## Optimal control of perfect plasticity. II: Displacement tracking.(English)Zbl 1468.49004

### MSC:

 49J20 Existence theories for optimal control problems involving partial differential equations 49J40 Variational inequalities 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
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 [1] L. Adam and J. Outrata, On optimal control of a sweeping process coupled with an ordinary differential equation, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), pp. 2709-2738. · Zbl 1304.49052 [2] C. E. Arroud and G. Colombo, A maximum principle for the controlled sweeping process, Set-Valued Var. Anal., 26 (2018), pp. 607-629. · Zbl 1398.49015 [3] H. Attouch, G. Buttazzo, and G. Michaille, Variational Analysis in Sobolev and BV Spaces-Applications to PDEs and Optimization, 2nd ed., MOS-SIAM Ser. Optim., SIAM, MOS, Philadelphia, 2014. · Zbl 1311.49001 [4] S. Bartels, A. Mielke, and T. Roubíček, Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation, SIAM J. Numer. Anal., 50 (2012), pp. 951-976. · Zbl 1248.35105 [5] M. Brokate, Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ, Methoden und Verfahren der Mathematischen Physik 35, Verlag Peter D. Lang, Frankfurt am Main, Germany, 1987. · Zbl 0691.49025 [6] M. Brokate and P. Krejčí, Optimal control of ODE systems involving a rate independent variational inequality, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), pp. 331-348. · Zbl 1260.49002 [7] M. Brokate and P. Krejčí, Weak differentiability of scalar hysteresis operators, Discrete Contin. Dyn. Syst., 35 (2015), pp. 2405-2421. · Zbl 1338.47118 [8] G. Colombo, R. Henrion, N. D. Hoang, and B. S. Mordukhovich, Optimal control of the sweeping process over polyhedral controlled sets, J. Differential Equations, 260 (2016), pp. 3397-3447. · Zbl 1334.49070 [9] G. Dal Maso, A. DeSimone, and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), pp. 237-291. · Zbl 1093.74007 [10] R. E. Edwards, Functional Analysis: Theory and Applications, Dover, New York, 1994. [11] M. Eleuteri and L. Lussardi, Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials, Evol. Equ. Control Theory, 3 (2014), pp. 411-427. · Zbl 1310.74012 [12] M. Eleuteri, L. Lussardi, and U. Stefanelli, Thermal control of the Souza-Auricchio model for shape memory alloys, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), pp. 369-386. · Zbl 1352.74095 [13] G. A. Francfort and A. Giacomini, Small-strain heterogeneous elastoplasticity revisited, Commun. Pure Appl. Math., 65 (2012), pp. 1185-1241. · Zbl 1396.74036 [14] G. A. Francfort, A. Giacomini, and J.-J. Marigo, The elasto-plastic exquisite corpse: A Suquet legacy, J. Mech. Phys. Solids, 97 (2016), pp. 125-139. · Zbl 1445.74011 [15] T. Geiger and D. Wachsmuth, Optimal Control of an Evolution Equation with Non-Smooth Dissipation, preprint, arXiv:1801.04077, 2018. [16] K. Gröger, A $$W^{1,p}$$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), pp. 679-687. · Zbl 0646.35024 [17] J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), pp. 353-384. · Zbl 0694.76014 [18] C. Johnson, Existence theorems for plasticity problems, J. Math. Pures Appl., 55 (1976), pp. 431-444. · Zbl 0351.73049 [19] S. May, R. Rannacher, and B. Vexler, Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems, SIAM J. Control Optim., 51 (2013), pp. 2585-2611. · Zbl 1273.65087 [20] H. Meinlschmidt, C. Meyer, and J. Rehberg, Regularization for optimal control problems associated to nonlinear evolution equations, J. Convex Anal., 27 (2020), pp. 443-485. · Zbl 1439.49039 [21] C. Meyer and S. Walther, Optimal Control of Perfect Plasticity, Part I: Stress Tracking, preprint, arXiv:2001.02969, 2020. [22] A. Mielke and F. Rindler, Reverse approximation of energetic solutions to rate-independent processes, Nonlinear Differential Equations Appl., 16 (2009), pp. 17-40. · Zbl 1163.49030 [23] A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Appl. Math. Sci. 193, Springer, New York, 2015. · Zbl 1339.35006 [24] M. G. Mora, Relaxation of the Hencky model in perfect plasticity, J. Math. Pures Appl., 106 (2016), pp. 725-743. · Zbl 1345.49013 [25] C. Münch, Optimal control of reaction-diffusion systems with hysteresis, ESAIM Control Optim. Calc. Var., 24 (2018), pp. 1453-1488. · Zbl 1414.49002 [26] N. Ottosen and M. Ristinmaa, The Mechanics of Constitutive Modeling, Elsevier, Amsterdam, 2005. [27] F. Rindler, Optimal control for nonconvex rate-independent evolution processes, SIAM J. Control Optim., 47 (2008), pp. 2773-2794. · Zbl 1176.49005 [28] F. Rindler, Approximation of rate-independent optimal control problems, SIAM J. Numer. Anal., 47 (2009), pp. 3884-3909. · Zbl 1206.65173 [29] J. Simo and T. Hughes, Computational Inelasticity, Interdiscip. Appl. Math. 7, Springer, New York, 1998. [30] J. Souček, Spaces of functions on domain $$\Omega$$, whose $$k$$-th derivatives are measures defined on $$\overline{\Omega}$$, Cas. Pro. Pest. Mat., 97 (1972), pp. 10-46. · Zbl 0247.46051 [31] U. Stefanelli, Magnetic control of magnetic shape-memory single crystals, Phys. B, 407 (2012), pp. 1316-1321. [32] U. Stefanelli, D. Wachsmuth, and G. Wachsmuth, Optimal control of a rate-independent evolution equation via viscous regularization, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), pp. 1467-1485. · Zbl 1369.49027 [33] P.-M. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions, J. Méc., 20 (1981), pp. 3-39. · Zbl 0474.73030 [34] R. Temam, Mathematical Problems in Plasticity, Courier Dover Publications, New York, 2018. · Zbl 1406.74002 [35] F. Tröltzsch, Optimal Control of Partial Differential Equations, Grad. Stud. Math. 112, American Mathematical Society, Providence, RI, 2010. [36] G. Wachsmuth, Optimal Control of Quasistatic Plasticity, Verlag Dr. Hut, Munich, Germany, 2011. · Zbl 1318.49071 [37] G. Wachsmuth, Optimal control of quasi-static plasticity with linear kinematic hardening, part I: Existence and discretization in time, SIAM J. Control Optim., 50 (2012), pp. 2836-2861. · Zbl 1258.49008 [38] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part II: Regularization and differentiability, J. Anal. Appl., 34 (2015), pp. 391-418. · Zbl 1342.49012 [39] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part III: Optimality conditions, J. Anal. Appl., 35 (2016), pp. 81-118. · Zbl 1345.49029 [40] S. Walther, Optimal Control of Plasticity Systems, Ph.D. thesis, TU Dortmund, Dortmund, Germany, 2021. [41] S. Walther, C. Meyer, and H. Meinlschmidt, Optimal control of an abstract evolution variational inequality with application to homogenized plasticity, J. Nonsmooth Anal. Optim., 1 (2020), pp. 1-41. [42] E. Zeidler, Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization, Springer, New York, 1985. · Zbl 0583.47051 [43] W. P. Ziemer, Weakly Differentiable Functions, Springer, New York, 1989. · Zbl 0692.46022
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