November  2014, 19(9): 2709-2738. doi: 10.3934/dcdsb.2014.19.2709

On optimal control of a sweeping process coupled with an ordinary differential equation

1. 

ÚTIA, Czech Academy of Sciences, Pod Vodárenskou věží 4, Prague 8, Czech Republic, Czech Republic

Received  June 2013 Revised  March 2014 Published  September 2014

We study a special case of an optimal control problem governed by a differential equation and a differential rate--independent variational inequality, both with given initial conditions. Under certain conditions, the variational inequality can be reformulated as a differential inclusion with discontinuous right-hand side. This inclusion is known as sweeping process.
    We perform a discretization scheme and prove the convergence of optimal solutions of the discretized problems to the optimal solution of the original problem. For the discretized problems we study the properties of the solution map and compute its coderivative. Employing an appropriate chain rule, this enables us to compute the subdifferential of the objective function and to apply a suitable optimization technique to solve the discretized problems. The investigated problem is used to model a situation arising in the area of queuing theory.
Citation: 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
References:
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K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations,, SIAM Publications Classics in Applied Mathematics, (1996).   Google Scholar

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M. Brokate, Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ,, Peter Lang GmbH, (1987).   Google Scholar

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

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M. Brokate and P. Krejčí, Optimal control of ODE systems involving a rate independent variational inequality,, Discrete and Continuous Dynamical Systems - Series B, 18 (2013), 331.  doi: 10.3934/dcdsb.2013.18.331.  Google Scholar

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H. S. Chung, R. D. Weaver and T. L. Friesz, Oligopolies in pollution permit markets: A dynamic game approach,, International Journal of Production Economics, 140 (2012), 48.  doi: 10.1016/j.ijpe.2012.01.017.  Google Scholar

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F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley, (1983).   Google Scholar

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F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer, (1998).   Google Scholar

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G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process,, Dynamics of Continuous, 19 (2012), 117.   Google Scholar

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B. Dacorogna, Direct methods in the calculus of variations,, vol. 78, (2008).   Google Scholar

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G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in finite elasticity,, Arch. Ration. Mech. Anal., 176 (2005), 165.   Google Scholar

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T. Donchev, E. Farkhi and B. S. Mordukhovich, Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces,, Journal of Differential Equations, 243 (2007), 301.  doi: 10.1016/j.jde.2007.05.011.  Google Scholar

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A. Dontchev and R. T. Rockafellar, Characterizations of strong regularity for variational inequalities over polyhedral convex sets,, SIAM Journal on Optimization, 6 (1996), 1087.  doi: 10.1137/S1052623495284029.  Google Scholar

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

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E. Emmrich, Discrete versions of Gronwall's lemma and their application to the numerical analysis of parabolic problems,, Preprint series of the Institute of Mathematics, 637 ().   Google Scholar

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A. D. Ioffe and J. Outrata, On metric and calmness qualification conditions in subdifferential calculus,, Set-Valued Analysis, 16 (2008), 199.  doi: 10.1007/s11228-008-0076-x.  Google Scholar

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M. Kočvara, M. Kružík and J. Outrata, On the control of an evolutionary equilibrium in micromagnetics,, in Optimization with Multivalued Mappings, 2 (2006), 143.  doi: 10.1007/0-387-34221-4_8.  Google Scholar

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P. Krejčí and J. Sprekels, Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity,, Appl. Math, 43 (1998), 173.  doi: 10.1023/A:1023224507448.  Google Scholar

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P. Krejčí and A. Vladimirov, Lipschitz continuity of polyhedral Skorokhod maps,, Zeitschrift für Analysis und Ihre Anwendungen, 20 (2000), 817.  doi: 10.4171/ZAA/1047.  Google Scholar

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P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators,, in Nonlinear differential equations, (1999), 47.   Google Scholar

[22]

A. S. Lewis and M. L. Overton, Nonsmooth optimization via quasi-Newton methods,, Mathematical Programming, (): 1.  doi: 10.1007/s10107-012-0514-2.  Google Scholar

[23]

S. Lu and S. Robinson, Normal fans of polyhedral convex sets,, Set-Valued Analysis, 16 (2008), 281.  doi: 10.1007/s11228-008-0077-9.  Google Scholar

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Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints,, Cambridge University Press, (1996).  doi: 10.1017/CBO9780511983658.  Google Scholar

[25]

B. Maury and J. Venel, A discrete contact model for crowd motion,, ESAIM: Mathematical Modelling and Numerical Analysis, 45 (2011), 145.  doi: 10.1051/m2an/2010035.  Google Scholar

[26]

Y. Moon, T. Yao and T. L. Friesz, Dynamic pricing and inventory policies: A strategic analysis of dual channel supply chain design,, Service Science, 2 (2010), 196.  doi: 10.1287/serv.2.3.196.  Google Scholar

[27]

B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings,, Journal of Mathematical Analysis and Applications, 183 (1994), 250.  doi: 10.1006/jmaa.1994.1144.  Google Scholar

[28]

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

[29]

J. J. Moreau, On unilateral constraints, friction and plasticity,, in New Variational Techniques in Mathematical Physics, (1974), 171.   Google Scholar

[30]

J. Outrata, M. Kočvara and J. Zowe, Nonsmooth approach to Optimization Problems with Equilibrium Constraints,, Kluwer Academic Publishers, (1998).  doi: 10.1007/978-1-4757-2825-5.  Google Scholar

[31]

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

[32]

R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions,, Ann. Inst. Henri Poincaré, 2 (1985), 167.   Google Scholar

[33]

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

[34]

H. Schramm and J. Zowe, A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results,, SIAM Journal on Optimization, 2 (1992), 121.  doi: 10.1137/0802008.  Google Scholar

[35]

A. Skajaa, Limited Memory BFGS for Nonsmooth Optimization,, Master thesis, (2010).   Google Scholar

[36]

A. Tasora, M. Anitescu, S. Negrini and D. Negrut, A compliant visco-plastic particle contact model based on differential variational inequalities,, International Journal of Non-Linear Mechanics, 53 (2013), 2.  doi: 10.1016/j.ijnonlinmec.2013.01.010.  Google Scholar

[37]

L. Thibault, Sweeping process with regular and nonregular sets,, Journal of Differential Equations, 193 (2003), 1.  doi: 10.1016/S0022-0396(03)00129-3.  Google Scholar

[38]

J. Venel, A numerical scheme for a class of sweeping processes,, Numerische Mathematik, 118 (2011), 367.  doi: 10.1007/s00211-010-0329-0.  Google Scholar

[39]

A. Visintin, Differential Models of Hysteresis,, Springer, (1994).  doi: 10.1007/978-3-662-11557-2.  Google Scholar

show all references

References:
[1]

A. Bergqvist, Magnetic vector hysteresis model with dry friction-like pinning,, Physica B: Condensed Matter, 233 (1997), 342.  doi: 10.1016/S0921-4526(97)00319-0.  Google Scholar

[2]

J. F. Bonnans, J. C. Gilbert, C. Lemaréchal and C. A. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects,, Springer-Verlag, (2006).   Google Scholar

[3]

K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations,, SIAM Publications Classics in Applied Mathematics, (1996).   Google Scholar

[4]

M. Brokate, Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ,, Peter Lang GmbH, (1987).   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]

M. Brokate and P. Krejčí, Optimal control of ODE systems involving a rate independent variational inequality,, Discrete and Continuous Dynamical Systems - Series B, 18 (2013), 331.  doi: 10.3934/dcdsb.2013.18.331.  Google Scholar

[7]

H. S. Chung, R. D. Weaver and T. L. Friesz, Oligopolies in pollution permit markets: A dynamic game approach,, International Journal of Production Economics, 140 (2012), 48.  doi: 10.1016/j.ijpe.2012.01.017.  Google Scholar

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley, (1983).   Google Scholar

[9]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer, (1998).   Google Scholar

[10]

G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process,, Dynamics of Continuous, 19 (2012), 117.   Google Scholar

[11]

B. Dacorogna, Direct methods in the calculus of variations,, vol. 78, (2008).   Google Scholar

[12]

G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in finite elasticity,, Arch. Ration. Mech. Anal., 176 (2005), 165.   Google Scholar

[13]

T. Donchev, E. Farkhi and B. S. Mordukhovich, Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces,, Journal of Differential Equations, 243 (2007), 301.  doi: 10.1016/j.jde.2007.05.011.  Google Scholar

[14]

A. Dontchev and R. T. Rockafellar, Characterizations of strong regularity for variational inequalities over polyhedral convex sets,, SIAM Journal on Optimization, 6 (1996), 1087.  doi: 10.1137/S1052623495284029.  Google Scholar

[15]

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

[16]

E. Emmrich, Discrete versions of Gronwall's lemma and their application to the numerical analysis of parabolic problems,, Preprint series of the Institute of Mathematics, 637 ().   Google Scholar

[17]

A. D. Ioffe and J. Outrata, On metric and calmness qualification conditions in subdifferential calculus,, Set-Valued Analysis, 16 (2008), 199.  doi: 10.1007/s11228-008-0076-x.  Google Scholar

[18]

M. Kočvara, M. Kružík and J. Outrata, On the control of an evolutionary equilibrium in micromagnetics,, in Optimization with Multivalued Mappings, 2 (2006), 143.  doi: 10.1007/0-387-34221-4_8.  Google Scholar

[19]

P. Krejčí and J. Sprekels, Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity,, Appl. Math, 43 (1998), 173.  doi: 10.1023/A:1023224507448.  Google Scholar

[20]

P. Krejčí and A. Vladimirov, Lipschitz continuity of polyhedral Skorokhod maps,, Zeitschrift für Analysis und Ihre Anwendungen, 20 (2000), 817.  doi: 10.4171/ZAA/1047.  Google Scholar

[21]

P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators,, in Nonlinear differential equations, (1999), 47.   Google Scholar

[22]

A. S. Lewis and M. L. Overton, Nonsmooth optimization via quasi-Newton methods,, Mathematical Programming, (): 1.  doi: 10.1007/s10107-012-0514-2.  Google Scholar

[23]

S. Lu and S. Robinson, Normal fans of polyhedral convex sets,, Set-Valued Analysis, 16 (2008), 281.  doi: 10.1007/s11228-008-0077-9.  Google Scholar

[24]

Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints,, Cambridge University Press, (1996).  doi: 10.1017/CBO9780511983658.  Google Scholar

[25]

B. Maury and J. Venel, A discrete contact model for crowd motion,, ESAIM: Mathematical Modelling and Numerical Analysis, 45 (2011), 145.  doi: 10.1051/m2an/2010035.  Google Scholar

[26]

Y. Moon, T. Yao and T. L. Friesz, Dynamic pricing and inventory policies: A strategic analysis of dual channel supply chain design,, Service Science, 2 (2010), 196.  doi: 10.1287/serv.2.3.196.  Google Scholar

[27]

B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings,, Journal of Mathematical Analysis and Applications, 183 (1994), 250.  doi: 10.1006/jmaa.1994.1144.  Google Scholar

[28]

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

[29]

J. J. Moreau, On unilateral constraints, friction and plasticity,, in New Variational Techniques in Mathematical Physics, (1974), 171.   Google Scholar

[30]

J. Outrata, M. Kočvara and J. Zowe, Nonsmooth approach to Optimization Problems with Equilibrium Constraints,, Kluwer Academic Publishers, (1998).  doi: 10.1007/978-1-4757-2825-5.  Google Scholar

[31]

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

[32]

R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions,, Ann. Inst. Henri Poincaré, 2 (1985), 167.   Google Scholar

[33]

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

[34]

H. Schramm and J. Zowe, A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results,, SIAM Journal on Optimization, 2 (1992), 121.  doi: 10.1137/0802008.  Google Scholar

[35]

A. Skajaa, Limited Memory BFGS for Nonsmooth Optimization,, Master thesis, (2010).   Google Scholar

[36]

A. Tasora, M. Anitescu, S. Negrini and D. Negrut, A compliant visco-plastic particle contact model based on differential variational inequalities,, International Journal of Non-Linear Mechanics, 53 (2013), 2.  doi: 10.1016/j.ijnonlinmec.2013.01.010.  Google Scholar

[37]

L. Thibault, Sweeping process with regular and nonregular sets,, Journal of Differential Equations, 193 (2003), 1.  doi: 10.1016/S0022-0396(03)00129-3.  Google Scholar

[38]

J. Venel, A numerical scheme for a class of sweeping processes,, Numerische Mathematik, 118 (2011), 367.  doi: 10.1007/s00211-010-0329-0.  Google Scholar

[39]

A. Visintin, Differential Models of Hysteresis,, Springer, (1994).  doi: 10.1007/978-3-662-11557-2.  Google Scholar

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