# American Institute of Mathematical Sciences

September  2021, 11(3): 479-498. doi: 10.3934/mcrf.2021009

## Optimal control of elliptic variational inequalities with bounded and unbounded operators

 University of Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, Essen, 45127, Germany

* Corresponding author: irwin.yousept@uni-due.de

Received  November 2018 Revised  November 2019 Published  September 2021 Early access  March 2021

Fund Project: This work was supported by the DFG under the grant YO 159/2-2 within the priority programme SPP 1962 'Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization'

This paper examines optimal control problems governed by elliptic variational inequalities of the second kind with bounded and unbounded operators. To tackle the bounded case, we employ the polyhedricity of the test set appearing in the dual formulation of the governing variational inequality. Based thereon, we are able to prove the directional differentiability of the associated solution operator, which leads to a strong stationary optimality system. The second part of the paper deals with the unbounded case. Due to the non-smoothness of the variational inequality and the unboundedness of the governing elliptic operator, the directional differentiability of the solution operator becomes difficult to handle. Our strategy is to apply the Yosida approximation to the unbounded operator, while the non-smoothness of the variational inequality is still preserved. Based on the developed strong stationary result for the bounded case, we are able to derive optimality conditions for the unbounded case by passing to the limit in the Yosida approximation. Finally, we apply the developed results to Maxwell-type variational inequalities arising in superconductivity.

Citation: Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control and Related Fields, 2021, 11 (3) : 479-498. doi: 10.3934/mcrf.2021009
##### References:
 [1] V. Barbu, Necessary conditions for distributed control problems governed by parabolic variational inequalities, SIAM Journal on Control and Optimization, 19 (1981), 64-86.  doi: 10.1137/0319006. [2] V. Barbu, Optimal Control of Variational Inequalities, Pitman (Advanced Publishing Program), Boston, 1984. [3] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, 2017. doi: 10.1007/978-3-319-48311-5. [4] L. M. Betz, Strong stationarity for optimal control of a non-smooth coupled system: Application to a viscous evolutionary VI coupled with an elliptic PDE, SIAM J. on Optimization (to appear), 29 (2019), 3069–3099. https://spp1962.wias-berlin.de/preprints/083r.pdf. doi: 10.1137/18M1216778. [5] J. F. Bonnans and A. Shapiro, Pertubation Analysis of Optimization Problems, Springer, New York, 2000. doi: 10.1007/978-1-4612-1394-9. [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [7] C. Christof and C. Meyer, Sensitivity analysis for a class of $H_0^1$-elliptic variational inequalities of the second kind, Set-Valued and Variational Analysis, 27 (2019), 469-502.  doi: 10.1007/s11228-018-0495-2. [8] C. Christof and G Wachsmuth, Differential sensitivity analysis of variational inequalities with locally Lipschitz continuous solution operators, Appl. Math. Optim., 81 (2020), 23-62.  doi: 10.1007/s00245-018-09553-y. [9] C. Christof and G Wachsmuth, On the non-polyhedricity of sets with upper and lower bounds in dual spaces, GAMM-Mitt, 40 (2018), 339-350.  doi: 10.1002/gamm.201740005. [10] J. C. De los Reyes, Optimal control of a class of variational inequalities of the second kind, SIAM Journal on Control and Optimization, 49 (2011), 1629-1658.  doi: 10.1137/090764438. [11] J. C. De los Reyes and C. Meyer, Strong stationarity conditions for a class of optimization problems governed by variational inequalities of the second kind, Journal of Optimization Theory and Applications, 168 (2016), 375-409.  doi: 10.1007/s10957-015-0748-2. [12] J. C. De los Reyes and I. Yousept, Optimal control of electrorheological fluids through the action of electric fields, Comput. Optim. Appl., 62 (2015), 241-270.  doi: 10.1007/s10589-014-9705-5. [13] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, Berlin Heidelberg, 2008. [14] Z.-X. He, State constrained control problems governed by variational inequalities, SIAM Journal on Control and Optimization, 25 (1987), 1119-1144.  doi: 10.1137/0325061. [15] R. Herzog, C. Meyer and G. Wachsmuth, B- and strong stationarity for optimal control of static plasticity with hardening, SIAM J. Optim., 23 (2013), 321-352.  doi: 10.1137/110821147. [16] M. Hintermüller and I. Kopacka, Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm, SIAM Journal on Optimization, 20 (2009), 868-902.  doi: 10.1137/080720681. [17] M. Hintermüller and T. M. Surowiec, On the directional differentiability of the solution mapping for a class of variational inequalities of the second kind, Set-Valued Var. Anal., 26 (2018), 631-642.  doi: 10.1007/s11228-017-0408-9. [18] M. Hintermüller and D. Wegner, Optimal control of a semi-discrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control and Optimization, 52 (2014), 747-772.  doi: 10.1137/120865628. [19] K. Ito and K. Kunisch, Optimal control of parabolic variational inequalities, Journal de Mathématiques Pures et Appliqués, 93 (2010), 329-360.  doi: 10.1016/j.matpur.2009.10.005. [20] J.-L. Lions and G. Stampacchia, Variational inequalities, Communications on Pure and Applied Mathematics, 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302. [21] C. Meyer and L. M. Susu, Optimal control of nonsmooth, semilinear parabolic equations, SIAM Journal on Control and Optimization, 55 (2017), 2206-2234.  doi: 10.1137/15M1040426. [22] A. Mielke and T. Roubíček, Rate-Independent Systems, Theory and Application., Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7. [23] F. Mignot, Contrôle dans les inéquations variationelles elliptiques, Journal of Functional Analysis, 22 (1976), 130-185.  doi: 10.1016/0022-1236(76)90017-3. [24] F. Mignot and J.-P. Puel, Optimal control in some variational inequalities, SIAM Journal on Control and Optimization, 22 (1984), 466-476.  doi: 10.1137/0322028. [25] A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [26] H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity, Mathematics of Operations Research, 25 (2000), 1-22.  doi: 10.1287/moor.25.1.1.15213. [27] G. Wachsmuth, A guided tour of polyhedric sets., Journal of Convex Analysis, 26 (2019), 153-188. [28] M. Winckler and I. Yousept, Fully discrete scheme for Bean's critical-state model with temperature effects in superconductivity, SIAM Journal on Numerical Analysis, 57 (2019), 2685-2706.  doi: 10.1137/18M1231407. [29] I. Yousept, Optimal control of quasilinear $\boldsymbol H(\mathbf {curl})$-elliptic partial differential equations in magnetostatic field problems, SIAM J. Control Optim., 51 (2013), 3624-3651.  doi: 10.1137/120904299. [30] I. Yousept, Hyperbolic Maxwell variational inequalities for Bean's critical-state model in type-Ⅱ superconductivity, SIAM J. Numer. Anal., 55 (2017), 2444-2464.  doi: 10.1137/16M1091939. [31] I. Yousept, Optimal control of non-smooth hyperbolic evolution Maxwell equations in type-II superconductivity, SIAM J. Control Optim., 55 (2017), 2305-2332.  doi: 10.1137/16M1074229. [32] I. Yousept, Hyperbolic Maxwell variational inequalities of the second kind, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 34, 23 pp. doi: 10.1051/cocv/2019015.

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##### References:
 [1] V. Barbu, Necessary conditions for distributed control problems governed by parabolic variational inequalities, SIAM Journal on Control and Optimization, 19 (1981), 64-86.  doi: 10.1137/0319006. [2] V. Barbu, Optimal Control of Variational Inequalities, Pitman (Advanced Publishing Program), Boston, 1984. [3] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, 2017. doi: 10.1007/978-3-319-48311-5. [4] L. M. Betz, Strong stationarity for optimal control of a non-smooth coupled system: Application to a viscous evolutionary VI coupled with an elliptic PDE, SIAM J. on Optimization (to appear), 29 (2019), 3069–3099. https://spp1962.wias-berlin.de/preprints/083r.pdf. doi: 10.1137/18M1216778. [5] J. F. Bonnans and A. Shapiro, Pertubation Analysis of Optimization Problems, Springer, New York, 2000. doi: 10.1007/978-1-4612-1394-9. [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [7] C. Christof and C. Meyer, Sensitivity analysis for a class of $H_0^1$-elliptic variational inequalities of the second kind, Set-Valued and Variational Analysis, 27 (2019), 469-502.  doi: 10.1007/s11228-018-0495-2. [8] C. Christof and G Wachsmuth, Differential sensitivity analysis of variational inequalities with locally Lipschitz continuous solution operators, Appl. Math. Optim., 81 (2020), 23-62.  doi: 10.1007/s00245-018-09553-y. [9] C. Christof and G Wachsmuth, On the non-polyhedricity of sets with upper and lower bounds in dual spaces, GAMM-Mitt, 40 (2018), 339-350.  doi: 10.1002/gamm.201740005. [10] J. C. De los Reyes, Optimal control of a class of variational inequalities of the second kind, SIAM Journal on Control and Optimization, 49 (2011), 1629-1658.  doi: 10.1137/090764438. [11] J. C. De los Reyes and C. Meyer, Strong stationarity conditions for a class of optimization problems governed by variational inequalities of the second kind, Journal of Optimization Theory and Applications, 168 (2016), 375-409.  doi: 10.1007/s10957-015-0748-2. [12] J. C. De los Reyes and I. Yousept, Optimal control of electrorheological fluids through the action of electric fields, Comput. Optim. Appl., 62 (2015), 241-270.  doi: 10.1007/s10589-014-9705-5. [13] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, Berlin Heidelberg, 2008. [14] Z.-X. He, State constrained control problems governed by variational inequalities, SIAM Journal on Control and Optimization, 25 (1987), 1119-1144.  doi: 10.1137/0325061. [15] R. Herzog, C. Meyer and G. Wachsmuth, B- and strong stationarity for optimal control of static plasticity with hardening, SIAM J. Optim., 23 (2013), 321-352.  doi: 10.1137/110821147. [16] M. Hintermüller and I. Kopacka, Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm, SIAM Journal on Optimization, 20 (2009), 868-902.  doi: 10.1137/080720681. [17] M. Hintermüller and T. M. Surowiec, On the directional differentiability of the solution mapping for a class of variational inequalities of the second kind, Set-Valued Var. Anal., 26 (2018), 631-642.  doi: 10.1007/s11228-017-0408-9. [18] M. Hintermüller and D. Wegner, Optimal control of a semi-discrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control and Optimization, 52 (2014), 747-772.  doi: 10.1137/120865628. [19] K. Ito and K. Kunisch, Optimal control of parabolic variational inequalities, Journal de Mathématiques Pures et Appliqués, 93 (2010), 329-360.  doi: 10.1016/j.matpur.2009.10.005. [20] J.-L. Lions and G. Stampacchia, Variational inequalities, Communications on Pure and Applied Mathematics, 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302. [21] C. Meyer and L. M. Susu, Optimal control of nonsmooth, semilinear parabolic equations, SIAM Journal on Control and Optimization, 55 (2017), 2206-2234.  doi: 10.1137/15M1040426. [22] A. Mielke and T. Roubíček, Rate-Independent Systems, Theory and Application., Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7. [23] F. Mignot, Contrôle dans les inéquations variationelles elliptiques, Journal of Functional Analysis, 22 (1976), 130-185.  doi: 10.1016/0022-1236(76)90017-3. [24] F. Mignot and J.-P. Puel, Optimal control in some variational inequalities, SIAM Journal on Control and Optimization, 22 (1984), 466-476.  doi: 10.1137/0322028. [25] A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [26] H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity, Mathematics of Operations Research, 25 (2000), 1-22.  doi: 10.1287/moor.25.1.1.15213. [27] G. Wachsmuth, A guided tour of polyhedric sets., Journal of Convex Analysis, 26 (2019), 153-188. [28] M. Winckler and I. Yousept, Fully discrete scheme for Bean's critical-state model with temperature effects in superconductivity, SIAM Journal on Numerical Analysis, 57 (2019), 2685-2706.  doi: 10.1137/18M1231407. [29] I. Yousept, Optimal control of quasilinear $\boldsymbol H(\mathbf {curl})$-elliptic partial differential equations in magnetostatic field problems, SIAM J. Control Optim., 51 (2013), 3624-3651.  doi: 10.1137/120904299. [30] I. Yousept, Hyperbolic Maxwell variational inequalities for Bean's critical-state model in type-Ⅱ superconductivity, SIAM J. Numer. Anal., 55 (2017), 2444-2464.  doi: 10.1137/16M1091939. [31] I. Yousept, Optimal control of non-smooth hyperbolic evolution Maxwell equations in type-II superconductivity, SIAM J. Control Optim., 55 (2017), 2305-2332.  doi: 10.1137/16M1074229. [32] I. Yousept, Hyperbolic Maxwell variational inequalities of the second kind, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 34, 23 pp. doi: 10.1051/cocv/2019015.
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