2014, 10(1): 311-336. doi: 10.3934/jimo.2014.10.311

Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation

1. 

Universität der Bundeswehr München, Institut für Mathematik und Rechneranwendung, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany

2. 

Elektrobit Automotive GmbH, Am Wolfsmantel 46, 91058 Erlangen, Germany

Received  October 2012 Revised  July 2013 Published  October 2013

We study convergence properties of Euler discretization of optimal control problems with ordinary differential equations and mixed control-state constraints. Under suitable consistency and stability assumptions a convergence rate of order $1/p$ of the discretized control to the continuous control is established in the $L^p$-norm. Throughout it is assumed that the optimal control is of bounded variation. The convergence proof exploits the reformulation of first order necessary optimality conditions as nonsmooth equations.
Citation: Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial & Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311
References:
[1]

W. Alt, Discretization and mesh-independence of Newton's method for generalized equations,, in, 195 (1997), 1.

[2]

W. Alt, Mesh-independence of the Lagrange-Newton method for nonlinear optimal control problems and their discretizations,, Optimization with data perturbations, 101 (2001), 101. doi: 10.1023/A:1010912305365.

[3]

W. Alt, R. Baier, M. Gerdts and F. Lempio, Approximation of linear control problems with bang-bang solutions,, Optimization, 62 (2013), 9. doi: 10.1080/02331934.2011.568619.

[4]

W. Alt, R. Baier, M. Gerdts and F. Lempio, Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions,, Numerical Algebra, 2 (2012), 547. doi: 10.3934/naco.2012.2.547.

[5]

N. Banihashemi and C. Y. Kaya, Inexact restoration for Euler discretization of box-constrained optimal control problems,, Journal of Optimization Theory and Applications, 156 (2013), 726. doi: 10.1007/s10957-012-0140-4.

[6]

C. Büskens, M. Gerdts, T. Nikolayzik, P. Kalmbach, M. Kunkel and D. Wassel, Homepage of the WORHP solver,, , (2010).

[7]

B. Chen, X. Chen and C. Kanzow, A penalized Fischer-Burmeister NCP-function,, Mathematical Programming, 88 (2000), 211. doi: 10.1007/PL00011375.

[8]

F. H. Clarke, "Optimization and Nonsmooth Analysis,'', Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).

[9]

A. L. Dontchev, W. W. Hager and K. Malanowski, Error bounds for Euler approximation of a state and control constrained optimal control problem,, Numerical Functional Analysis and Optimization, 21 (2000), 653. doi: 10.1080/01630560008816979.

[10]

A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order runge-kutta approximations in control constrained optimal control,, SIAM Journal on Numerical Analysis, 38 (2000), 202. doi: 10.1137/S0036142999351765.

[11]

I. S. Duff, MA57 - A code for the solution of sparse symmetric definite and indefinite systems,, ACM Transactions on Mathematical Software, 30 (2004), 118. doi: 10.1145/992200.992202.

[12]

C. Geiger and C. Kanzow, "Theorie und Numerik Restringierter Optimierungsaufgaben,'', Springer, (2002). doi: 10.1007/978-3-642-56004-0.

[13]

M. Gerdts, Global convergence of a nonsmooth Newton method for control-state constrained optimal control problems,, SIAM Journal on Optimization, 19 (2008), 326. doi: 10.1137/060657546.

[14]

M. Gerdts and B. Hüpping, Virtual control regularization of state constrained linear quadratic optimal control problems.,, Comput. Optim. Appl., 51 (2012), 867. doi: 10.1007/s10589-010-9353-3.

[15]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system,, Numerische Mathematik, 87 (2000), 247. doi: 10.1007/s002110000178.

[16]

H. Heuser, "Funktionalanalysis: Theorie und Anwendung,'', B. G. Teubner, (2006).

[17]

M. Josephy, Composing functions of bounded variation,, Proceedings of the American Mathematical Society, 83 (1981), 354. doi: 10.1090/S0002-9939-1981-0624930-9.

[18]

M. Kunkel, "Nonsmooth Newton Methods and Convergence of Discretized Optimal Control Problems Subject to DAEs,", PhD thesis, (2012), 706.

[19]

F. Lempio, Numerische mathematik II - methoden der analysis,, Bayreuther Mathematische Schriften, 55 (1998).

[20]

L. A. Ljusternik and W. I. Sobolew, "Elemente Der Funktionalanalysis,'', Fünfte Auflage. Übersetzung der zweiten russischen Auflage von Klaus Fiedler und herausgegeben von Konrad Gröger. Mathematische Lehrbücher und Monographien, (1976).

[21]

K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems,, Optimization, 52 (2003), 75. doi: 10.1080/0233193021000058940.

[22]

K. Malanowski, Ch. Büskens and H. Maurer, Convergence of approximations to nonlinear optimal control problems,, in, (1998), 253.

[23]

M. McAsey, L. Mou and W. Han, Convergence of the forward-backward sweep method in optimal control,, Computational Optimization and Applications, 53 (2012), 207. doi: 10.1007/s10589-011-9454-7.

[24]

I. P. Natanson, "Theorie der Funktionen Einer Reellen Veränderlichen,'', Verlag Harri Deutsch, (1981).

[25]

R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parametrization for optimal control problems with continuous inequality constraints: New convergence results,, Numerical Algebra, 2 (2012), 571. doi: 10.3934/naco.2012.2.571.

[26]

H. J. Stetter, Analysis of discretization methods for ordinary differential equations,, In, 23 (1973).

[27]

D. Sun and L. Qi, On NCP-functions,, Computational optimization—a tribute to Olvi Mangasarian, 13 (1999), 201. doi: 10.1023/A:1008669226453.

[28]

V. M. Veliov, Error analysis of discrete approximations to bang-bang optimal control problems: The linear case,, Control Cybern., 34 (2005), 967.

show all references

References:
[1]

W. Alt, Discretization and mesh-independence of Newton's method for generalized equations,, in, 195 (1997), 1.

[2]

W. Alt, Mesh-independence of the Lagrange-Newton method for nonlinear optimal control problems and their discretizations,, Optimization with data perturbations, 101 (2001), 101. doi: 10.1023/A:1010912305365.

[3]

W. Alt, R. Baier, M. Gerdts and F. Lempio, Approximation of linear control problems with bang-bang solutions,, Optimization, 62 (2013), 9. doi: 10.1080/02331934.2011.568619.

[4]

W. Alt, R. Baier, M. Gerdts and F. Lempio, Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions,, Numerical Algebra, 2 (2012), 547. doi: 10.3934/naco.2012.2.547.

[5]

N. Banihashemi and C. Y. Kaya, Inexact restoration for Euler discretization of box-constrained optimal control problems,, Journal of Optimization Theory and Applications, 156 (2013), 726. doi: 10.1007/s10957-012-0140-4.

[6]

C. Büskens, M. Gerdts, T. Nikolayzik, P. Kalmbach, M. Kunkel and D. Wassel, Homepage of the WORHP solver,, , (2010).

[7]

B. Chen, X. Chen and C. Kanzow, A penalized Fischer-Burmeister NCP-function,, Mathematical Programming, 88 (2000), 211. doi: 10.1007/PL00011375.

[8]

F. H. Clarke, "Optimization and Nonsmooth Analysis,'', Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).

[9]

A. L. Dontchev, W. W. Hager and K. Malanowski, Error bounds for Euler approximation of a state and control constrained optimal control problem,, Numerical Functional Analysis and Optimization, 21 (2000), 653. doi: 10.1080/01630560008816979.

[10]

A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order runge-kutta approximations in control constrained optimal control,, SIAM Journal on Numerical Analysis, 38 (2000), 202. doi: 10.1137/S0036142999351765.

[11]

I. S. Duff, MA57 - A code for the solution of sparse symmetric definite and indefinite systems,, ACM Transactions on Mathematical Software, 30 (2004), 118. doi: 10.1145/992200.992202.

[12]

C. Geiger and C. Kanzow, "Theorie und Numerik Restringierter Optimierungsaufgaben,'', Springer, (2002). doi: 10.1007/978-3-642-56004-0.

[13]

M. Gerdts, Global convergence of a nonsmooth Newton method for control-state constrained optimal control problems,, SIAM Journal on Optimization, 19 (2008), 326. doi: 10.1137/060657546.

[14]

M. Gerdts and B. Hüpping, Virtual control regularization of state constrained linear quadratic optimal control problems.,, Comput. Optim. Appl., 51 (2012), 867. doi: 10.1007/s10589-010-9353-3.

[15]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system,, Numerische Mathematik, 87 (2000), 247. doi: 10.1007/s002110000178.

[16]

H. Heuser, "Funktionalanalysis: Theorie und Anwendung,'', B. G. Teubner, (2006).

[17]

M. Josephy, Composing functions of bounded variation,, Proceedings of the American Mathematical Society, 83 (1981), 354. doi: 10.1090/S0002-9939-1981-0624930-9.

[18]

M. Kunkel, "Nonsmooth Newton Methods and Convergence of Discretized Optimal Control Problems Subject to DAEs,", PhD thesis, (2012), 706.

[19]

F. Lempio, Numerische mathematik II - methoden der analysis,, Bayreuther Mathematische Schriften, 55 (1998).

[20]

L. A. Ljusternik and W. I. Sobolew, "Elemente Der Funktionalanalysis,'', Fünfte Auflage. Übersetzung der zweiten russischen Auflage von Klaus Fiedler und herausgegeben von Konrad Gröger. Mathematische Lehrbücher und Monographien, (1976).

[21]

K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems,, Optimization, 52 (2003), 75. doi: 10.1080/0233193021000058940.

[22]

K. Malanowski, Ch. Büskens and H. Maurer, Convergence of approximations to nonlinear optimal control problems,, in, (1998), 253.

[23]

M. McAsey, L. Mou and W. Han, Convergence of the forward-backward sweep method in optimal control,, Computational Optimization and Applications, 53 (2012), 207. doi: 10.1007/s10589-011-9454-7.

[24]

I. P. Natanson, "Theorie der Funktionen Einer Reellen Veränderlichen,'', Verlag Harri Deutsch, (1981).

[25]

R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parametrization for optimal control problems with continuous inequality constraints: New convergence results,, Numerical Algebra, 2 (2012), 571. doi: 10.3934/naco.2012.2.571.

[26]

H. J. Stetter, Analysis of discretization methods for ordinary differential equations,, In, 23 (1973).

[27]

D. Sun and L. Qi, On NCP-functions,, Computational optimization—a tribute to Olvi Mangasarian, 13 (1999), 201. doi: 10.1023/A:1008669226453.

[28]

V. M. Veliov, Error analysis of discrete approximations to bang-bang optimal control problems: The linear case,, Control Cybern., 34 (2005), 967.

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