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Noether's symmetry Theorem for variational and optimal control problems with time delay
1. | Department of Science and Technology, University of Cape Verde, Praia, Santiago, Cape Verde, Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal |
2. | CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal |
References:
[1] |
O. P. Agrawal, J. Gregory and K. Pericak-Spector, A Bliss-type multiplier rule for constrained variational problems with time delay, J. Math. Anal. Appl., 210 (1997), 702-711.
doi: 10.1006/jmaa.1997.5427. |
[2] |
Z. Bartosiewicz and D. F. M. Torres, Noether's theorem on time scales, J. Math. Anal. Appl., 342 (2008), 1220-1226.
doi: 10.1016/j.jmaa.2008.01.018. |
[3] |
M. Basin, "New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems," Lecture Notes in Control and Information Sciences, 380, Springer, Berlin, 2008. |
[4] |
G. V. Bokov, Pontryagin's maximum principle in a problem with time delay, J. Math. Sci. (N. Y.), 172 (2011), 623-634.
doi: 10.1007/s10958-011-0208-y. |
[5] |
J. Cresson, G. S. F. Frederico and D. F. M. Torres, Constants of motion for non-differentiable quantum variational problems, Topol. Methods Nonlinear Anal., 33 (2009), 217-231. |
[6] |
G. S. F. Frederico, "Generalizations of Noether's Theorem in the Calculus of Variations and Optimal Control," Ph.D. thesis, University of Cape Verde, 2009. |
[7] |
G. S. F. Frederico and D. F. M. Torres, Nonconservative Noether's theorem in optimal control, Int. J. Tomogr. Stat., 5 (2007), 109-114. |
[8] |
G. S. F. Frederico and D. F. M. Torres, A formulation of Noether's theorem for fractional problems of the calculus of variations, J. Math. Anal. Appl., 334 (2007), 834-846.
doi: 10.1016/j.jmaa.2007.01.013. |
[9] |
G. S. F. Frederico and D. F. M. Torres, Fractional Noether's theorem in the Riesz-Caputo sense, Appl. Math. Comput., 217 (2010), 1023-1033.
doi: 10.1016/j.amc.2010.01.100. |
[10] |
L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Appl. Methods, 30 (2009), 341-365.
doi: 10.1002/oca.843. |
[11] |
P. D. F. Gouveia and D. F. M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math., 5 (2005), 387-409. |
[12] |
P. D. F. Gouveia and D. F. M. Torres, Computing ODE symmetries as abnormal variational symmetries, Nonlinear Anal., 71 (2009), e138-e146.
doi: 10.1016/j.na.2008.10.009. |
[13] |
P. D. F. Gouveia, D. F. M. Torres and E. A. M. Rocha, Symbolic computation of variational symmetries in optimal control, Control Cybernet., 35 (2006), 831-849. |
[14] |
D. K. Hughes, Variational and optimal control problems with delayed argument, J. Optimization Theory Appl., 2 (1968), 1-14.
doi: 10.1007/BF00927159. |
[15] |
G. L. Kharatishvili, A maximum principle in extremal problems with delays, in "Mathematical Theory of Control (Proc. Conf., Los Angeles, Calif., 1967)", 26-34, Academic Press, New York, 1967. |
[16] |
G. L. Kharatishvili and T. A. Tadumadze, Formulas for the variation of a solution and optimal control problems for differential equations with retarded arguments, J. Math. Sci. (N. Y.), 140 (2007), 1-175.
doi: 10.1007/s10958-007-0412-y. |
[17] |
N. Martins and D. F. M. Torres, Noether's symmetry theorem for nabla problems of the calculus of variations, Appl. Math. Lett., 23 (2010), 1432-1438.
doi: 10.1016/j.aml.2010.07.013. |
[18] |
M. N. Oğuztöreli, "Time-Lag Control Systems," Mathematics in Science and Engineering, 24 Academic Press, New York, 1966. |
[19] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, Inc., New York, 1962. |
[20] |
E. A. M. Rocha and D. F. M. Torres, Quadratures for Pontryagin extremals for optimal control problems, Control Cybernet., 35 (2006), 947-963. |
[21] |
D. F. M. Torres, Conservation laws in optimal control, in "Dynamics, bifurcations, and control (Kloster Irsee, 2001)", 287-296, Lecture Notes in Control and Inform. Sci., 273 Springer, Berlin, 2002. |
[22] |
D. F. M. Torres, The role of symmetry in the regularity properties of optimal controls, in "Symmetry in Nonlinear Mathematical Physics. Part 1, 2, 3", 1488-1495, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 50, Part 1, 2, 3 Natsīonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2004. |
[23] |
D. F. M. Torres, Carathéodory equivalence, Noether theorems, and Tonelli full-regularity in the calculus of variations and optimal control, J. Math. Sci. (N. Y.), 120 (2004), 1032-1050.
doi: 10.1023/B:JOTH.0000013565.78376.fb. |
[24] |
D. F. M. Torres, Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations, Commun. Pure Appl. Anal., 3 (2004), 491-500.
doi: 10.3934/cpaa.2004.3.491. |
show all references
References:
[1] |
O. P. Agrawal, J. Gregory and K. Pericak-Spector, A Bliss-type multiplier rule for constrained variational problems with time delay, J. Math. Anal. Appl., 210 (1997), 702-711.
doi: 10.1006/jmaa.1997.5427. |
[2] |
Z. Bartosiewicz and D. F. M. Torres, Noether's theorem on time scales, J. Math. Anal. Appl., 342 (2008), 1220-1226.
doi: 10.1016/j.jmaa.2008.01.018. |
[3] |
M. Basin, "New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems," Lecture Notes in Control and Information Sciences, 380, Springer, Berlin, 2008. |
[4] |
G. V. Bokov, Pontryagin's maximum principle in a problem with time delay, J. Math. Sci. (N. Y.), 172 (2011), 623-634.
doi: 10.1007/s10958-011-0208-y. |
[5] |
J. Cresson, G. S. F. Frederico and D. F. M. Torres, Constants of motion for non-differentiable quantum variational problems, Topol. Methods Nonlinear Anal., 33 (2009), 217-231. |
[6] |
G. S. F. Frederico, "Generalizations of Noether's Theorem in the Calculus of Variations and Optimal Control," Ph.D. thesis, University of Cape Verde, 2009. |
[7] |
G. S. F. Frederico and D. F. M. Torres, Nonconservative Noether's theorem in optimal control, Int. J. Tomogr. Stat., 5 (2007), 109-114. |
[8] |
G. S. F. Frederico and D. F. M. Torres, A formulation of Noether's theorem for fractional problems of the calculus of variations, J. Math. Anal. Appl., 334 (2007), 834-846.
doi: 10.1016/j.jmaa.2007.01.013. |
[9] |
G. S. F. Frederico and D. F. M. Torres, Fractional Noether's theorem in the Riesz-Caputo sense, Appl. Math. Comput., 217 (2010), 1023-1033.
doi: 10.1016/j.amc.2010.01.100. |
[10] |
L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Appl. Methods, 30 (2009), 341-365.
doi: 10.1002/oca.843. |
[11] |
P. D. F. Gouveia and D. F. M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math., 5 (2005), 387-409. |
[12] |
P. D. F. Gouveia and D. F. M. Torres, Computing ODE symmetries as abnormal variational symmetries, Nonlinear Anal., 71 (2009), e138-e146.
doi: 10.1016/j.na.2008.10.009. |
[13] |
P. D. F. Gouveia, D. F. M. Torres and E. A. M. Rocha, Symbolic computation of variational symmetries in optimal control, Control Cybernet., 35 (2006), 831-849. |
[14] |
D. K. Hughes, Variational and optimal control problems with delayed argument, J. Optimization Theory Appl., 2 (1968), 1-14.
doi: 10.1007/BF00927159. |
[15] |
G. L. Kharatishvili, A maximum principle in extremal problems with delays, in "Mathematical Theory of Control (Proc. Conf., Los Angeles, Calif., 1967)", 26-34, Academic Press, New York, 1967. |
[16] |
G. L. Kharatishvili and T. A. Tadumadze, Formulas for the variation of a solution and optimal control problems for differential equations with retarded arguments, J. Math. Sci. (N. Y.), 140 (2007), 1-175.
doi: 10.1007/s10958-007-0412-y. |
[17] |
N. Martins and D. F. M. Torres, Noether's symmetry theorem for nabla problems of the calculus of variations, Appl. Math. Lett., 23 (2010), 1432-1438.
doi: 10.1016/j.aml.2010.07.013. |
[18] |
M. N. Oğuztöreli, "Time-Lag Control Systems," Mathematics in Science and Engineering, 24 Academic Press, New York, 1966. |
[19] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, Inc., New York, 1962. |
[20] |
E. A. M. Rocha and D. F. M. Torres, Quadratures for Pontryagin extremals for optimal control problems, Control Cybernet., 35 (2006), 947-963. |
[21] |
D. F. M. Torres, Conservation laws in optimal control, in "Dynamics, bifurcations, and control (Kloster Irsee, 2001)", 287-296, Lecture Notes in Control and Inform. Sci., 273 Springer, Berlin, 2002. |
[22] |
D. F. M. Torres, The role of symmetry in the regularity properties of optimal controls, in "Symmetry in Nonlinear Mathematical Physics. Part 1, 2, 3", 1488-1495, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 50, Part 1, 2, 3 Natsīonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2004. |
[23] |
D. F. M. Torres, Carathéodory equivalence, Noether theorems, and Tonelli full-regularity in the calculus of variations and optimal control, J. Math. Sci. (N. Y.), 120 (2004), 1032-1050.
doi: 10.1023/B:JOTH.0000013565.78376.fb. |
[24] |
D. F. M. Torres, Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations, Commun. Pure Appl. Anal., 3 (2004), 491-500.
doi: 10.3934/cpaa.2004.3.491. |
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