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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. Cesari, Optimization-theory and Applications, Springer, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[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.  Google Scholar

[6]

L. E. El'sgol'c, Qualitative Methods of Mathematical Analysis, Translations of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

B. A. Georgieva, Noether-type Theorems for the Generalized Variational Principle of Herglotz, Ph.D. thesis, ProQuest LLC, Ann Arbor, MI, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[22]

G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930. Google Scholar

[23]

D. K. Hughes, Variational and optimal control problems with delayed argument, J. Optim. Theory Appl., 2 (1968), 1-14. doi: 10.1007/BF00927159.  Google Scholar

[24]

A. B. Malinowska, On fractional variational problems which admit local transformations, J. Vib. Control, 19 (2013), 1161-1169. doi: 10.1177/1077546312442697.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

E. Noether, Invariante Variationsprobleme, Nachr. v. d. Ges. d. Wiss. zu Göttingen, (1918), 235-257. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

L. D. Sabbagh, Variational problems with lags, J. Optim. Theory Appl., 3 (1969), 34-51. doi: 10.1007/BF00929540.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

D. F. M. Torres, Gauge symmetries and Noether currents in optimal control, Appl. Math. E-Notes, 3 (2003), 49-57.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. Cesari, Optimization-theory and Applications, Springer, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[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.  Google Scholar

[6]

L. E. El'sgol'c, Qualitative Methods of Mathematical Analysis, Translations of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

B. A. Georgieva, Noether-type Theorems for the Generalized Variational Principle of Herglotz, Ph.D. thesis, ProQuest LLC, Ann Arbor, MI, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[22]

G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930. Google Scholar

[23]

D. K. Hughes, Variational and optimal control problems with delayed argument, J. Optim. Theory Appl., 2 (1968), 1-14. doi: 10.1007/BF00927159.  Google Scholar

[24]

A. B. Malinowska, On fractional variational problems which admit local transformations, J. Vib. Control, 19 (2013), 1161-1169. doi: 10.1177/1077546312442697.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

E. Noether, Invariante Variationsprobleme, Nachr. v. d. Ges. d. Wiss. zu Göttingen, (1918), 235-257. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

L. D. Sabbagh, Variational problems with lags, J. Optim. Theory Appl., 3 (1969), 34-51. doi: 10.1007/BF00929540.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

D. F. M. Torres, Gauge symmetries and Noether currents in optimal control, Appl. Math. E-Notes, 3 (2003), 49-57.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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