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The control parameterization method for nonlinear optimal control: A survey
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 |
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). 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. |
[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. Google Scholar |
[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). 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. |
[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. Google Scholar |
[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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