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Optimal control of a perturbed sweeping process via discrete approximations
1. | Department of Mathematics, Wayne State University, Detroit, Michigan 48202, United States |
2. | Department of Mathematics, Wayne State University, Detroit, MI 48202 |
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] |
S. Adly, T. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program., 148 (2014), 5-47.
doi: 10.1007/s10107-014-0754-4. |
[3] |
J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, Springer, 2005. |
[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] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[6] |
C. Castaing, M. D. P. Monteiro Marques and P. Raynaud de Fitte, Some problems in optimal control governed by the sweeping process, J. Nonlinear Convex Anal., 15 (2014), 1043-1070. |
[7] |
T. H. Cao and B. S. Mordukhovich, Optimality conditions for a controlled sweeping process with applications to the crowd motion model,, preprint, ().
|
[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, In Y. Gao and D. Motreanu, editors Handbook of Nonconvex Analysis, International Press, (2010), 99-182. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Springer, 1989.
doi: 10.1007/978-3-642-61302-9. |
[15] |
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. |
[16] |
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.
doi: 10.1007/3-540-45501-9_1. |
[17] |
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. |
[18] |
M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, Birkhäuser, 1993.
doi: 10.1007/978-3-0348-7614-8. |
[19] |
B. S. Mordukhovich, Sensitivity analysis in nonsmooth optimization, In D. A. Field and V. Komkov, editors, Theoretical Aspects of Industrial Design, SIAM, (1992), pages 32-46. |
[20] |
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. |
[21] |
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Springer, 2006. |
[22] |
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Springer, 2006. |
[23] |
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. |
[24] |
J. S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program., 113 (2008), 345-424.
doi: 10.1007/s10107-006-0052-x. |
[25] |
F. Rindler, Optimal control for nonconvex rate-independent evolution processes, SIAM J. Control Optim., 47 (2008), 2773-2794.
doi: 10.1137/080718711. |
[26] |
F. Rindler, Approximation of rate-independent optimal control problems, SIAM J. Numer. Anal., 47 (2009), 3884-3909.
doi: 10.1137/080744050. |
[27] |
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
[28] |
A. H. Siddiqi, P. Manchanda and M. Brokate, On some recent developments concerning Moreau's sweeping process, Trends in Industrial and Applied Mathematics, Kluwer, 72 (2002), 339-354.
doi: 10.1007/978-1-4613-0263-6_15. |
[29] |
D. E. Stewart, Dynamics with Inequalities: Impacts and Hard Constraints, SIAM, 2011.
doi: 10.1137/1.9781611970715. |
[30] |
A. A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer, 2000.
doi: 10.1007/978-94-015-9490-5. |
[31] |
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] |
S. Adly, T. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program., 148 (2014), 5-47.
doi: 10.1007/s10107-014-0754-4. |
[3] |
J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, Springer, 2005. |
[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] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[6] |
C. Castaing, M. D. P. Monteiro Marques and P. Raynaud de Fitte, Some problems in optimal control governed by the sweeping process, J. Nonlinear Convex Anal., 15 (2014), 1043-1070. |
[7] |
T. H. Cao and B. S. Mordukhovich, Optimality conditions for a controlled sweeping process with applications to the crowd motion model,, preprint, ().
|
[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, In Y. Gao and D. Motreanu, editors Handbook of Nonconvex Analysis, International Press, (2010), 99-182. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Springer, 1989.
doi: 10.1007/978-3-642-61302-9. |
[15] |
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. |
[16] |
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.
doi: 10.1007/3-540-45501-9_1. |
[17] |
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. |
[18] |
M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, Birkhäuser, 1993.
doi: 10.1007/978-3-0348-7614-8. |
[19] |
B. S. Mordukhovich, Sensitivity analysis in nonsmooth optimization, In D. A. Field and V. Komkov, editors, Theoretical Aspects of Industrial Design, SIAM, (1992), pages 32-46. |
[20] |
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. |
[21] |
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Springer, 2006. |
[22] |
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Springer, 2006. |
[23] |
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. |
[24] |
J. S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program., 113 (2008), 345-424.
doi: 10.1007/s10107-006-0052-x. |
[25] |
F. Rindler, Optimal control for nonconvex rate-independent evolution processes, SIAM J. Control Optim., 47 (2008), 2773-2794.
doi: 10.1137/080718711. |
[26] |
F. Rindler, Approximation of rate-independent optimal control problems, SIAM J. Numer. Anal., 47 (2009), 3884-3909.
doi: 10.1137/080744050. |
[27] |
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
[28] |
A. H. Siddiqi, P. Manchanda and M. Brokate, On some recent developments concerning Moreau's sweeping process, Trends in Industrial and Applied Mathematics, Kluwer, 72 (2002), 339-354.
doi: 10.1007/978-1-4613-0263-6_15. |
[29] |
D. E. Stewart, Dynamics with Inequalities: Impacts and Hard Constraints, SIAM, 2011.
doi: 10.1137/1.9781611970715. |
[30] |
A. A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer, 2000.
doi: 10.1007/978-94-015-9490-5. |
[31] |
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