February  2018, 11(1): 91-102. doi: 10.3934/dcdss.2018006

Noether currents for higher-order variational problems of Herglotz type with time delay

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

* Corresponding author: N. Martins

Received  September 2016 Revised  March 2017 Published  January 2018

We study, from an optimal control perspective, Noether currents for higher-order problems of Herglotz type with time delay. Main result provides new Noether currents for such generalized variational problems, which are particularly useful in the search of extremals. The proof is based on the idea of rewriting the higher-order delayed generalized variational problem as a first-order optimal control problem without time delays.

Citation: 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
References:
[1]

L. AbrunheiroL. Machado and N. Martins, The Herglotz variational problem on spheres and its optimal control approach, J. Math. Anal., 7 (2016), 12-22. Google Scholar

[2]

R. Almeida and A. B. Malinowska, Fractional variational principle of Herglotz, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2367-2381. doi: 10.3934/dcdsb.2014.19.2367. Google Scholar

[3]

M. Bañados and I. Reyes, A short review on Noether's theorems, gauge symmetries and boundary terms Internat. J. Modern Phys. D 25 (2016), 1630021, 74 pp. doi: 10.1142/S0218271816300214. Google Scholar

[4]

M. Benharrat and D. F. M. Torres, Optimal control with time delays via the penalty method Math. Probl. Eng. 2014 (2014), Art. ID 250419, 9 pp. doi: 10.1155/2014/250419. Google Scholar

[5]

X. Dupuis, Optimal control of leukemic cell population dynamics, Math. Model. Nat. Phenom., 9 (2014), 4-26. doi: 10.1051/mmnp/20149102. Google Scholar

[6]

G. S. F. Frederico and D. F. M. Torres, Noether's symmetry theorem for variational and optimal control problems with time delay, Numer. Algebra Control Optim., 2 (2012), 619-630. doi: 10.3934/naco.2012.2.619. Google Scholar

[7]

G. S. F. Frederico and D. F. M. Torres, A nondifferentiable quantum variational embedding in presence of time delays, Int. J. Difference Equ., 8 (2013), 49-62. Google Scholar

[8]

S. Friederich, Symmetry, empirical equivalence, and identity, British J. Philos. Sci., 66 (2015), 537-559. doi: 10.1093/bjps/axt046. Google Scholar

[9]

B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273. doi: 10.12775/TMNA.2002.036. Google Scholar

[10]

B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314. doi: 10.12775/TMNA.2005.034. Google Scholar

[11]

B. GeorgievaR. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables, J. Math. Phys., 44 (2003), 3911-3927. doi: 10.1063/1.1597419. Google Scholar

[12]

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

[13]

R. B. GuentherJ. 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

[14]

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

[15]

T. Guinn, Reduction of delayed optimal control problems to nondelayed problems, J. Optimization Theory Appl., 18 (1976), 371-377. doi: 10.1007/BF00933818. Google Scholar

[16]

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

[17]

S. M. Hoseini and H. R. Marzban, Costate computation by an adaptive pseudospectral method for solving optimal control problems with piecewise constant time lag, J. Optim. Theory Appl., 170 (2016), 735-755. doi: 10.1007/s10957-016-0957-3. Google Scholar

[18]

Y. Kosmann-Schwarzbach, The Noether Theorems. Invariance and Conservation Laws in the Twentieth Century Translated, revised and augmented from the 2006 French edition by B. E. Schwarzbach, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, 2011. doi: 10.1007/978-0-387-87868-3. Google Scholar

[19]

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

[20]

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

[21]

A. B. Malinowska and T. Odzijewicz, Second Noether's theorem with time delay, Appl. Anal., 96 (2017), 1358-1378. doi: 10.1080/00036811.2016.1192136. Google Scholar

[22]

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

[23]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes Interscience Publishers, John Wiley and Sons Inc, New York, London, 1962. Google Scholar

[24]

S. P. S. SantosN. 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

[25]

S. P. S. SantosN. Martins and D. F. M. Torres, Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem, Discrete Contin. Dyn. Syst., 35 (2015), 4593-4610. doi: 10.3934/dcds.2015.35.4593. Google Scholar

[26]

S. P. S. SantosN. Martins and D. F. M. Torres, Noether's theorem for higher-order variational problems of Herglotz type, Discrete Contin. Dyn. Syst., 2015 (2015), Dynamical Systems, Differential Equations and Applications, 10th AIMS Conference. Suppl., 990-999. doi: 10.3934/proc.2015.990. Google Scholar

[27]

S. P. S. SantosN. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz with time delay, Pure and Applied Functional Analysis, 1 (2016), 291-307. Google Scholar

[28]

G. Sardanashvily, Noether's Theorems. Applications in Mechanics and Field Theory Atlantis Studies in Variational Geometry, 3, Atlantis Press, Paris, 2016. doi: 10.2991/978-94-6239-171-0. Google Scholar

[29]

C. J. SilvaH. Maurer and D. F. M. Torres, Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321-337. doi: 10.3934/mbe.2017021. Google Scholar

[30]

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

[31]

D. F. M. Torres, Conservation laws in optimal control, in Dynamics, bifurcations, and control (Kloster Irsee, 2001), 287–296, Lecture Notes in Control and Inform. Sci. , 273, Springer, Berlin, 2002. doi: 10.1007/3-540-45606-6_20. Google Scholar

[32]

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

[33]

D. F. M. Torres, Quasi-invariant optimal control problems, Port. Math.(N.S.), 61 (2004), 97-114. Google Scholar

[34]

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]

L. AbrunheiroL. Machado and N. Martins, The Herglotz variational problem on spheres and its optimal control approach, J. Math. Anal., 7 (2016), 12-22. Google Scholar

[2]

R. Almeida and A. B. Malinowska, Fractional variational principle of Herglotz, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2367-2381. doi: 10.3934/dcdsb.2014.19.2367. Google Scholar

[3]

M. Bañados and I. Reyes, A short review on Noether's theorems, gauge symmetries and boundary terms Internat. J. Modern Phys. D 25 (2016), 1630021, 74 pp. doi: 10.1142/S0218271816300214. Google Scholar

[4]

M. Benharrat and D. F. M. Torres, Optimal control with time delays via the penalty method Math. Probl. Eng. 2014 (2014), Art. ID 250419, 9 pp. doi: 10.1155/2014/250419. Google Scholar

[5]

X. Dupuis, Optimal control of leukemic cell population dynamics, Math. Model. Nat. Phenom., 9 (2014), 4-26. doi: 10.1051/mmnp/20149102. Google Scholar

[6]

G. S. F. Frederico and D. F. M. Torres, Noether's symmetry theorem for variational and optimal control problems with time delay, Numer. Algebra Control Optim., 2 (2012), 619-630. doi: 10.3934/naco.2012.2.619. Google Scholar

[7]

G. S. F. Frederico and D. F. M. Torres, A nondifferentiable quantum variational embedding in presence of time delays, Int. J. Difference Equ., 8 (2013), 49-62. Google Scholar

[8]

S. Friederich, Symmetry, empirical equivalence, and identity, British J. Philos. Sci., 66 (2015), 537-559. doi: 10.1093/bjps/axt046. Google Scholar

[9]

B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273. doi: 10.12775/TMNA.2002.036. Google Scholar

[10]

B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314. doi: 10.12775/TMNA.2005.034. Google Scholar

[11]

B. GeorgievaR. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables, J. Math. Phys., 44 (2003), 3911-3927. doi: 10.1063/1.1597419. Google Scholar

[12]

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

[13]

R. B. GuentherJ. 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

[14]

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

[15]

T. Guinn, Reduction of delayed optimal control problems to nondelayed problems, J. Optimization Theory Appl., 18 (1976), 371-377. doi: 10.1007/BF00933818. Google Scholar

[16]

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

[17]

S. M. Hoseini and H. R. Marzban, Costate computation by an adaptive pseudospectral method for solving optimal control problems with piecewise constant time lag, J. Optim. Theory Appl., 170 (2016), 735-755. doi: 10.1007/s10957-016-0957-3. Google Scholar

[18]

Y. Kosmann-Schwarzbach, The Noether Theorems. Invariance and Conservation Laws in the Twentieth Century Translated, revised and augmented from the 2006 French edition by B. E. Schwarzbach, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, 2011. doi: 10.1007/978-0-387-87868-3. Google Scholar

[19]

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

[20]

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

[21]

A. B. Malinowska and T. Odzijewicz, Second Noether's theorem with time delay, Appl. Anal., 96 (2017), 1358-1378. doi: 10.1080/00036811.2016.1192136. Google Scholar

[22]

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

[23]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes Interscience Publishers, John Wiley and Sons Inc, New York, London, 1962. Google Scholar

[24]

S. P. S. SantosN. 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

[25]

S. P. S. SantosN. Martins and D. F. M. Torres, Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem, Discrete Contin. Dyn. Syst., 35 (2015), 4593-4610. doi: 10.3934/dcds.2015.35.4593. Google Scholar

[26]

S. P. S. SantosN. Martins and D. F. M. Torres, Noether's theorem for higher-order variational problems of Herglotz type, Discrete Contin. Dyn. Syst., 2015 (2015), Dynamical Systems, Differential Equations and Applications, 10th AIMS Conference. Suppl., 990-999. doi: 10.3934/proc.2015.990. Google Scholar

[27]

S. P. S. SantosN. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz with time delay, Pure and Applied Functional Analysis, 1 (2016), 291-307. Google Scholar

[28]

G. Sardanashvily, Noether's Theorems. Applications in Mechanics and Field Theory Atlantis Studies in Variational Geometry, 3, Atlantis Press, Paris, 2016. doi: 10.2991/978-94-6239-171-0. Google Scholar

[29]

C. J. SilvaH. Maurer and D. F. M. Torres, Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321-337. doi: 10.3934/mbe.2017021. Google Scholar

[30]

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

[31]

D. F. M. Torres, Conservation laws in optimal control, in Dynamics, bifurcations, and control (Kloster Irsee, 2001), 287–296, Lecture Notes in Control and Inform. Sci. , 273, Springer, Berlin, 2002. doi: 10.1007/3-540-45606-6_20. Google Scholar

[32]

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

[33]

D. F. M. Torres, Quasi-invariant optimal control problems, Port. Math.(N.S.), 61 (2004), 97-114. Google Scholar

[34]

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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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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