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December  2017, 10(6): 1467-1485. doi: 10.3934/dcdss.2017076

Optimal control of a rate-independent evolution equation via viscous regularization

Ulisse Stefanelli 1,2, , Daniel Wachsmuth 3,, and  Gerd Wachsmuth 4,

1. 

University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
2. 

Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes - CNR, via Ferrata 1, 27100 Pavia, Italy
3. 

Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
4. 

Technische Universität Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany

* Corresponding author: D. Wachsmuth.

Received  July 2016 Revised  October 2016 Published  June 2017

Fund Project: The second and third author were supported by DFG grants within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization), which is gratefully acknowledged.

We study the optimal control of a rate-independent system that is driven by a convex quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem.

Keywords: Rate-independent system, optimal control, necessary optimality conditions.
Mathematics Subject Classification: Primary: 49K20; Secondary: 35K87.
Citation: Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076
References:
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L. Adam, J. Outrata and T. Roubíček, Identification of some nonsmooth evolution systems with illustration on adhesive contacts at small strains, Optimization, (2015), 1-25.   Google Scholar

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J.-F. Babadjian, G. A. Francfort and M. G. Mora, Quasi-static evolution in nonassociative plasticity: The cap model, SIAM Journal on Mathematical Analysis, 44 (2012), 245-292.  doi: 10.1137/110823511.  Google Scholar

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M. Brokate, Optimale Steuerung Von Gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ Number 35 in Methoden und Verfahren der mathematischen Physik. Verlag Peter Lang, Frankfurt, 1987.  Google Scholar

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M. Brokate, Optimal control of ODE systems with hysteresis nonlinearities, In Trends in mathematical optimization (Irsee, 1986), volume 84 of Internat. Schriftenreihe Numer. Math. , pages 25–41. Birkhäuser, Basel, 1988. Google Scholar

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M. Brokate and P. Krejčí, Optimal control of ODE systems involving a rate independent variational inequality, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 18 (2013), 331-348.  doi: 10.3934/dcdsb.2013.18.331.  Google Scholar

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M. Brokate and J. Sprekels, Hysteresis and Phase Transitions volume 121 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

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F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1027-1071.  doi: 10.1142/S0218202508002942.  Google Scholar

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C. Castaing, M. D. P. Monteiro Marques and P. Raynaud de Fitte, Some problems in optimal control governed by the sweeping process, Journal of Nonlinear and Convex Analysis. An International Journal, 15 (2014), 1043-1070.   Google Scholar

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G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process, Dynamics of Continuous, Discrete & Impulsive Systems. Series B. Applications & Algorithms, 19 (2012), 117-159.   Google Scholar

[10]

G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Discrete approximations of a controlled sweeping process, Set-Valued and Variational Analysis, 23 (2015), 69-86.  doi: 10.1007/s11228-014-0299-y.  Google Scholar

[11]

G. Colombo, R. Henrion, Nguyen D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process over polyhedral controlled sets, Journal of Differential Equations, 260 (2016), 3397-3447.  doi: 10.1016/j.jde.2015.10.039.  Google Scholar

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G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Archive for Rational Mechanics and Analysis, 189 (2008), 469-544.  doi: 10.1007/s00205-008-0117-5.  Google Scholar

[13]

G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: A weak formulation via viscoplastic regularization and time rescaling, Calculus of Variations and Partial Differential Equations, 40 (2011), 125-181.  doi: 10.1007/s00526-010-0336-0.  Google Scholar

[14]

A. DeSimone and R. D. James, A constrained theory of magnetoelasticity, J. Mech. Phys. Solids, 50 (2002), 283-320.  doi: 10.1016/S0022-5096(01)00050-3.  Google Scholar

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J. Diestel and J. J. Uhl, Vector Measures Mathematical Surveys and Monographs. American Mathematical Society, Providence, 1977.  Google Scholar

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M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, Journal of Convex Analysis, 13 (2006), 151-167.   Google Scholar

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M. Eleuteri and L. Lussardi, Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials, Evolution Equations and Control Theory, 3 (2014), 411-427.  doi: 10.3934/eect.2014.3.411.  Google Scholar

[18]

M. Eleuteri, L. Lussardi and U. Stefanelli, Thermal control of the Souza-Auricchio model for shape memory alloys, Discrete and Continuous Dynamical Systems. Series S, 6 (2013), 369-386.   Google Scholar

[19]

A. Fiaschi, A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies, ESAIM. Control, Optimisation and Calculus of Variations, 15 (2009), 245-278.  doi: 10.1051/cocv:2008030.  Google Scholar

[20]

G. A. Francfort and U. Stefanelli, Quasi-static evolution for the Armstrong-Frederick hardening-plasticity model, Applied Mathematics Research Express. AMRX, (2013), 297-344.   Google Scholar

[21]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1974.  Google Scholar

[22]

R. Herzog, C. Meyer and G. Wachsmuth, C-stationarity for optimal control of static plasticity with linear kinematic hardening, SIAM Journal on Control and Optimization, 50 (2012), 3052-3082.  doi: 10.1137/100809325.  Google Scholar

[23]

R. Herzog, C. Meyer and G. Wachsmuth, B-and strong stationarity for optimal control of static plasticity with hardening, SIAM Journal on Optimization, 23 (2013), 321-352.  doi: 10.1137/110821147.  Google Scholar

[24]

R. Herzog, C. Meyer and G. Wachsmuth, Optimal control of elastoplastic processes: Analysis, algorithms, numerical analysis and applications, In Trends in PDE constrained optimization, volume 165 of Internat. Ser. Numer. Math. , pages 27–41. Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-05083-6_4.  Google Scholar

[25]

D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1529-1569.  doi: 10.1142/S0218202508003121.  Google Scholar

[26]

D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Mathematical Models and Methods in Applied Sciences, 23 (2013), 565-616.  doi: 10.1142/S021820251250056X.  Google Scholar

[27]

D. Knees, R. Rossi and C. Zanini, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 24 (2015), 126-162.  doi: 10.1016/j.nonrwa.2015.02.001.  Google Scholar

[28]

D. Knees, C. Zanini and A. Mielke, Crack growth in polyconvex materials, Physica D. Nonlinear Phenomena, 239 (2010), 1470-1484.  doi: 10.1016/j.physd.2009.02.008.  Google Scholar

[29]

M. Kočvara and J. V. Outrata, On the modeling and control of delamination processes, In Control and boundary analysis, volume 240 of Lect. Notes Pure Appl. Math. , pages 169–187. Chapman & Hall/CRC, Boca Raton, FL, 2005.  Google Scholar

[30]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations volume 8 of GAKUTO International Series Mathematical Sciences and Applications, Gakkōtosho, 1996.  Google Scholar

[31]

P. Krejčí and M. Liero, Rate independent Kurzweil processes, Applications of Mathematics, 54 (2009), 117-145.  doi: 10.1007/s10492-009-0009-5.  Google Scholar

[32]

G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation, Math. Models Methods Appl. Sci., 21 (2011), 2019-2047.  doi: 10.1142/S0218202511005647.  Google Scholar

[33]

G. Lazzaroni and R. Toader, Some remarks on the viscous approximation of crack growth, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 131-146.  doi: 10.3934/dcdss.2013.6.131.  Google Scholar

[34]

A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615.  doi: 10.3934/dcds.2009.25.585.  Google Scholar

[35]

A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80.  doi: 10.1051/cocv/2010054.  Google Scholar

[36]

A. Mielke and T. Roubíček, Rate-independent Systems volume 193 of Applied Mathematical Sciences, Springer, New York, 2015. Theory and application. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[37]

A. Mielke and S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 67-135.   Google Scholar

[38]

H.-B. Mühlhaus and E. C. Aifantis, A variational principle for gradient plasticity, International Journal of Solids and Structures, 28 (1991), 845-857.  doi: 10.1016/0020-7683(91)90004-Y.  Google Scholar

[39]

M. Negri, A comparative analysis on variational models for quasi-static brittle crack propagation, Advances in Calculus of Variations, 3 (2010), 149-212.  doi: 10.1515/ACV.2010.008.  Google Scholar

[40]

F. Rindler, Optimal control for nonconvex rate-independent evolution processes, SIAM Journal on Control and Optimization, 47 (2008), 2773-2794.  doi: 10.1137/080718711.  Google Scholar

[41]

F. Rindler, Approximation of rate-independent optimal control problems, SIAM Journal on Numerical Analysis, 47 (2009), 3884-3909.  doi: 10.1137/080744050.  Google Scholar

[42]

T. Roubíček, Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity, SIAM Journal on Mathematical Analysis, 45 (2013), 101-126.  doi: 10.1137/12088286X.  Google Scholar

[43]

F. Solombrino, Quasistatic evolution in perfect plasticity for general heterogeneous materials, Archive for Rational Mechanics and Analysis, 212 (2014), 283-330.  doi: 10.1007/s00205-013-0703-z.  Google Scholar

[44]

U. Stefanelli, Magnetic control of magnetic shape-memory crystals, Phys. B, 407 (2012), 1316-1321.  doi: 10.1016/j.physb.2011.06.043.  Google Scholar

[45]

R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Bollettino della Unione Matematica Italiana. Serie 9, 2 (2009), 1-35.   Google Scholar

[46]

A. Visintin, Differential Models of Hysteresis volume 111 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

[47]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part Ⅰ: Existence and discretization in time, SIAM Journal on Control and Optimization, 50 (2012), 2836-2861.  doi: 10.1137/110839187.  Google Scholar

[48]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part Ⅱ: Regularization and differentiability, Zeitschrift für Analysis und ihre Anwendungen, 34 (2015), 391-418.  doi: 10.4171/ZAA/1546.  Google Scholar

[49]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening Ⅲ: Optimality conditions, Zeitschrift für Analysis und ihre Anwendungen, 35 (2016), 81-118.  doi: 10.4171/ZAA/1556.  Google Scholar

show all references

References:
[1]

L. Adam, J. Outrata and T. Roubíček, Identification of some nonsmooth evolution systems with illustration on adhesive contacts at small strains, Optimization, (2015), 1-25.   Google Scholar

[2]

J.-F. Babadjian, G. A. Francfort and M. G. Mora, Quasi-static evolution in nonassociative plasticity: The cap model, SIAM Journal on Mathematical Analysis, 44 (2012), 245-292.  doi: 10.1137/110823511.  Google Scholar

[3]

M. Brokate, Optimale Steuerung Von Gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ Number 35 in Methoden und Verfahren der mathematischen Physik. Verlag Peter Lang, Frankfurt, 1987.  Google Scholar

[4]

M. Brokate, Optimal control of ODE systems with hysteresis nonlinearities, In Trends in mathematical optimization (Irsee, 1986), volume 84 of Internat. Schriftenreihe Numer. Math. , pages 25–41. Birkhäuser, Basel, 1988. Google Scholar

[5]

M. Brokate and P. Krejčí, Optimal control of ODE systems involving a rate independent variational inequality, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 18 (2013), 331-348.  doi: 10.3934/dcdsb.2013.18.331.  Google Scholar

[6]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions volume 121 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[7]

F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1027-1071.  doi: 10.1142/S0218202508002942.  Google Scholar

[8]

C. Castaing, M. D. P. Monteiro Marques and P. Raynaud de Fitte, Some problems in optimal control governed by the sweeping process, Journal of Nonlinear and Convex Analysis. An International Journal, 15 (2014), 1043-1070.   Google Scholar

[9]

G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process, Dynamics of Continuous, Discrete & Impulsive Systems. Series B. Applications & Algorithms, 19 (2012), 117-159.   Google Scholar

[10]

G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Discrete approximations of a controlled sweeping process, Set-Valued and Variational Analysis, 23 (2015), 69-86.  doi: 10.1007/s11228-014-0299-y.  Google Scholar

[11]

G. Colombo, R. Henrion, Nguyen D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process over polyhedral controlled sets, Journal of Differential Equations, 260 (2016), 3397-3447.  doi: 10.1016/j.jde.2015.10.039.  Google Scholar

[12]

G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Archive for Rational Mechanics and Analysis, 189 (2008), 469-544.  doi: 10.1007/s00205-008-0117-5.  Google Scholar

[13]

G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: A weak formulation via viscoplastic regularization and time rescaling, Calculus of Variations and Partial Differential Equations, 40 (2011), 125-181.  doi: 10.1007/s00526-010-0336-0.  Google Scholar

[14]

A. DeSimone and R. D. James, A constrained theory of magnetoelasticity, J. Mech. Phys. Solids, 50 (2002), 283-320.  doi: 10.1016/S0022-5096(01)00050-3.  Google Scholar

[15]

J. Diestel and J. J. Uhl, Vector Measures Mathematical Surveys and Monographs. American Mathematical Society, Providence, 1977.  Google Scholar

[16]

M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, Journal of Convex Analysis, 13 (2006), 151-167.   Google Scholar

[17]

M. Eleuteri and L. Lussardi, Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials, Evolution Equations and Control Theory, 3 (2014), 411-427.  doi: 10.3934/eect.2014.3.411.  Google Scholar

[18]

M. Eleuteri, L. Lussardi and U. Stefanelli, Thermal control of the Souza-Auricchio model for shape memory alloys, Discrete and Continuous Dynamical Systems. Series S, 6 (2013), 369-386.   Google Scholar

[19]

A. Fiaschi, A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies, ESAIM. Control, Optimisation and Calculus of Variations, 15 (2009), 245-278.  doi: 10.1051/cocv:2008030.  Google Scholar

[20]

G. A. Francfort and U. Stefanelli, Quasi-static evolution for the Armstrong-Frederick hardening-plasticity model, Applied Mathematics Research Express. AMRX, (2013), 297-344.   Google Scholar

[21]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1974.  Google Scholar

[22]

R. Herzog, C. Meyer and G. Wachsmuth, C-stationarity for optimal control of static plasticity with linear kinematic hardening, SIAM Journal on Control and Optimization, 50 (2012), 3052-3082.  doi: 10.1137/100809325.  Google Scholar

[23]

R. Herzog, C. Meyer and G. Wachsmuth, B-and strong stationarity for optimal control of static plasticity with hardening, SIAM Journal on Optimization, 23 (2013), 321-352.  doi: 10.1137/110821147.  Google Scholar

[24]

R. Herzog, C. Meyer and G. Wachsmuth, Optimal control of elastoplastic processes: Analysis, algorithms, numerical analysis and applications, In Trends in PDE constrained optimization, volume 165 of Internat. Ser. Numer. Math. , pages 27–41. Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-05083-6_4.  Google Scholar

[25]

D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1529-1569.  doi: 10.1142/S0218202508003121.  Google Scholar

[26]

D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Mathematical Models and Methods in Applied Sciences, 23 (2013), 565-616.  doi: 10.1142/S021820251250056X.  Google Scholar

[27]

D. Knees, R. Rossi and C. Zanini, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 24 (2015), 126-162.  doi: 10.1016/j.nonrwa.2015.02.001.  Google Scholar

[28]

D. Knees, C. Zanini and A. Mielke, Crack growth in polyconvex materials, Physica D. Nonlinear Phenomena, 239 (2010), 1470-1484.  doi: 10.1016/j.physd.2009.02.008.  Google Scholar

[29]

M. Kočvara and J. V. Outrata, On the modeling and control of delamination processes, In Control and boundary analysis, volume 240 of Lect. Notes Pure Appl. Math. , pages 169–187. Chapman & Hall/CRC, Boca Raton, FL, 2005.  Google Scholar

[30]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations volume 8 of GAKUTO International Series Mathematical Sciences and Applications, Gakkōtosho, 1996.  Google Scholar

[31]

P. Krejčí and M. Liero, Rate independent Kurzweil processes, Applications of Mathematics, 54 (2009), 117-145.  doi: 10.1007/s10492-009-0009-5.  Google Scholar

[32]

G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation, Math. Models Methods Appl. Sci., 21 (2011), 2019-2047.  doi: 10.1142/S0218202511005647.  Google Scholar

[33]

G. Lazzaroni and R. Toader, Some remarks on the viscous approximation of crack growth, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 131-146.  doi: 10.3934/dcdss.2013.6.131.  Google Scholar

[34]

A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615.  doi: 10.3934/dcds.2009.25.585.  Google Scholar

[35]

A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80.  doi: 10.1051/cocv/2010054.  Google Scholar

[36]

A. Mielke and T. Roubíček, Rate-independent Systems volume 193 of Applied Mathematical Sciences, Springer, New York, 2015. Theory and application. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[37]

A. Mielke and S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 67-135.   Google Scholar

[38]

H.-B. Mühlhaus and E. C. Aifantis, A variational principle for gradient plasticity, International Journal of Solids and Structures, 28 (1991), 845-857.  doi: 10.1016/0020-7683(91)90004-Y.  Google Scholar

[39]

M. Negri, A comparative analysis on variational models for quasi-static brittle crack propagation, Advances in Calculus of Variations, 3 (2010), 149-212.  doi: 10.1515/ACV.2010.008.  Google Scholar

[40]

F. Rindler, Optimal control for nonconvex rate-independent evolution processes, SIAM Journal on Control and Optimization, 47 (2008), 2773-2794.  doi: 10.1137/080718711.  Google Scholar

[41]

F. Rindler, Approximation of rate-independent optimal control problems, SIAM Journal on Numerical Analysis, 47 (2009), 3884-3909.  doi: 10.1137/080744050.  Google Scholar

[42]

T. Roubíček, Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity, SIAM Journal on Mathematical Analysis, 45 (2013), 101-126.  doi: 10.1137/12088286X.  Google Scholar

[43]

F. Solombrino, Quasistatic evolution in perfect plasticity for general heterogeneous materials, Archive for Rational Mechanics and Analysis, 212 (2014), 283-330.  doi: 10.1007/s00205-013-0703-z.  Google Scholar

[44]

U. Stefanelli, Magnetic control of magnetic shape-memory crystals, Phys. B, 407 (2012), 1316-1321.  doi: 10.1016/j.physb.2011.06.043.  Google Scholar

[45]

R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Bollettino della Unione Matematica Italiana. Serie 9, 2 (2009), 1-35.   Google Scholar

[46]

A. Visintin, Differential Models of Hysteresis volume 111 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

[47]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part Ⅰ: Existence and discretization in time, SIAM Journal on Control and Optimization, 50 (2012), 2836-2861.  doi: 10.1137/110839187.  Google Scholar

[48]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part Ⅱ: Regularization and differentiability, Zeitschrift für Analysis und ihre Anwendungen, 34 (2015), 391-418.  doi: 10.4171/ZAA/1546.  Google Scholar

[49]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening Ⅲ: Optimality conditions, Zeitschrift für Analysis und ihre Anwendungen, 35 (2016), 81-118.  doi: 10.4171/ZAA/1556.  Google Scholar

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