# American Institute of Mathematical Sciences

March  2017, 22(2): 267-306. doi: 10.3934/dcdsb.2017014

## Optimality conditions for a controlled sweeping process with applications to the crowd motion model

 1 Vietnamese-German University, Le Lai Street, Hoa Phu Ward, Thu Dau Mot City, Binh Duong Province, Vietnam 2 Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA and RUDN University, Moscow 117198, Russia

Received  November 2015 Revised  November 2016 Published  December 2016

Fund Project: This research was partly supported by the National Science Foundation under grants DMS-1007132 and DMS-1512846, by the Air Force Office of Scientific Research grant #15RT0462, and by the Ministry of Education and Science of the Russian Federation (the Agreement number 02.a03.21.0008 of 24 June 2016).

The paper concerns the study and applications of a new class of optimal control problems governed by a perturbed sweeping process of the hysteresis type with control functions acting in both play-and-stop operator and additive perturbations. Such control problems can be reduced to optimization of discontinuous and unbounded differential inclusions with pointwise state constraints, which are immensely challenging in control theory and prevent employing conventional variation techniques to derive necessary optimality conditions. We develop the method of discrete approximations married with appropriate generalized differential tools of modern variational analysis to overcome principal difficulties in passing to the limit from optimality conditions for finite-difference systems. This approach leads us to nondegenerate necessary conditions for local minimizers of the controlled sweeping process expressed entirely via the problem data. Besides illustrative examples, we apply the obtained results to an optimal control problem associated with of the crowd motion model of traffic flow in a corridor, which is formulated in this paper. The derived optimality conditions allow us to develop an effective procedure to solve this problem in a general setting and completely calculate optimal solutions in particular situations.

Citation: Tan H. Cao, Boris S. Mordukhovich. Optimality conditions for a controlled sweeping process with applications to the crowd motion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 267-306. doi: 10.3934/dcdsb.2017014
##### 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] A. V. Arutyunov and S. M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim., 35 (1997), 930-952.  doi: 10.1137/S036301299426996X. [3] A. V. Arutyunov and D. Yu. Karamzin, Nondegenerate necessary optimality conditions for the optimal control problems with equality type state constraints, J. Global Optim., 64 (2016), 623-647.  doi: 10.1007/s10898-015-0272-9. [4] 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. [5] 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. [6] F. H. Clarke, Yu. S Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory Springer, 1998. [7] G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst.-Ser. B, 19 (2012), 117-159. [8] 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. [9] G. Colombo and L. Thibault, Prox-regular sets and applications, In Y. Gao and D. Motreanu, editors Handbook of Nonconvex Analysis, International Press (2010), 99-182. [10] 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. [11] 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. [12] R. Henrion, B. S. Mordukhovich and N. M. Nam, Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities, SIAM J. Optim., 20 (2010), 2199-2227.  doi: 10.1137/090766413. [13] P. Krečí, Evolution variational inequalities and multidimensional hysteresis operators, In P. Drabek, P. Krečí and P. Takac, editors, Nonlinear Differential Equations, Res. Notes Math. 404, pages 47-110. Chapman & Hall, CRC, 1999. [14] M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, In B. Brogliato, editor, Impacts in Mechanical Systems, Lecture Notes in Phys. 551, pages 1-60, Springer 2000. [15] B. Maury and J. Venel, A mathematical framework for a crowd motion model, C. R. Acad. Sci. Paris Ser. I, 346 (2008), 1245-1250.  doi: 10.1016/j.crma.2008.10.014. [16] B. Maury and J. Venel, Handling of contacts in crowd motion simulations, In C. AppertRolland et al. , editors, Traffic and Granular Flow'07, pages 171-180, Springer 2009. [17] A. Mielke, Evolution of rate-independent systems, Evolutionary equations, In C. M. Dafermos and E. Feireis, editors, Handbook of Differential Equations, Elsevier, 2 (2005), 461-559. [18] 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. [19] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅰ: Basic Theory Springer, 2006. [20] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications Springer, 2006. [21] J. J. Moreau, On unilateral constraints, friction and plasticity, In G. Capriz and G. Stampacchia, editors, New Variational Techniques in Mathematical Physics, Proceedings of C. I. M. E. Summer Schools, pages 173-322. Cremonese, 1974. [22] R. T. Rockafellar and R. J. -B. Wets, Variational Analysis. Springer-Verlag, Berlin, 1998. [23] A. H. Siddiqi, P. Manchanda and M. Brokate, On some recent developments concerning Moreau's sweeping process, In A. H. Siddiqi and M. Kočvara, editors, Trends in Industrial and Applied Mathematics, Kluwer, 72 (2002), 339-354. [24] J. Venel, A numerical scheme for a class of sweeping process, Numerische Mathematik, 118 (2011), 367-400.  doi: 10.1007/s00211-010-0329-0. [25] 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] A. V. Arutyunov and S. M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim., 35 (1997), 930-952.  doi: 10.1137/S036301299426996X. [3] A. V. Arutyunov and D. Yu. Karamzin, Nondegenerate necessary optimality conditions for the optimal control problems with equality type state constraints, J. Global Optim., 64 (2016), 623-647.  doi: 10.1007/s10898-015-0272-9. [4] 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. [5] 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. [6] F. H. Clarke, Yu. S Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory Springer, 1998. [7] G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst.-Ser. B, 19 (2012), 117-159. [8] 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. [9] G. Colombo and L. Thibault, Prox-regular sets and applications, In Y. Gao and D. Motreanu, editors Handbook of Nonconvex Analysis, International Press (2010), 99-182. [10] 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. [11] 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. [12] R. Henrion, B. S. Mordukhovich and N. M. Nam, Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities, SIAM J. Optim., 20 (2010), 2199-2227.  doi: 10.1137/090766413. [13] P. Krečí, Evolution variational inequalities and multidimensional hysteresis operators, In P. Drabek, P. Krečí and P. Takac, editors, Nonlinear Differential Equations, Res. Notes Math. 404, pages 47-110. Chapman & Hall, CRC, 1999. [14] M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, In B. Brogliato, editor, Impacts in Mechanical Systems, Lecture Notes in Phys. 551, pages 1-60, Springer 2000. [15] B. Maury and J. Venel, A mathematical framework for a crowd motion model, C. R. Acad. Sci. Paris Ser. I, 346 (2008), 1245-1250.  doi: 10.1016/j.crma.2008.10.014. [16] B. Maury and J. Venel, Handling of contacts in crowd motion simulations, In C. AppertRolland et al. , editors, Traffic and Granular Flow'07, pages 171-180, Springer 2009. [17] A. Mielke, Evolution of rate-independent systems, Evolutionary equations, In C. M. Dafermos and E. Feireis, editors, Handbook of Differential Equations, Elsevier, 2 (2005), 461-559. [18] 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. [19] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅰ: Basic Theory Springer, 2006. [20] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications Springer, 2006. [21] J. J. Moreau, On unilateral constraints, friction and plasticity, In G. Capriz and G. Stampacchia, editors, New Variational Techniques in Mathematical Physics, Proceedings of C. I. M. E. Summer Schools, pages 173-322. Cremonese, 1974. [22] R. T. Rockafellar and R. J. -B. Wets, Variational Analysis. Springer-Verlag, Berlin, 1998. [23] A. H. Siddiqi, P. Manchanda and M. Brokate, On some recent developments concerning Moreau's sweeping process, In A. H. Siddiqi and M. Kočvara, editors, Trends in Industrial and Applied Mathematics, Kluwer, 72 (2002), 339-354. [24] J. Venel, A numerical scheme for a class of sweeping process, Numerische Mathematik, 118 (2011), 367-400.  doi: 10.1007/s00211-010-0329-0. [25] R. B. Vinter, Optimal Control Birkhaüser, 2000.
Direction of optimal control
Two-dimensional motion.
Crowd motion model in a corridor
Two participants out of contact for $t<t_1$.
Two participants in contact for $t\ge t_1$.
Out of contact situation for two adjacent participants when $t<t_1$
All the participants in contact for $t\ge t_1$
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