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

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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

Abstract
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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.
[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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