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 & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619
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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.  Google Scholar

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show all references

References:
[1]

J. Math. Anal. Appl., 210 (1997), 702-711. doi: 10.1006/jmaa.1997.5427.  Google Scholar

[2]

J. Math. Anal. Appl., 342 (2008), 1220-1226. doi: 10.1016/j.jmaa.2008.01.018.  Google Scholar

[3]

Lecture Notes in Control and Information Sciences, 380, Springer, Berlin, 2008.  Google Scholar

[4]

J. Math. Sci. (N. Y.), 172 (2011), 623-634. doi: 10.1007/s10958-011-0208-y.  Google Scholar

[5]

Topol. Methods Nonlinear Anal., 33 (2009), 217-231.  Google Scholar

[6]

Ph.D. thesis, University of Cape Verde, 2009. Google Scholar

[7]

Int. J. Tomogr. Stat., 5 (2007), 109-114.  Google Scholar

[8]

J. Math. Anal. Appl., 334 (2007), 834-846. doi: 10.1016/j.jmaa.2007.01.013.  Google Scholar

[9]

Appl. Math. Comput., 217 (2010), 1023-1033. doi: 10.1016/j.amc.2010.01.100.  Google Scholar

[10]

Optimal Control Appl. Methods, 30 (2009), 341-365. doi: 10.1002/oca.843.  Google Scholar

[11]

Comput. Methods Appl. Math., 5 (2005), 387-409.  Google Scholar

[12]

Nonlinear Anal., 71 (2009), e138-e146. doi: 10.1016/j.na.2008.10.009.  Google Scholar

[13]

Control Cybernet., 35 (2006), 831-849.  Google Scholar

[14]

J. Optimization Theory Appl., 2 (1968), 1-14. doi: 10.1007/BF00927159.  Google Scholar

[15]

in "Mathematical Theory of Control (Proc. Conf., Los Angeles, Calif., 1967)", 26-34, Academic Press, New York, 1967.  Google Scholar

[16]

J. Math. Sci. (N. Y.), 140 (2007), 1-175. doi: 10.1007/s10958-007-0412-y.  Google Scholar

[17]

Appl. Math. Lett., 23 (2010), 1432-1438. doi: 10.1016/j.aml.2010.07.013.  Google Scholar

[18]

Mathematics in Science and Engineering, 24 Academic Press, New York, 1966.  Google Scholar

[19]

Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, Inc., New York, 1962.  Google Scholar

[20]

Control Cybernet., 35 (2006), 947-963.  Google Scholar

[21]

in "Dynamics, bifurcations, and control (Kloster Irsee, 2001)", 287-296, Lecture Notes in Control and Inform. Sci., 273 Springer, Berlin, 2002.  Google Scholar

[22]

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.  Google Scholar

[23]

J. Math. Sci. (N. Y.), 120 (2004), 1032-1050. doi: 10.1023/B:JOTH.0000013565.78376.fb.  Google Scholar

[24]

Commun. Pure Appl. Anal., 3 (2004), 491-500. doi: 10.3934/cpaa.2004.3.491.  Google Scholar

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