Geodesic
Newton and Bouligand derivatives of the scalar play and stop operator
Martin Brokate1, 2
1 Department of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85747 Garching, Germany.
2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 51

Voir la notice de l'article provenant de la source EDP Sciences

We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strengthened form, and a corresponding chain rule is developed. The construction of the Newton derivative ensures that the mappings involved are measurable.
DOI : 10.1051/mmnp/2020013

Martin Brokate 1, 2

1 Department of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85747 Garching, Germany.
2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany. 
@article{10_1051_mmnp_2020013,
     author = {Martin Brokate},
     title = {Newton and {Bouligand} derivatives of the scalar play and stop operator},
     journal = {Mathematical modelling of natural phenomena},
     eid = {51},
     publisher = {mathdoc},
     volume = {15},
     year = {2020},
     doi = {10.1051/mmnp/2020013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2020013/}
}
TY  - JOUR
AU  - Martin Brokate
TI  - Newton and Bouligand derivatives of the scalar play and stop operator
JO  - Mathematical modelling of natural phenomena
PY  - 2020
VL  - 15
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2020013/
DO  - 10.1051/mmnp/2020013
LA  - en
ID  - 10_1051_mmnp_2020013
ER  - 
%0 Journal Article
%A Martin Brokate
%T Newton and Bouligand derivatives of the scalar play and stop operator
%J Mathematical modelling of natural phenomena
%D 2020
%V 15
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2020013/
%R 10.1051/mmnp/2020013
%G en
%F 10_1051_mmnp_2020013
Martin Brokate. Newton and Bouligand derivatives of the scalar play and stop operator. Mathematical modelling of natural phenomena, Tome 15 (2020), article  no. 51. doi: 10.1051/mmnp/2020013

[1] H. Bauschke and P. Combettes, Convex analysis and monotone operator theory in Hilbert spaces. Springer, Berlin (2011).

[2] J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer, New York (2000).

[3] M. Brokate, P. Krejci Weak Differentiability of Scalar Hysteresis Operators Discrete Contin. Dyn. Syst 2015 2405 2421

[4] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions. Springer, New York (1996).

[5] C. Christof Sensitivity analysis and optimal control of obstacle-type evolution variational inequalities SIAM J. Control Opt 2019 192 218

[6] I.V. Girsanov, Lectures on Mathematical Theory of Extremum Problems. Springer, Berlin (1972).

[7] M. Hintermüller, K. Kunisch PDE-constrained optimization subject to pointwise constraints on the control, the state, and its derivatives SIAM J. Opt 2009 1133 1156

[8] M. Hintermüller, K. Ito, K. Kunisch The primal-dual active set strategy as a semismooth Newton method SIAM J. Opt 2003 865 888

[9] S. Hu and N.S. Papageorgiu, Handbook of multivalued analysis, volume I: Theory. Kluwer, South Holland (1997).

[10] K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. SIAM Series Advances in Design and Control. SIAM, Philadelphia, (2008).

[11] M.A. Krasnosel’Skiĭ, B.M. Darinskiĭ, I.V. Emelin, P.P. Zabrejko, E.A. Lifshits, A.V. Pokrovskiĭ An operator-hysterant Dokl. Akad. Nauk SSSR 1970 34 37

[12] M.A. Krasnosel’Skiĭ, B.M. Darinskiĭ, I.V. Emelin, P.P. Zabrejko, E.A. Lifshits, A.V. Pokrovskiĭ Soviet Math. Dokl. 1970 29 33

[13] P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gakkōtosho, Tokyo (1996).

[14] A. Mielke and T. Roubíček, Rate-Independent Systems. Springer, Berlin (2015).

[15] F. Mignot Contrôle dans les inéquations variationelles elliptiques J. Funct. Anal 1976 130 185

[16] N.S. Papageorgiu and S.T. Kyritsi-Yiallourou, Handbook of applied analysis. Springer, Berlin (2009).

[17] M. Ulbrich Semismooth Newton methods for operator equations in function spaces SIAM J. Optim 2003 805 841

[18] M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM, Philadelphia, (2011).

[19] A. Visintin, Differential Models of Hysteresis. Springer, Berlin (1994).

Cité par Sources :