Second-order sufficient optimality conditions for optimal control of static elastoplasticity with hardening
ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 1, pp. 271-300.

The paper is concerned with the optimal control of static elastoplasticity with linear kinematic hardening. This leads to an optimal control problem governed by an elliptic variational inequality (VI) of first kind in mixed form. Based on L p -regularity results for the state equation, it is shown that the control-to-state operator is Bouligand differentiable. This enables to establish second-order sufficient optimality conditions by means of a Taylor expansion of a particularly chosen Lagrange function.

Received:
DOI: 10.1051/cocv/2014024
Classification: 49K20, 74C05, 74P10, 35R45
Keywords: Second-order sufficient conditions, optimal control of variational inequalities, bouligand differentiability
Betz, Thomas 1; Meyer, Christian 1

1 TU Dortmund, Faculty of Mathematics, Vogelpothsweg 87, 44227 Dortmund, Germany.
@article{COCV_2015__21_1_271_0,
     author = {Betz, Thomas and Meyer, Christian},
     title = {Second-order sufficient optimality conditions for optimal control of static elastoplasticity with hardening},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {271--300},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {1},
     year = {2015},
     doi = {10.1051/cocv/2014024},
     zbl = {1311.49017},
     mrnumber = {3348423},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2014024/}
}
TY  - JOUR
AU  - Betz, Thomas
AU  - Meyer, Christian
TI  - Second-order sufficient optimality conditions for optimal control of static elastoplasticity with hardening
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2015
SP  - 271
EP  - 300
VL  - 21
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2014024/
DO  - 10.1051/cocv/2014024
LA  - en
ID  - COCV_2015__21_1_271_0
ER  - 
%0 Journal Article
%A Betz, Thomas
%A Meyer, Christian
%T Second-order sufficient optimality conditions for optimal control of static elastoplasticity with hardening
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2015
%P 271-300
%V 21
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2014024/
%R 10.1051/cocv/2014024
%G en
%F COCV_2015__21_1_271_0
Betz, Thomas; Meyer, Christian. Second-order sufficient optimality conditions for optimal control of static elastoplasticity with hardening. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 1, pp. 271-300. doi : 10.1051/cocv/2014024. http://archive.numdam.org/articles/10.1051/cocv/2014024/

V. Barbu, Optimal Control of Variational Inequalities, vol. 100 of Res. Notes Math. Pitman, Boston (1984). | MR | Zbl

M. Bergounioux, Optimal control problems governed by abstract elliptic variational inequalities with state constraints. SIAM J. Control Optim. 36 (1998) 273–289. | DOI | MR | Zbl

F. Bonnans and D. Tiba, Pontryagin’s principle in the control of semilinear elliptic variational inequalities. Appl. Math. Optim. 23 (1991) 299–312. | DOI | MR | Zbl

E. Casas, J.C. De Los Reyes, and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19 (2008) 616–643. | DOI | MR | Zbl

E. Casas and M. Mateos, Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431–1454. | DOI | MR | Zbl

E. Casas and F. Tröltzsch, First- and second order optimality conditions for a class of optimal control problems with quasilinear ellitpic equations. SIAM J. Control Optim. 48 (2009) 688–718. | DOI | MR | Zbl

E. Casas, F. Tröltzsch, and A. Unger, Second order sufficient optimality conditions for a nonlinear elliptic control problem. Zeitschrift für Analysis und ihre Anwendungen 15 (1996) 687–707. | DOI | MR | Zbl

K. Gröger, A W 1,p -estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Mathematische Annalen 283 (1989) 679–687. | DOI | MR | Zbl

P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR | Zbl

R. Haller-Dintelmann, C. Meyer, J. Rehberg, and A. Schiela, Hölder continuity and optimal control for nonsmooth elliptic problems. Appl. Math. Optim. 60 (2009) 397–428. | DOI | MR | Zbl

W. Han and B.D. Reddy, Plasticity. Springer, New York (1999). | MR | Zbl

R. Herzog and C. Meyer, Optimal control of static plasticity with linear kinematic hardening. J. Appl. Math. Mech. 91 (2011) 777–794. | MR | Zbl

R. Herzog, C. Meyer, and G. Wachsmuth, Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions. J. Math. Anal. Appl. 382 (2011) 802–813. | DOI | MR | Zbl

R. Herzog, C. Meyer, and G. Wachsmuth, C-stationarity for optimal control of static plasticity with linear kinematic hardening. SIAM J. Control Optim. 50 (2012) 3052–3082. | DOI | MR | Zbl

R. Herzog, C. Meyer, and G. Wachsmuth, B- and strong stationarity for optimal control of static plasticity with hardening. SIAM J. Optim. 23 (2013) 321–352. | DOI | MR | Zbl

M. Hintermüller and I. Kopacka, Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm. SIAM J. Optim. 20 (2009) 868–902. | DOI | MR | Zbl

M. Hintermüller and Th. Surowiec, First order optimality conditions for elliptic mathematical programs with equilibrium constraints via variational analysis. SIAM J. Optim. 21 (2012) 1561–1593. | DOI | MR | Zbl

K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343–364. | DOI | MR | Zbl

C. Kanzow and A. Schwartz, Mathematical programs with equilibrium constraints: enhanced Fritz John-conditions, new constraint qualifications, and improved exact penalty results. SIAM J. Optim. 20 (2010) 2730–2753. | DOI | MR | Zbl

K. Kunisch and D. Wachsmuth, Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities. ESAIM: COCV 18 (2012) 520–547. | Numdam | MR | Zbl

K. Kunisch and D. Wachsmuth, Path-following for optimal control of stationary variational inequalities. Comput. Optim. Appl. (2012) 1–29. | MR | Zbl

F. Mignot, Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22 (1976) 130–185. | DOI | MR | Zbl

F. Mignot and J.-P. Puel, Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466–476. | DOI | MR | Zbl

P. Neff and D. Knees, Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity. SIAM J. Math. Anal. 40 (2008) 21–43. | DOI | MR | Zbl

J. Outrata, J. Jarušek and J. Stará, On optimality conditions in control of elliptic variational inequalities. Set-Valued Var. Anal. 19 (2011) 23–42. | DOI | MR | Zbl

H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity. Math. Oper. Res. 25 (2000) 1–22. | DOI | MR | Zbl

G. Wachsmuth, Optimal control of quasistatic plasticity – An MPCC in function space. Ph.D. thesis, Chemnitz University of Technology, Germany (2011).

G. Wachsmuth, Differentiability of implicit functions: Beyond the implicit function theorem. Technical Report SPP1253-137, Priority Program 1253, German Research Foundation (2012). | MR

Cited by Sources: