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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

Tan H. Cao 1, and  Boris S. Mordukhovich 2,

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.
Keywords: Controlled sweeping process, hysteresis, generalized differentiation, variational analysis, optimality conditions., discrete approximations, play-and-stop operator.
Mathematics Subject Classification: Primary: 49M25, 47J40; Secondary: 90C30, 49J5.
Citation: Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3331-3358. doi: 10.3934/dcdsb.2016100
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show all references

References:
[1]

Discrete Contin. Dyn. Syst.-Ser. B, 19 (2014), 2709-2738. doi: 10.3934/dcdsb.2014.19.2709.  Google Scholar

[2]

Math. Program., 148 (2014), 5-47. doi: 10.1007/s10107-014-0754-4.  Google Scholar

[3]

Springer, 2005.  Google Scholar

[4]

Discrete Contin. Dyn. Syst.-Ser. B, 18 (2013), 331-348. doi: 10.3934/dcdsb.2013.18.331.  Google Scholar

[5]

Springer, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[6]

J. Nonlinear Convex Anal., 15 (2014), 1043-1070.  Google Scholar

[7]

T. H. Cao and B. S. Mordukhovich, Optimality conditions for a controlled sweeping process with applications to the crowd motion model,, preprint, ().   Google Scholar

[8]

Dyn. Contin. Discrete Impuls. Syst.-Ser. B, 19 (2012), 117-159.  Google Scholar

[9]

J. Diff. Eqs., 260 (2016), 3397-3447. doi: 10.1016/j.jde.2015.10.039.  Google Scholar

[10]

In Y. Gao and D. Motreanu, editors Handbook of Nonconvex Analysis, International Press, (2010), 99-182.  Google Scholar

[11]

J. Diff. Eqs., 243 (2007), 301-328. doi: 10.1016/j.jde.2007.05.011.  Google Scholar

[12]

Math. Program., 104 (2005), 347-373. doi: 10.1007/s10107-005-0619-y.  Google Scholar

[13]

SIAM J. Optim., 20 (2010), 2199-2227. doi: 10.1137/090766413.  Google Scholar

[14]

Springer, 1989. doi: 10.1007/978-3-642-61302-9.  Google Scholar

[15]

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

[16]

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.  Google Scholar

[17]

C. R. Acad. Sci. Paris Ser. I, 346 (2008), 1245-1250. doi: 10.1016/j.crma.2008.10.014.  Google Scholar

[18]

Birkhäuser, 1993. doi: 10.1007/978-3-0348-7614-8.  Google Scholar

[19]

In D. A. Field and V. Komkov, editors, Theoretical Aspects of Industrial Design, SIAM, (1992), pages 32-46.  Google Scholar

[20]

SIAM J. Control Optim., 33 (1995), 882-915. doi: 10.1137/S0363012993245665.  Google Scholar

[21]

Springer, 2006.  Google Scholar

[22]

Springer, 2006.  Google Scholar

[23]

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

[24]

Math. Program., 113 (2008), 345-424. doi: 10.1007/s10107-006-0052-x.  Google Scholar

[25]

SIAM J. Control Optim., 47 (2008), 2773-2794. doi: 10.1137/080718711.  Google Scholar

[26]

SIAM J. Numer. Anal., 47 (2009), 3884-3909. doi: 10.1137/080744050.  Google Scholar

[27]

Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[28]

Trends in Industrial and Applied Mathematics, Kluwer, 72 (2002), 339-354. doi: 10.1007/978-1-4613-0263-6_15.  Google Scholar

[29]

SIAM, 2011. doi: 10.1137/1.9781611970715.  Google Scholar

[30]

Kluwer, 2000. doi: 10.1007/978-94-015-9490-5.  Google Scholar

[31]

Birkhaüser, 2000.  Google Scholar

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