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

Ulisse Stefanelli; Daniel Wachsmuth; Gerd Wachsmuth

doi:10.3934/dcdss.2017076

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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 and Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076

References:

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[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	J. Diestel and J. J. Uhl, Vector Measures Mathematical Surveys and Monographs. American Mathematical Society, Providence, 1977.
[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.
[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.
[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.
[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.
[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.
[21]	H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1974.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[30]	P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations volume 8 of GAKUTO International Series Mathematical Sciences and Applications, Gakkōtosho, 1996.
[31]	P. Krejčí and M. Liero, Rate independent Kurzweil processes, Applications of Mathematics, 54 (2009), 117-145. doi: 10.1007/s10492-009-0009-5.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[41]	F. Rindler, Approximation of rate-independent optimal control problems, SIAM Journal on Numerical Analysis, 47 (2009), 3884-3909. doi: 10.1137/080744050.
[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.
[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.
[44]	U. Stefanelli, Magnetic control of magnetic shape-memory crystals, Phys. B, 407 (2012), 1316-1321. doi: 10.1016/j.physb.2011.06.043.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	J. Diestel and J. J. Uhl, Vector Measures Mathematical Surveys and Monographs. American Mathematical Society, Providence, 1977.
[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.
[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.
[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.
[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.
[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.
[21]	H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1974.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[30]	P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations volume 8 of GAKUTO International Series Mathematical Sciences and Applications, Gakkōtosho, 1996.
[31]	P. Krejčí and M. Liero, Rate independent Kurzweil processes, Applications of Mathematics, 54 (2009), 117-145. doi: 10.1007/s10492-009-0009-5.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[41]	F. Rindler, Approximation of rate-independent optimal control problems, SIAM Journal on Numerical Analysis, 47 (2009), 3884-3909. doi: 10.1137/080744050.
[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.
[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.
[44]	U. Stefanelli, Magnetic control of magnetic shape-memory crystals, Phys. B, 407 (2012), 1316-1321. doi: 10.1016/j.physb.2011.06.043.
[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.
[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.
[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.
[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.
[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.

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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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