Geodesic
Optimal control of differential inclusions, II: sweeping
The Bulletin of Irkutsk State University. Series Mathematics, Tome 31 (2020), pp. 62-77
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper is devoted to optimal control of dynamical systems governed by differential inclusions with discontinuous velocity mappings. This framework mostly concerns a new class of optimal control problems described by various versions of the so-called sweeping/Moreau processes that are very challenging mathematically and highly important in applications to mechanics, engineering, economics, robotics, etc. Our approach is based on developing the method of discrete approximations for optimal control problems of such differential inclusions that addresses both numerical and qualitative aspects of optimal control. In this way we establish necessary optimality conditions for optimal solutions to sweeping differential inclusions and discuss their various applications. Deriving necessary optimality conditions strongly involves advanced tools of first-order and second-order variational analysis and generalized differentiation.
Keywords: optimal control, differential inclusions, variational analysis, sweeping processes, discrete approximations, generalized differentiation. 
@article{IIGUM_2020_31_a4,
     author = {B. Sh. Mordukhovich},
     title = {Optimal control of differential {inclusions,~II:~sweeping}},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {62--77},
     year = {2020},
     volume = {31},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2020_31_a4/}
}
TY  - JOUR
AU  - B. Sh. Mordukhovich
TI  - Optimal control of differential inclusions, II: sweeping
JO  - The Bulletin of Irkutsk State University. Series Mathematics
PY  - 2020
SP  - 62
EP  - 77
VL  - 31
UR  - http://geodesic.mathdoc.fr/item/IIGUM_2020_31_a4/
LA  - en
ID  - IIGUM_2020_31_a4
ER  - 
%0 Journal Article
%A B. Sh. Mordukhovich
%T Optimal control of differential inclusions, II: sweeping
%J The Bulletin of Irkutsk State University. Series Mathematics
%D 2020
%P 62-77
%V 31
%U http://geodesic.mathdoc.fr/item/IIGUM_2020_31_a4/
%G en
%F IIGUM_2020_31_a4
B. Sh. Mordukhovich. Optimal control of differential inclusions, II: sweeping. The Bulletin of Irkutsk State University. Series Mathematics, Tome 31 (2020), pp. 62-77. http://geodesic.mathdoc.fr/item/IIGUM_2020_31_a4/

[1] Arround C. E., Colombo G., “A maximum principle for the controlled sweeping process”, Set-Valued Var. Anal., 26 (2018), 607–629 | DOI | MR

[2] Brokate M., Krejčí P., “Optimal control of ODE systems involving a rate independent variational inequality”, Disc. Contin. Dyn. Syst. Ser. B, 18 (2013), 331–348 | DOI | MR | Zbl

[3] Cao T. H., Mordukhovich B. S., “Optimal control of a perturbed sweeping process via discrete approximations”, Disc. Contin. Dyn. Syst. Ser. B, 21 (2016), 3331–3358 | DOI | MR | Zbl

[4] Cao T. H., Mordukhovich B. S., “Optimality conditions for a controlled sweeping process with applications to the crowd motion model”, Disc. Contin. Dyn. Syst. Ser. B, 22 (2017), 267–306 | DOI | MR | Zbl

[5] Cao T. H., Mordukhovich B. S., “Optimal control of a nonconvex perturbed sweeping process”, J. Diff. Eqs., 266 (2019), 1003–1050 | DOI | MR | Zbl

[6] Cao T. H., Mordukhovich B. S., “Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model”, Disc. Contin. Dyn. Syst., Ser. B, 24 (2019), 4191–4216 | MR | Zbl

[7] Colombo G., Henrion R., Hoang N. D., Mordukhovich B. S., “Optimal control of the sweeping process”, Dyn. Contin. Discrete Impuls. Syst. Ser. B, 19 (2012), 117–159 | MR | Zbl

[8] Colombo G., Henrion R., Hoang N. D., Mordukhovich B. S., “Optimal control of the sweeping process over polyhedral controlled sets”, J. Diff. Eqs., 260 (2016), 3397–3447 | DOI | MR | Zbl

[9] Colombo G., Mordukhovich B. S., Nguyen D., “Optimization of a perturbed sweeping process by discontinuous controls”, SIAM J. Control Optim. (to appear) , arXiv: 1808.04041

[10] Colombo G., Mordukhovich B. S., Nguyen D., “Optimal control of sweeping processes in robotics and traffic flow models”, J. Optim. Theory Appl., 182 (2019), 439–472 | DOI | MR | Zbl

[11] Colombo G., Thibault L., “Prox-regular sets and applications”, Handbook of Nonconvex Analysis, eds. D.Y. Gao, D. Motreanu, International Press, Boston, 2010, 99–182 | MR | Zbl

[12] De Pinho M.d.R., Ferreira M. M.A., Smirnov G. V., “Optimal control involving sweeping processes”, Set-Valued Var. Anal., 27 (2019), 523–548 | DOI | MR | Zbl

[13] Edmond J. F., Thibault L., “Relaxation of an optimal control problem involving a perturbed sweeping process”, Math. Program., 104 (2005), 347–373 | DOI | MR | Zbl

[14] Hedjar R., Bounkhel M., “Real-time obstacle avoidance for a swarm of autonomous mobile robots”, Int. J. Adv. Robot. Syst., 11 (2014), 1–12 | DOI

[15] Hoang N. D., Mordukhovich B. S., “Extended Euler-Lagrange and Hamiltonian formalisms in optimal control of sweeping processes with controlled sweeping sets”, J. Optim. Theory Appl., 180 (2019), 256–289 | DOI | MR | Zbl

[16] Krasnosel'skii A. M., Pokrovskii A. V., Systems with Hysteresis, Springer, Berlin, 1989 | DOI | MR | Zbl

[17] Lovas G. G., “Modeling and simulation of pedestrian traffic flow”, Transpn. Res.-B, 28B (1994), 429–443 | DOI

[18] Mordukhovich B. S., “Sensitivity analysis in nonsmooth optimization”, Theoretical Aspects of Industrial Design (Philadelphia, Pennsylvania), SIAM Proc. Appl. Math., 58, eds. D.A. Field, V. Komkov, 32–46 | MR

[19] Mordukhovich B. S., Variational Analysis and Generalized Differentiation, v. I, Basic Theory, Springer, Berlin, 2006 | DOI | MR

[20] Mordukhovich B. S., Variational Analysis and Generalized Differentiation, v. II, Applications, Springer, Berlin, 2006 | DOI | MR

[21] Mordukhovich B. S., Variational Analysis and Applications, Springer, New York, 2018 | DOI | MR | Zbl

[22] Mordukhovich B. S., “Optimal control of differential inclusions, I: Lipshitzian case”, The Bulletin of Irkutsk State University. Series Mathematics, 30 (2019), 45–58 | DOI | Zbl

[23] Mordukhovich B. S., Outrata J. V., “Coderivative analysis of quasi-variational inequalities with mapplications to stability and optimization”, SIAM J. Optim., 18 (2007), 389–412 | DOI | MR | Zbl

[24] Mordukhovich B. S., Rockafellar R. T., “Second-order subdifferential calculus with applications to tilt stability in optimization”, SIAM J. Optim., 22 (2012), 953–986 | DOI | MR | Zbl

[25] Moreau J. J., “On unilateral constraints, friction and plasticity”, New Variational Techniques in Mathematical Physics, Proceedings from CIME (Cremonese, Rome, 1974), eds. G. Capriz, G. Stampacchia, 173–322 | MR

[26] Tolstonogov A. A., “Control sweeping process”, J. Convex Anal., 23 (2016), 1099–1123 | MR | Zbl

[27] Venel J., “A numerical scheme for a class of sweeping process”, Numerische Mathematik, 118 (2011), 451–484 | DOI | MR