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Regularity of minimizers for second order variational problems in one independent variable
On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems
1. | CMAT e Departamento de Matemática e Aplicações, Universidade do Minho, 4800-058 Guimarães, Portugal |
2. | ISR-Porto, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal, Portugal |
References:
[1] |
A. V. Arutyunov and S. M. Aseev, State constraints in optimal control. The degeneracy phenomenon,, Systems Control Lett., 26 (1995), 267.
doi: doi:10.1016/0167-6911(95)00021-Z. |
[2] |
A. V. Arutyunov and S. M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints,, SIAM J. Control Optim., 35 (1997), 930.
doi: doi:10.1137/S036301299426996X. |
[3] |
J. Abadie, On the Kuhn-Tucker theorem,, in, (1967), 21. Google Scholar |
[4] |
A. V. Arutyunov, On necessary conditions for optimality in a problem with phase constraints,, Dokl. Akad. Nauk SSSR, 280 (1985), 1033.
|
[5] |
Aram V. Arutyunov, "Optimality Conditions. Abnormal and Degenerate Problems,", Mathematics and its Applications, 526 (2000).
|
[6] |
A. V. Arutyunov and N. T. Tynyanskiy, The maximum principle in a problem with phase constraints,, Izv. Akad. Nauk SSSR Tekhn. Kibernet, (1984), 60.
|
[7] |
P. Bettiol and H. Frankowska, Normality of the maximum principle for nonconvex constrained Bolza problems,, J. Differential Equations, 243 (2007), 256.
doi: doi:10.1016/j.jde.2007.05.005. |
[8] |
A. Cernea and H. Frankowska, A connection between the maximum principle and dynamic programming for constrained control problems,, SIAM J. Control Optim., 44 (2005), 673.
doi: doi:10.1137/S0363012903430585. |
[9] |
F. H. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).
|
[10] |
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998).
|
[11] |
A. Ya. Dubovitskii and A. A. Milyutin, Extremum problems under constraints,, Dokl. Akad. Nauk SSSR, 149 (1963), 759. Google Scholar |
[12] |
M. d. R. de Pinho, R. B. Vinter and H. Zheng, A maximum principle for optimal control problems with mixed constraints,, IMA J. Math. Control Inform., 18 (2001), 189.
doi: doi:10.1093/imamci/18.2.189. |
[13] |
M. M. A. Ferreira, F. A. C. C. Fontes and R. B. Vinter, Nondegenerate necessary conditions for nonconvex optimal control problems with state constraints,, J. Math. Anal. Appl., 233 (1999), 116.
doi: doi:10.1006/jmaa.1999.6270. |
[14] |
F. A. C. C. Fontes, "Optimisation-based Control of Constrained Nonlinear Systems,", Ph.D. thesis, (1999). Google Scholar |
[15] |
F. A. C. C. Fontes, "Normality in the Necessary Conditions of Optimality for Control Problems with State Constraints,", Proceedings of the IASTED Conference on Control and Applications (Cancun, (2000). Google Scholar |
[16] |
F. A. C. C. Fontes, A general framework to design stabilizing nonlinear model predictive controllers,, Systems Control Lett., 42 (2001), 127.
doi: doi:10.1016/S0167-6911(00)00084-0. |
[17] |
F. A. C. C. Fontes, Nondegenerate necessary conditions of optimality for control problems with state constraints,, in, (2002), 45. Google Scholar |
[18] |
M. M. A. Ferreira and R. B. Vinter, When is the maximum principle for state constrained problems nondegenerate?,, J. Math. Anal. Appl., 187 (1994), 438.
doi: doi:10.1006/jmaa.1994.1366. |
[19] |
F. John, "Extremum Problems as Inequalities as Subsidiary Conditions,", Studies and Essays: Courant Anniversary Volume (K. O. Friedrichs, (1948). Google Scholar |
[20] |
H. W. Kuhn and A. W. Tucker, Nonlinear programming,, in, (1951), 481. Google Scholar |
[21] |
K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems,, Optimization, 52 (2003), 75.
doi: doi:10.1080/0233193021000058940. |
[22] |
O. L. Mangasarian, "Nonlinear Programming,", McGraw-Hill, (1969). Google Scholar |
[23] |
B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation. I,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 330 (2006).
|
[24] |
L. W. Neustadt, A general theory of extremals,, Journal of Computer and System Sciences, 3 (1969), 57. Google Scholar |
[25] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", Wiley Interscience, (1962). Google Scholar |
[26] |
F. Rampazzo and R. B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control,, IMA J. Math. Control Inform., 16 (1999), 335.
doi: doi:10.1093/imamci/16.4.335. |
[27] |
F. Rampazzo and R. Vinter, Degenerate optimal control problems with state constraints,, SIAM J. Control Optim., 39 (2000), 989.
doi: doi:10.1137/S0363012998340223. |
[28] |
R. Vinter, "Optimal Control,", Systems & Control: Foundations & Applications, (2000).
|
[29] |
R. B. Vinter and G. Pappas, A maximum principle for nonsmooth optimal-control problems with state constraints,, J. Math. Anal. Appl., 89 (1982), 212.
doi: doi:10.1016/0022-247X(82)90099-3. |
[30] |
R. B. Vinter and H. Zheng, Necessary conditions for optimal control problems with state constraints,, Trans. Amer. Math. Soc., 350 (1998), 1181.
doi: doi:10.1090/S0002-9947-98-02129-1. |
[31] |
J. Warga, "Optimal Control of Differential and Functional Equations,", Academic Press, (1972).
|
show all references
References:
[1] |
A. V. Arutyunov and S. M. Aseev, State constraints in optimal control. The degeneracy phenomenon,, Systems Control Lett., 26 (1995), 267.
doi: doi:10.1016/0167-6911(95)00021-Z. |
[2] |
A. V. Arutyunov and S. M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints,, SIAM J. Control Optim., 35 (1997), 930.
doi: doi:10.1137/S036301299426996X. |
[3] |
J. Abadie, On the Kuhn-Tucker theorem,, in, (1967), 21. Google Scholar |
[4] |
A. V. Arutyunov, On necessary conditions for optimality in a problem with phase constraints,, Dokl. Akad. Nauk SSSR, 280 (1985), 1033.
|
[5] |
Aram V. Arutyunov, "Optimality Conditions. Abnormal and Degenerate Problems,", Mathematics and its Applications, 526 (2000).
|
[6] |
A. V. Arutyunov and N. T. Tynyanskiy, The maximum principle in a problem with phase constraints,, Izv. Akad. Nauk SSSR Tekhn. Kibernet, (1984), 60.
|
[7] |
P. Bettiol and H. Frankowska, Normality of the maximum principle for nonconvex constrained Bolza problems,, J. Differential Equations, 243 (2007), 256.
doi: doi:10.1016/j.jde.2007.05.005. |
[8] |
A. Cernea and H. Frankowska, A connection between the maximum principle and dynamic programming for constrained control problems,, SIAM J. Control Optim., 44 (2005), 673.
doi: doi:10.1137/S0363012903430585. |
[9] |
F. H. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).
|
[10] |
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998).
|
[11] |
A. Ya. Dubovitskii and A. A. Milyutin, Extremum problems under constraints,, Dokl. Akad. Nauk SSSR, 149 (1963), 759. Google Scholar |
[12] |
M. d. R. de Pinho, R. B. Vinter and H. Zheng, A maximum principle for optimal control problems with mixed constraints,, IMA J. Math. Control Inform., 18 (2001), 189.
doi: doi:10.1093/imamci/18.2.189. |
[13] |
M. M. A. Ferreira, F. A. C. C. Fontes and R. B. Vinter, Nondegenerate necessary conditions for nonconvex optimal control problems with state constraints,, J. Math. Anal. Appl., 233 (1999), 116.
doi: doi:10.1006/jmaa.1999.6270. |
[14] |
F. A. C. C. Fontes, "Optimisation-based Control of Constrained Nonlinear Systems,", Ph.D. thesis, (1999). Google Scholar |
[15] |
F. A. C. C. Fontes, "Normality in the Necessary Conditions of Optimality for Control Problems with State Constraints,", Proceedings of the IASTED Conference on Control and Applications (Cancun, (2000). Google Scholar |
[16] |
F. A. C. C. Fontes, A general framework to design stabilizing nonlinear model predictive controllers,, Systems Control Lett., 42 (2001), 127.
doi: doi:10.1016/S0167-6911(00)00084-0. |
[17] |
F. A. C. C. Fontes, Nondegenerate necessary conditions of optimality for control problems with state constraints,, in, (2002), 45. Google Scholar |
[18] |
M. M. A. Ferreira and R. B. Vinter, When is the maximum principle for state constrained problems nondegenerate?,, J. Math. Anal. Appl., 187 (1994), 438.
doi: doi:10.1006/jmaa.1994.1366. |
[19] |
F. John, "Extremum Problems as Inequalities as Subsidiary Conditions,", Studies and Essays: Courant Anniversary Volume (K. O. Friedrichs, (1948). Google Scholar |
[20] |
H. W. Kuhn and A. W. Tucker, Nonlinear programming,, in, (1951), 481. Google Scholar |
[21] |
K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems,, Optimization, 52 (2003), 75.
doi: doi:10.1080/0233193021000058940. |
[22] |
O. L. Mangasarian, "Nonlinear Programming,", McGraw-Hill, (1969). Google Scholar |
[23] |
B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation. I,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 330 (2006).
|
[24] |
L. W. Neustadt, A general theory of extremals,, Journal of Computer and System Sciences, 3 (1969), 57. Google Scholar |
[25] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", Wiley Interscience, (1962). Google Scholar |
[26] |
F. Rampazzo and R. B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control,, IMA J. Math. Control Inform., 16 (1999), 335.
doi: doi:10.1093/imamci/16.4.335. |
[27] |
F. Rampazzo and R. Vinter, Degenerate optimal control problems with state constraints,, SIAM J. Control Optim., 39 (2000), 989.
doi: doi:10.1137/S0363012998340223. |
[28] |
R. Vinter, "Optimal Control,", Systems & Control: Foundations & Applications, (2000).
|
[29] |
R. B. Vinter and G. Pappas, A maximum principle for nonsmooth optimal-control problems with state constraints,, J. Math. Anal. Appl., 89 (1982), 212.
doi: doi:10.1016/0022-247X(82)90099-3. |
[30] |
R. B. Vinter and H. Zheng, Necessary conditions for optimal control problems with state constraints,, Trans. Amer. Math. Soc., 350 (1998), 1181.
doi: doi:10.1090/S0002-9947-98-02129-1. |
[31] |
J. Warga, "Optimal Control of Differential and Functional Equations,", Academic Press, (1972).
|
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