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. ColomboR. HenrionN. 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. ColomboR. HenrionN. 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. DonchevE. 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.

show all references

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. ColomboR. HenrionN. 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. ColomboR. HenrionN. 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. DonchevE. 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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Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3331-3358. doi: 10.3934/dcdsb.2016100

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