American Institute of Mathematical Sciences

Advanced
2012, 2(3): 619-630. doi: 10.3934/naco.2012.2.619

Noether's symmetry Theorem for variational and optimal control problems with time delay

Gastão S. F. Frederico 1, and  Delfim F. M. Torres 2,

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

Received  March 2012 Revised  March 2012 Published  August 2012

We extend the DuBois--Reymond necessary optimality condition and Noether's symmetry theorem to the time delay variational setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations and optimal control with delays.
Keywords: DuBois--Reymond necessary optimality condition, Noether's theorem., Time delay, constants of motion, invariance, symmetries.
Mathematics Subject Classification: Primary: 49K05, 49S0.
Citation: Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619
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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