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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

F. H. Clarke, Yu. S Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory Springer, 1998. Google Scholar

[7]

G. ColomboR. HenrionN. D. Hoang and B. S. Mordukhovich, Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst.-Ser. B, 19 (2012), 117-159.   Google Scholar

[8]

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.  Google Scholar

[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. Google Scholar

[10]

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.  Google Scholar

[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.  Google Scholar

[12]

R. HenrionB. 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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[19]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅰ: Basic Theory Springer, 2006. Google Scholar

[20]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications Springer, 2006. Google Scholar

[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. Google Scholar

[22]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis. Springer-Verlag, Berlin, 1998. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[25]

R. B. Vinter, Optimal Control Birkhaüser, 2000. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

F. H. Clarke, Yu. S Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory Springer, 1998. Google Scholar

[7]

G. ColomboR. HenrionN. D. Hoang and B. S. Mordukhovich, Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst.-Ser. B, 19 (2012), 117-159.   Google Scholar

[8]

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.  Google Scholar

[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. Google Scholar

[10]

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.  Google Scholar

[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.  Google Scholar

[12]

R. HenrionB. 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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[19]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅰ: Basic Theory Springer, 2006. Google Scholar

[20]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications Springer, 2006. Google Scholar

[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. Google Scholar

[22]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis. Springer-Verlag, Berlin, 1998. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[25]

R. B. Vinter, Optimal Control Birkhaüser, 2000. Google Scholar

Figure 1.  Direction of optimal control
Figure 2.  Two-dimensional motion.
Figure 3.  Crowd motion model in a corridor
Figure 4.  Two participants out of contact for $t<t_1$.
Figure 5.  Two participants in contact for $t\ge t_1$.
Figure 6.  Out of contact situation for two adjacent participants when $t<t_1$
Figure 7.  All the participants in contact for $t\ge t_1$
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