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Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity
Optimal control of a rate-independent evolution equation via viscous regularization
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 |
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.
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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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