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January  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

Matthias Gerdts 1, and  Martin Kunkel 2,

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.
Keywords: bounded variation, control-state constraints, Optimal control, nonsmooth equation., discretization, convergence.
Mathematics Subject Classification: Primary: 49J15, 49J52, 49M25; Secondary: 49K4.
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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[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).   Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[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).   Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

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