteresis
Martin Brokate
Department of Mathematics, TU München
INdAM Meeting Ocerto 2016, Cortona,
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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 differentiabilityw w r v v
(accumulated maximum)
M. Brokate, P. Krejci, Weak differentiability of scalar hysteresis operators,
Discrete Cont. Dyn. Syst. Ser. A 35 (2015),

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

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. either completely to the right of the red line, 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 Assume: is directionally differentiable
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.

30. 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

31. Parabolic problem
Solution operator
Assumptions:
Then

32. Parabolic problem
Theorem: (Visintin)
Solution operator

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

34. Auxiliary estimates
Assume
Then

35. First order problem
Theorem:

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

37. Parabolic control problem with hysteresis
Minimize

38. Optimality condition
Reduced cost functional

39. 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).

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
local in time,
possibly nonlocal
in space
and directionally differentiable

41. 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

42. 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