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

Ulisse Stefanelli; Daniel Wachsmuth; Gerd Wachsmuth

doi:10.3934/dcdss.2017076

Article Contents

2017, Volume 10, Issue 6: 1467-1485. Doi: 10.3934/dcdss.2017076

This issue Previous Article Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity Next Article Cohesive zone-type delamination in visco-elasticity

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

* Corresponding author: D. Wachsmuth.
* Corresponding author: D. Wachsmuth.

Received: July 2016

Revised: October 2016

Published: December 2017

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 49K20; Secondary: 35K87.

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References

	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	J. Diestel and J. J. Uhl, Vector Measures Mathematical Surveys and Monographs. American Mathematical Society, Providence, 1977.
	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.
	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.
	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.
	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.
	G. A. Francfort and U. Stefanelli , Quasi-static evolution for the Armstrong-Frederick hardening-plasticity model, Applied Mathematics Research Express. AMRX, (2013) , 297-344.
	H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1974.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations volume 8 of GAKUTO International Series Mathematical Sciences and Applications, Gakkōtosho, 1996.
	P. Krejčí and M. Liero , Rate independent Kurzweil processes, Applications of Mathematics, 54 (2009) , 117-145. doi: 10.1007/s10492-009-0009-5.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	F. Rindler , Optimal control for nonconvex rate-independent evolution processes, SIAM Journal on Control and Optimization, 47 (2008) , 2773-2794. doi: 10.1137/080718711.
	F. Rindler , Approximation of rate-independent optimal control problems, SIAM Journal on Numerical Analysis, 47 (2009) , 3884-3909. doi: 10.1137/080744050.
	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.
	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.
	U. Stefanelli , Magnetic control of magnetic shape-memory crystals, Phys. B, 407 (2012) , 1316-1321. doi: 10.1016/j.physb.2011.06.043.
	R. Toader and C. Zanini , An artificial viscosity approach to quasistatic crack growth, Bollettino della Unione Matematica Italiana. Serie 9, 2 (2009) , 1-35.
	A. Visintin, Differential Models of Hysteresis volume 111 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.
	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.
	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.
	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.