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

Previous Article
When are minimizing controls also minimizing relaxed controls?
DCDS Home
This Issue
Next Article
Fields of extremals and sensitivity analysis for multi-input bilinear optimal control problems

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
Related Papers

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

[1]	Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491
[2]	Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577
[3]	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
[4]	Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990
[5]	Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417
[6]	Colin J. Cotter, Michael John Priestley Cullen. Particle relabelling symmetries and Noether's theorem for vertical slice models. Journal of Geometric Mechanics, 2019, 11 (2) : 139-151. doi: 10.3934/jgm.2019007
[7]	Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 91-102. doi: 10.3934/dcdss.2018006
[8]	Jehad O. Alzabut. A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Conference Publications, 2007, 2007 (Special) : 35-43. doi: 10.3934/proc.2007.2007.35
[9]	Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017
[10]	Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533
[11]	Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161
[12]	Clara Carlota, António Ornelas. The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 467-484. doi: 10.3934/dcds.2011.29.467
[13]	RazIye Mert, A. Zafer. A necessary and sufficient condition for oscillation of second order sublinear delay dynamic equations. Conference Publications, 2011, 2011 (Special) : 1061-1067. doi: 10.3934/proc.2011.2011.1061
[14]	Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018140
[15]	Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351
[16]	John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7.
[17]	Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. Optimality of (s, S) policies with nonlinear processes. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 161-185. doi: 10.3934/dcdsb.2017008
[18]	Hermann Brunner, Stefano Maset. Time transformations for delay differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751
[19]	David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319
[20]	Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics & Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences