American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process
  • DCDS-B Home
  • This Issue
  • Next Article
    Minimizing $\mathcal R_0$ for in-host virus model with periodic combination antiviral therapy
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
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.  doi: 10.3934/dcdsb.2014.19.2709.  Google Scholar

[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.  doi: 10.1007/s10107-014-0754-4.  Google Scholar

[3]

J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis,, Springer, (2005).   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.  doi: 10.3934/dcdsb.2013.18.331.  Google Scholar

[5]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[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.   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]

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.   Google Scholar

[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.  doi: 10.1016/j.jde.2015.10.039.  Google Scholar

[10]

G. Colombo and L. Thibault, Prox-regular sets and applications,, In Y. Gao and D. Motreanu, (2010), 99.   Google Scholar

[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.  doi: 10.1016/j.jde.2007.05.011.  Google Scholar

[12]

J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process,, Math. Program., 104 (2005), 347.  doi: 10.1007/s10107-005-0619-y.  Google Scholar

[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.  doi: 10.1137/090766413.  Google Scholar

[14]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis,, Springer, (1989).  doi: 10.1007/978-3-642-61302-9.  Google Scholar

[15]

P. Krečí, Evolution variational inequalities and multidimensional hysteresis operators,, In P. Drabek, (1999), 47.   Google Scholar

[16]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process,, In B. Brogliato, (2000), 1.  doi: 10.1007/3-540-45501-9_1.  Google Scholar

[17]

B. Maury and J. Venel, A mathematical framework for a crowd motion model,, C. R. Acad. Sci. Paris Ser. I, 346 (2008), 1245.  doi: 10.1016/j.crma.2008.10.014.  Google Scholar

[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.  Google Scholar

[19]

B. S. Mordukhovich, Sensitivity analysis in nonsmooth optimization,, In D. A. Field and V. Komkov, (1992), 32.   Google Scholar

[20]

B. S. Mordukhovich, Discrete approximations and refined Euler-Lagrange conditions for differential inclusions,, SIAM J. Control Optim., 33 (1995), 882.  doi: 10.1137/S0363012993245665.  Google Scholar

[21]

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

[22]

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

[23]

J. J. Moreau, On unilateral constraints, friction and plasticity,, In G. Capriz and G. Stampacchia, (1974), 173.   Google Scholar

[24]

J. S. Pang and D. E. Stewart, Differential variational inequalities,, Math. Program., 113 (2008), 345.  doi: 10.1007/s10107-006-0052-x.  Google Scholar

[25]

F. Rindler, Optimal control for nonconvex rate-independent evolution processes,, SIAM J. Control Optim., 47 (2008), 2773.  doi: 10.1137/080718711.  Google Scholar

[26]

F. Rindler, Approximation of rate-independent optimal control problems,, SIAM J. Numer. Anal., 47 (2009), 3884.  doi: 10.1137/080744050.  Google Scholar

[27]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis,, Springer-Verlag, (1998).  doi: 10.1007/978-3-642-02431-3.  Google Scholar

[28]

A. H. Siddiqi, P. Manchanda and M. Brokate, On some recent developments concerning Moreau's sweeping process,, Trends in Industrial and Applied Mathematics, 72 (2002), 339.  doi: 10.1007/978-1-4613-0263-6_15.  Google Scholar

[29]

D. E. Stewart, Dynamics with Inequalities: Impacts and Hard Constraints,, SIAM, (2011).  doi: 10.1137/1.9781611970715.  Google Scholar

[30]

A. A. Tolstonogov, Differential Inclusions in a Banach Space,, Kluwer, (2000).  doi: 10.1007/978-94-015-9490-5.  Google Scholar

[31]

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.  doi: 10.3934/dcdsb.2014.19.2709.  Google Scholar

[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.  doi: 10.1007/s10107-014-0754-4.  Google Scholar

[3]

J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis,, Springer, (2005).   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.  doi: 10.3934/dcdsb.2013.18.331.  Google Scholar

[5]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[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.   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]

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.   Google Scholar

[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.  doi: 10.1016/j.jde.2015.10.039.  Google Scholar

[10]

G. Colombo and L. Thibault, Prox-regular sets and applications,, In Y. Gao and D. Motreanu, (2010), 99.   Google Scholar

[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.  doi: 10.1016/j.jde.2007.05.011.  Google Scholar

[12]

J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process,, Math. Program., 104 (2005), 347.  doi: 10.1007/s10107-005-0619-y.  Google Scholar

[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.  doi: 10.1137/090766413.  Google Scholar

[14]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis,, Springer, (1989).  doi: 10.1007/978-3-642-61302-9.  Google Scholar

[15]

P. Krečí, Evolution variational inequalities and multidimensional hysteresis operators,, In P. Drabek, (1999), 47.   Google Scholar

[16]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process,, In B. Brogliato, (2000), 1.  doi: 10.1007/3-540-45501-9_1.  Google Scholar

[17]

B. Maury and J. Venel, A mathematical framework for a crowd motion model,, C. R. Acad. Sci. Paris Ser. I, 346 (2008), 1245.  doi: 10.1016/j.crma.2008.10.014.  Google Scholar

[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.  Google Scholar

[19]

B. S. Mordukhovich, Sensitivity analysis in nonsmooth optimization,, In D. A. Field and V. Komkov, (1992), 32.   Google Scholar

[20]

B. S. Mordukhovich, Discrete approximations and refined Euler-Lagrange conditions for differential inclusions,, SIAM J. Control Optim., 33 (1995), 882.  doi: 10.1137/S0363012993245665.  Google Scholar

[21]

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

[22]

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

[23]

J. J. Moreau, On unilateral constraints, friction and plasticity,, In G. Capriz and G. Stampacchia, (1974), 173.   Google Scholar

[24]

J. S. Pang and D. E. Stewart, Differential variational inequalities,, Math. Program., 113 (2008), 345.  doi: 10.1007/s10107-006-0052-x.  Google Scholar

[25]

F. Rindler, Optimal control for nonconvex rate-independent evolution processes,, SIAM J. Control Optim., 47 (2008), 2773.  doi: 10.1137/080718711.  Google Scholar

[26]

F. Rindler, Approximation of rate-independent optimal control problems,, SIAM J. Numer. Anal., 47 (2009), 3884.  doi: 10.1137/080744050.  Google Scholar

[27]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis,, Springer-Verlag, (1998).  doi: 10.1007/978-3-642-02431-3.  Google Scholar

[28]

A. H. Siddiqi, P. Manchanda and M. Brokate, On some recent developments concerning Moreau's sweeping process,, Trends in Industrial and Applied Mathematics, 72 (2002), 339.  doi: 10.1007/978-1-4613-0263-6_15.  Google Scholar

[29]

D. E. Stewart, Dynamics with Inequalities: Impacts and Hard Constraints,, SIAM, (2011).  doi: 10.1137/1.9781611970715.  Google Scholar

[30]

A. A. Tolstonogov, Differential Inclusions in a Banach Space,, Kluwer, (2000).  doi: 10.1007/978-94-015-9490-5.  Google Scholar

[31]

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

[1]

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
[2]

Dmitrii Rachinskii. On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3361-3386. doi: 10.3934/dcdsb.2018246
[3]

Majid E. Abbasov. Generalized exhausters: Existence, construction, optimality conditions. Journal of Industrial & Management Optimization, 2015, 11 (1) : 217-230. doi: 10.3934/jimo.2015.11.217
[4]

Ricardo Almeida. Optimality conditions for fractional variational problems with free terminal time. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 1-19. doi: 10.3934/dcdss.2018001
[5]

Monika Dryl, Delfim F. M. Torres. Necessary optimality conditions for infinite horizon variational problems on time scales. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 145-160. doi: 10.3934/naco.2013.3.145
[6]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361
[7]

Viviana Alejandra Díaz, David Martín de Diego. Generalized variational calculus for continuous and discrete mechanical systems. Journal of Geometric Mechanics, 2018, 10 (4) : 373-410. doi: 10.3934/jgm.2018014
[8]

Dalila Azzam-Laouir, Fatiha Selamnia. On state-dependent sweeping process in Banach spaces. Evolution Equations & Control Theory, 2018, 7 (2) : 183-196. doi: 10.3934/eect.2018009
[9]

Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014
[10]

Laurent Pfeiffer. Optimality conditions in variational form for non-linear constrained stochastic control problems. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020008
[11]

Xiuhong Chen, Zhihua Li. On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity. Journal of Industrial & Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081
[12]

Ram U. Verma. General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 333-339. doi: 10.3934/naco.2011.1.333
[13]

Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235
[14]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709
[15]

Doria Affane, Meriem Aissous, Mustapha Fateh Yarou. Almost mixed semi-continuous perturbation of Moreau's sweeping process. Evolution Equations & Control Theory, 2020, 9 (1) : 27-38. doi: 10.3934/eect.2020015
[16]

Andrzej Nowakowski. Variational analysis of semilinear plate equation with free boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3133-3154. doi: 10.3934/dcds.2015.35.3133
[17]

Anthony M. Bloch, Melvin Leok, Jerrold E. Marsden, Dmitry V. Zenkov. Controlled Lagrangians and stabilization of discrete mechanical systems. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 19-36. doi: 10.3934/dcdss.2010.3.19
[18]

B. Bonnard, J.-B. Caillau, E. Trélat. Second order optimality conditions with applications. Conference Publications, 2007, 2007 (Special) : 145-154. doi: 10.3934/proc.2007.2007.145
[19]

Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231
[20]

José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences