Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem

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September 2015, 35(9): 4593-4610. doi: 10.3934/dcds.2015.35.4593

Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem

Simão P. S. Santos ^1, , Natália Martins ^2, and Delfim F. M. Torres ^3,

1.	Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
2.	Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro
3.	CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received May 2014 Revised October 2014 Published April 2015

Abstract
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We extend the DuBois--Reymond necessary optimality condition and Noether's first theorem to variational problems of Herglotz type with time delay. Our results provide, as corollaries, the DuBois--Reymond necessary optimality condition and the first Noether theorem for variational problems with time delay recently proved in [Numer. Algebra Control Optim. 2 (2012), no. 3, 619--630]. Our main result is also a generalization of the first Noether-type theorem for the generalized variational principle of Herglotz proved in [Topol. Methods Nonlinear Anal. 20 (2002), no. 2, 261--273].

Keywords: time delay, Noether's theorem., invariance, Herglotz's calculus of variations, Euler--Lagrange differential equations, DuBois--Reymond necessary optimality condition.

Mathematics Subject Classification: Primary: 49K15, 49S05; Secondary: 34H0.

Citation: Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4593-4610. doi: 10.3934/dcds.2015.35.4593

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, N. Martins and D. F. M. Torres, The second Euler-Lagrange equation of variational calculus on time scales, Eur. J. Control, 17 (2011), 9-18. doi: 10.3166/ejc.17.9-18.
[3]	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.
[4]	L. Cesari, Optimization-theory and Applications, Springer, New York, 1983. doi: 10.1007/978-1-4613-8165-5.
[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]	L. E. El'sgol'c, Qualitative Methods of Mathematical Analysis, Translations of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island, 1964.
[7]	G. S. F. Frederico, T. Odzijewicz and D. F. M. Torres, Noether's theorem for non-smooth extremals of variational problems with time delay, Appl. Anal., 93 (2014), 153-170. doi: 10.1080/00036811.2012.762090.
[8]	G. S. F. Frederico and D. F. M. Torres, Nonconservative Noether's theorem in optimal control, Int. J. Tomogr. Stat., 5 (2007), 109-114.
[9]	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.
[10]	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.
[11]	G. S. F. Frederico and D. F. M. Torres, Noether's symmetry theorem for variational and optimal control problems with time delay, Numer. Alg. Contr. Optim., 2 (2012), 619-630. doi: 10.3934/naco.2012.2.619.
[12]	I. M. Gelfand and S. V. Fomin, Calculus of Variations, Revised English edition translated and edited by Richard A. Silverman, Prentice Hall, Englewood Cliffs, NJ, 1963.
[13]	B. A. Georgieva, Noether-type Theorems for the Generalized Variational Principle of Herglotz, Ph.D. thesis, ProQuest LLC, Ann Arbor, MI, 2001.
[14]	B. Georgieva, Symmetries of the Herglotz variational principle in the case of one independent variable, Ann. Sofia Univ., Fac. Math and Inf., 100 (2011), 113-122.
[15]	B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273.
[16]	B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314.
[17]	B. Georgieva, R. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem, J. Math. Phys., 44 (2003), 3911-3927. doi: 10.1063/1.1597419.
[18]	L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, J. Ind. Manag. Optim., 10 (2014), 413-441. doi: 10.3934/jimo.2014.10.413.
[19]	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.
[20]	R. B. Guenther, J. A. Gottsch and D. B. Kramer, The Herglotz algorithm for constructing canonical transformations, SIAM Rev., 38 (1996), 287-293. doi: 10.1137/1038042.
[21]	R. B. Guenther, C. M. Guenther and J. A. Gottsch, The Herglotz Lectures on Contact Transformations and Hamiltonian Systems, Lecture Notes in Nonlinear Analysis, Vol. 1, Juliusz Schauder Center for Nonlinear Studies, Nicholas Copernicus University, Torún, 1996.
[22]	G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930.
[23]	D. K. Hughes, Variational and optimal control problems with delayed argument, J. Optim. Theory Appl., 2 (1968), 1-14. doi: 10.1007/BF00927159.
[24]	A. B. Malinowska, On fractional variational problems which admit local transformations, J. Vib. Control, 19 (2013), 1161-1169. doi: 10.1177/1077546312442697.
[25]	A. B. Malinowska and N. Martins, The second Noether theorem on time scales, Abstr. Appl. Anal., 2013 (2013), Art. ID 675127, 14 pp. doi: 10.1155/2013/675127.
[26]	A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imp. Coll. Press, London, 2012. doi: 10.1142/p871.
[27]	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.
[28]	E. Noether, Invariante Variationsprobleme, Nachr. v. d. Ges. d. Wiss. zu Göttingen, (1918), 235-257.
[29]	J. C. Orum, R. T. Hudspeth, W. Black and R. B. Guenther, Extension of the Herglotz algorithm to nonautonomous canonical transformations, SIAM Rev., 42 (2000), 83-90. doi: 10.1137/S003614459834762X.
[30]	W. J. Palm and W. E. Schmitendorf, Conjugate-point conditions for variational problems with delayed argument, J. Optim. Theory Appl., 14 (1974), 599-612. doi: 10.1007/BF00932963.
[31]	L. D. Sabbagh, Variational problems with lags, J. Optim. Theory Appl., 3 (1969), 34-51. doi: 10.1007/BF00929540.
[32]	S. P. S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type, Vietnam J. Math., 42 (2014), 409-419. doi: 10.1007/s10013-013-0048-9.
[33]	D. F. M. Torres, On the Noether theorem for optimal control, Eur. J. Control, 8 (2002), 56-63. doi: 10.3166/ejc.8.56-63.
[34]	D. F. M. Torres, Conservation laws in optimal control, in Dynamics, bifurcations, and control (Kloster Irsee, 2001), Lecture Notes in Control and Inform. Sci., Springer, Berlin, 273 (2002), 287-296. doi: 10.1007/3-540-45606-6_20.
[35]	D. F. M. Torres, Gauge symmetries and Noether currents in optimal control, Appl. Math. E-Notes, 3 (2003), 49-57.
[36]	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.
[37]	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, N. Martins and D. F. M. Torres, The second Euler-Lagrange equation of variational calculus on time scales, Eur. J. Control, 17 (2011), 9-18. doi: 10.3166/ejc.17.9-18.
[3]	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.
[4]	L. Cesari, Optimization-theory and Applications, Springer, New York, 1983. doi: 10.1007/978-1-4613-8165-5.
[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]	L. E. El'sgol'c, Qualitative Methods of Mathematical Analysis, Translations of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island, 1964.
[7]	G. S. F. Frederico, T. Odzijewicz and D. F. M. Torres, Noether's theorem for non-smooth extremals of variational problems with time delay, Appl. Anal., 93 (2014), 153-170. doi: 10.1080/00036811.2012.762090.
[8]	G. S. F. Frederico and D. F. M. Torres, Nonconservative Noether's theorem in optimal control, Int. J. Tomogr. Stat., 5 (2007), 109-114.
[9]	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.
[10]	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.
[11]	G. S. F. Frederico and D. F. M. Torres, Noether's symmetry theorem for variational and optimal control problems with time delay, Numer. Alg. Contr. Optim., 2 (2012), 619-630. doi: 10.3934/naco.2012.2.619.
[12]	I. M. Gelfand and S. V. Fomin, Calculus of Variations, Revised English edition translated and edited by Richard A. Silverman, Prentice Hall, Englewood Cliffs, NJ, 1963.
[13]	B. A. Georgieva, Noether-type Theorems for the Generalized Variational Principle of Herglotz, Ph.D. thesis, ProQuest LLC, Ann Arbor, MI, 2001.
[14]	B. Georgieva, Symmetries of the Herglotz variational principle in the case of one independent variable, Ann. Sofia Univ., Fac. Math and Inf., 100 (2011), 113-122.
[15]	B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273.
[16]	B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314.
[17]	B. Georgieva, R. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem, J. Math. Phys., 44 (2003), 3911-3927. doi: 10.1063/1.1597419.
[18]	L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, J. Ind. Manag. Optim., 10 (2014), 413-441. doi: 10.3934/jimo.2014.10.413.
[19]	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.
[20]	R. B. Guenther, J. A. Gottsch and D. B. Kramer, The Herglotz algorithm for constructing canonical transformations, SIAM Rev., 38 (1996), 287-293. doi: 10.1137/1038042.
[21]	R. B. Guenther, C. M. Guenther and J. A. Gottsch, The Herglotz Lectures on Contact Transformations and Hamiltonian Systems, Lecture Notes in Nonlinear Analysis, Vol. 1, Juliusz Schauder Center for Nonlinear Studies, Nicholas Copernicus University, Torún, 1996.
[22]	G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930.
[23]	D. K. Hughes, Variational and optimal control problems with delayed argument, J. Optim. Theory Appl., 2 (1968), 1-14. doi: 10.1007/BF00927159.
[24]	A. B. Malinowska, On fractional variational problems which admit local transformations, J. Vib. Control, 19 (2013), 1161-1169. doi: 10.1177/1077546312442697.
[25]	A. B. Malinowska and N. Martins, The second Noether theorem on time scales, Abstr. Appl. Anal., 2013 (2013), Art. ID 675127, 14 pp. doi: 10.1155/2013/675127.
[26]	A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imp. Coll. Press, London, 2012. doi: 10.1142/p871.
[27]	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.
[28]	E. Noether, Invariante Variationsprobleme, Nachr. v. d. Ges. d. Wiss. zu Göttingen, (1918), 235-257.
[29]	J. C. Orum, R. T. Hudspeth, W. Black and R. B. Guenther, Extension of the Herglotz algorithm to nonautonomous canonical transformations, SIAM Rev., 42 (2000), 83-90. doi: 10.1137/S003614459834762X.
[30]	W. J. Palm and W. E. Schmitendorf, Conjugate-point conditions for variational problems with delayed argument, J. Optim. Theory Appl., 14 (1974), 599-612. doi: 10.1007/BF00932963.
[31]	L. D. Sabbagh, Variational problems with lags, J. Optim. Theory Appl., 3 (1969), 34-51. doi: 10.1007/BF00929540.
[32]	S. P. S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type, Vietnam J. Math., 42 (2014), 409-419. doi: 10.1007/s10013-013-0048-9.
[33]	D. F. M. Torres, On the Noether theorem for optimal control, Eur. J. Control, 8 (2002), 56-63. doi: 10.3166/ejc.8.56-63.
[34]	D. F. M. Torres, Conservation laws in optimal control, in Dynamics, bifurcations, and control (Kloster Irsee, 2001), Lecture Notes in Control and Inform. Sci., Springer, Berlin, 273 (2002), 287-296. doi: 10.1007/3-540-45606-6_20.
[35]	D. F. M. Torres, Gauge symmetries and Noether currents in optimal control, Appl. Math. E-Notes, 3 (2003), 49-57.
[36]	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.
[37]	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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