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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 - A, 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.  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.  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.  doi: 10.1016/j.jmaa.2008.01.018.  Google Scholar

[4]

L. Cesari, Optimization-theory and Applications,, Springer, (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.   Google Scholar

[6]

L. E. El'sgol'c, Qualitative Methods of Mathematical Analysis, Translations of Mathematical Monographs,, American Mathematical Society, (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.  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.   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.  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.  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.  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, (1963).   Google Scholar

[13]

B. A. Georgieva, Noether-type Theorems for the Generalized Variational Principle of Herglotz,, Ph.D. thesis, (2001).   Google Scholar

[14]

B. Georgieva, Symmetries of the Herglotz variational principle in the case of one independent variable,, Ann. Sofia Univ., 100 (2011), 113.   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.   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.   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.  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.  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).  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.  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, (1996).   Google Scholar

[22]

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

[23]

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

[24]

A. B. Malinowska, On fractional variational problems which admit local transformations,, J. Vib. Control, 19 (2013), 1161.  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).  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, (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.  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.   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.  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.  doi: 10.1007/BF00932963.  Google Scholar

[31]

L. D. Sabbagh, Variational problems with lags,, J. Optim. Theory Appl., 3 (1969), 34.  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.  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.  doi: 10.3166/ejc.8.56-63.  Google Scholar

[34]

D. F. M. Torres, Conservation laws in optimal control,, in Dynamics, 273 (2002), 287.  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.   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.  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), 3 (2004), 491.  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.  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.  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.  doi: 10.1016/j.jmaa.2008.01.018.  Google Scholar

[4]

L. Cesari, Optimization-theory and Applications,, Springer, (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.   Google Scholar

[6]

L. E. El'sgol'c, Qualitative Methods of Mathematical Analysis, Translations of Mathematical Monographs,, American Mathematical Society, (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.  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.   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.  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.  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.  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, (1963).   Google Scholar

[13]

B. A. Georgieva, Noether-type Theorems for the Generalized Variational Principle of Herglotz,, Ph.D. thesis, (2001).   Google Scholar

[14]

B. Georgieva, Symmetries of the Herglotz variational principle in the case of one independent variable,, Ann. Sofia Univ., 100 (2011), 113.   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.   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.   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.  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.  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).  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.  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, (1996).   Google Scholar

[22]

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

[23]

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

[24]

A. B. Malinowska, On fractional variational problems which admit local transformations,, J. Vib. Control, 19 (2013), 1161.  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).  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, (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.  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.   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.  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.  doi: 10.1007/BF00932963.  Google Scholar

[31]

L. D. Sabbagh, Variational problems with lags,, J. Optim. Theory Appl., 3 (1969), 34.  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.  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.  doi: 10.3166/ejc.8.56-63.  Google Scholar

[34]

D. F. M. Torres, Conservation laws in optimal control,, in Dynamics, 273 (2002), 287.  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.   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.  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), 3 (2004), 491.  doi: 10.3934/cpaa.2004.3.491.  Google Scholar

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