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December  2016, 21(10): 3331-3358. doi: 10.3934/dcdsb.2016100

## 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

Received  November 2015 Revised  February 2016 Published  November 2016

The paper addresses an optimal control problem for a perturbed sweeping process of the rate-independent hysteresis type described by a controlled play-and stop" operator with separately controlled perturbations. This problem can be reduced to dynamic optimization of a state-constrained unbounded differential inclusion with highly irregular data that cannot be treated by means of known results in optimal control theory for differential inclusions. We develop the method of discrete approximations, which allows us to adequately replace the original optimal control problem by a sequence of well-posed finite-dimensional optimization problems whose optimal solutions strongly converge to that of the controlled perturbed sweeping process. To solve the discretized control systems, we derive effective necessary optimality conditions by using second-order generalized differential tools of variational analysis that explicitly calculated in terms of the given problem data.
Citation: 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
##### 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] 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] 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] R. B. Vinter, Optimal Control, Birkhaüser, 2000.
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