# American Institute of Mathematical Sciences

August  2019, 24(8): 4191-4216. doi: 10.3934/dcdsb.2019078

## Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model

 1 Department of Applied Mathematics and Statistics, State University of New York–Korea, Yeonsu-Gu, Incheon 21985, Republic of Korea, Springfield, MO 65801-2604, USA 2 Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA 3 RUDN University, Moscow 117198, Russia

Received  February 2018 Revised  October 2018 Published  August 2019 Early access  April 2019

Fund Project: Research of this author was partly supported by the MSIT (Ministry of Science and ICT), Korea, under the ICT Consilience Creative Program (IITP-2017-R0346-16-1007) supervised by the IITP (Institute for Information & Communications Technology Promotion).
Research of this author was partly supported by the US National Science Foundation under grant DMS-1512846, by the US Air Force Office of Scientific Research under grant #15RT0462, and by the RUDN University Program 5-100.

This paper concerns optimal control of a nonconvex perturbed sweeping process and its applications to optimization of the planar crowd motion model of traffic equilibria. The obtained theoretical results allow us to investigate a dynamic optimization problem for the microscopic planar crown motion model with finitely many participants and completely solve it analytically in the case of two participants.

Citation: Tan H. Cao, Boris S. Mordukhovich. Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4191-4216. doi: 10.3934/dcdsb.2019078
##### References:
 [1] L. Adam and J. V. Outrata, On optimal control of a sweeping process coupled with an ordinary differential equation, Discrete Contin. Dyn. Syst.–Ser. B, 19 (2014), 2709-2738.  doi: 10.3934/dcdsb.2014.19.2709. [2] C. E. Arround and G. Colombo, A maximum principle of the controlled sweeping process, Set-Valued Var. Anal., 26 (2018), 607-629.  doi: 10.1007/s11228-017-0400-4. [3] M. Brokate and P. Krejčí, Optimal control of ODE systems involving a rate independent variational inequality, Discrete Contin. Dyn. Syst.–Ser. B, 18 (2013), 331-348.  doi: 10.3934/dcdsb.2013.18.331. [4] T. H. Cao and B. S. Mordukhovich, Optimal control of a perturbed sweeping process via discrete approximations, Discrete Contin. Dyn. Sysy.–Ser. B, 21 (2016), 3331-3358.  doi: 10.3934/dcdsb.2016100. [5] T. H. Cao and B. S. Mordukhovich, Optimality conditions for a controlled sweeping process with applications to the crowd motion model, Discrete Contin. Dyn. Syst.-Ser. B, 22 (2017), 267-306.  doi: 10.3934/dcdsb.2017014. [6] T. H. Cao and B. S. Mordukhovich, Optimal control of a nonconvex perturbed sweeping process, J. Diff. Eqs., 266 (2019), 1003-1050.  doi: 10.1016/j.jde.2018.07.066. [7] F. H. Clarke, Yu. S Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, 1998. [8] G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst.–Ser. B, 19 (2012), 117-159. [9] G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process over polyhedral controlled sets, J. Diff. Eqs., 260 (2016), 3397-3447.  doi: 10.1016/j.jde.2015.10.039. [10] G. Colombo and L. Thibault, Prox-regular sets and applications, Handbook of Nonconvex Analysis, International Press, (2010), 99–182. [11] M. d. R. de Pinho, M. M. A. Ferreira and G. V. Smirnov, Optimal control involving sweeping processes, Set-Valued Var. Anal., 2018, 1–26. doi: 10.1007/s11228-018-0501-8. [12] T. Donchev, E. Farkhi and B. S. Mordukhovich, Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces, J. Diff. Eqs., 243 (2007), 301-328.  doi: 10.1016/j.jde.2007.05.011. [13] J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program., 104 (2005), 347-373.  doi: 10.1007/s10107-005-0619-y. [14] P. E. Kloeden and E. Platen., Numerical Solution of Stochastic Differential Equations., Springer, 1992. doi: 10.1007/978-3-662-12616-5. [15] B. Maury and J. Venel, A discrete model for crowd motion, ESAIM: M2AN, 45 (2011), 145-168.  doi: 10.1051/m2an/2010035. [16] B. S. Mordukhovich, Discrete approximations and refined Euler-Lagrange conditions for differential inclusions, SIAM J. Control Optim., 33 (1995), 882-915.  doi: 10.1137/S0363012993245665. [17] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅰ: Basic Theory, Springer, 2006. [18] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications, Springer, 2006. [19] J. J. Moreau, On unilateral constraints, friction and plasticity, New Variational Techniques in Mathematical Physics, Proceedings of C.I.M.E. Summer Schools, pages 173–322. Cremonese, 1974. [20] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3. [21] A. A. Tolstonogov, Control sweeping process, J. Convex Anal., 23 (2016), 1099-1123. [22] J. Venel, A numerical scheme for a class of sweeping process, Numerische Mathematik, 118 (2011), 367-400.  doi: 10.1007/s00211-010-0329-0. [23] R. B. Vinter, Optimal Control, Birkhaüser, 2000.

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##### References:
 [1] L. Adam and J. V. Outrata, On optimal control of a sweeping process coupled with an ordinary differential equation, Discrete Contin. Dyn. Syst.–Ser. B, 19 (2014), 2709-2738.  doi: 10.3934/dcdsb.2014.19.2709. [2] C. E. Arround and G. Colombo, A maximum principle of the controlled sweeping process, Set-Valued Var. Anal., 26 (2018), 607-629.  doi: 10.1007/s11228-017-0400-4. [3] M. Brokate and P. Krejčí, Optimal control of ODE systems involving a rate independent variational inequality, Discrete Contin. Dyn. Syst.–Ser. B, 18 (2013), 331-348.  doi: 10.3934/dcdsb.2013.18.331. [4] T. H. Cao and B. S. Mordukhovich, Optimal control of a perturbed sweeping process via discrete approximations, Discrete Contin. Dyn. Sysy.–Ser. B, 21 (2016), 3331-3358.  doi: 10.3934/dcdsb.2016100. [5] T. H. Cao and B. S. Mordukhovich, Optimality conditions for a controlled sweeping process with applications to the crowd motion model, Discrete Contin. Dyn. Syst.-Ser. B, 22 (2017), 267-306.  doi: 10.3934/dcdsb.2017014. [6] T. H. Cao and B. S. Mordukhovich, Optimal control of a nonconvex perturbed sweeping process, J. Diff. Eqs., 266 (2019), 1003-1050.  doi: 10.1016/j.jde.2018.07.066. [7] F. H. Clarke, Yu. S Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, 1998. [8] G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst.–Ser. B, 19 (2012), 117-159. [9] G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process over polyhedral controlled sets, J. Diff. Eqs., 260 (2016), 3397-3447.  doi: 10.1016/j.jde.2015.10.039. [10] G. Colombo and L. Thibault, Prox-regular sets and applications, Handbook of Nonconvex Analysis, International Press, (2010), 99–182. [11] M. d. R. de Pinho, M. M. A. Ferreira and G. V. Smirnov, Optimal control involving sweeping processes, Set-Valued Var. Anal., 2018, 1–26. doi: 10.1007/s11228-018-0501-8. [12] T. Donchev, E. Farkhi and B. S. Mordukhovich, Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces, J. Diff. Eqs., 243 (2007), 301-328.  doi: 10.1016/j.jde.2007.05.011. [13] J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program., 104 (2005), 347-373.  doi: 10.1007/s10107-005-0619-y. [14] P. E. Kloeden and E. Platen., Numerical Solution of Stochastic Differential Equations., Springer, 1992. doi: 10.1007/978-3-662-12616-5. [15] B. Maury and J. Venel, A discrete model for crowd motion, ESAIM: M2AN, 45 (2011), 145-168.  doi: 10.1051/m2an/2010035. [16] B. S. Mordukhovich, Discrete approximations and refined Euler-Lagrange conditions for differential inclusions, SIAM J. Control Optim., 33 (1995), 882-915.  doi: 10.1137/S0363012993245665. [17] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅰ: Basic Theory, Springer, 2006. [18] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications, Springer, 2006. [19] J. J. Moreau, On unilateral constraints, friction and plasticity, New Variational Techniques in Mathematical Physics, Proceedings of C.I.M.E. Summer Schools, pages 173–322. Cremonese, 1974. [20] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3. [21] A. A. Tolstonogov, Control sweeping process, J. Convex Anal., 23 (2016), 1099-1123. [22] J. Venel, A numerical scheme for a class of sweeping process, Numerische Mathematik, 118 (2011), 367-400.  doi: 10.1007/s00211-010-0329-0. [23] R. B. Vinter, Optimal Control, Birkhaüser, 2000.
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