March  2022, 15(3): 573-586. doi: 10.3934/dcdss.2021152

A non-standard class of variational problems of Herglotz type

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

* Corresponding author: natalia@ua.pt

Received  February 2020 Revised  February 2021 Published  March 2022 Early access  November 2021

Fund Project: This work is supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020

In this paper, we extend the variational problem of Herglotz considering the case where the Lagrangian depends not only on the independent variable, an unknown function $ x $ and its derivative and an unknown functional $ z $, but also on the end points conditions and a real parameter. Herglotz's problems of calculus of variations of this type cannot be solved using the standard theory. Main results of this paper are necessary optimality condition of Euler-Lagrange type, natural boundary conditions and the Dubois-Reymond condition for our non-standard variational problem of Herglotz type. We also prove a necessary optimality condition that arises as a consequence of the Lagrangian dependence of the parameter. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases. In addition, two examples are given in order to illustrate our results.

Citation: Natália Martins. A non-standard class of variational problems of Herglotz type. Discrete & Continuous Dynamical Systems - S, 2022, 15 (3) : 573-586. doi: 10.3934/dcdss.2021152
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

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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

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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

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B. GeorgievaR. 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

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R. B. Guenther and J. A. Gottsch, The Herglotz lectures on contact transformations and Hamiltonian systems, Juliusz Schauder Center for Nonlinear Studies, Nicholas Copernicus University, Torún, 1 (1996). Google Scholar

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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

[10]

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

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K. A. Hoffman, Stability results for constrained calculus of variations problems: An analysis of the twisted elastic loop, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 1357-1381.  doi: 10.1098/rspa.2004.1435.  Google Scholar

[12]

L. MachadoL. Abrunheiro and N. Martins, Variational and optimal control approaches for the second-order Herglotz problem on spheres, J. Optim. Theory Appl., 182 (2019), 965-983.  doi: 10.1007/s10957-018-1424-0.  Google Scholar

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A. B. Malinowska and D. F. M. Torres, Natural boundary conditions in the calculus of variations, Math. Methods Appl. Sci., 33 (2010), 1712-1722.  doi: 10.1002/mma.1289.  Google Scholar

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[15]

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

[16]

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

[17]

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., Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., 2015 (2015), 990-999.  doi: 10.3934/proc.2015.990.  Google Scholar

[18]

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

[19]

S. P. S. SantosN. Martins and D. F. M. Torres, Noether currents for higher-order variational problems of Herglotz type with time delay, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 91-102.  doi: 10.3934/dcdss.2018006.  Google Scholar

[20]

D. TavaresR. Almeida and D. F. M. Torres, Fractional Herglotz variational problems of variable order, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 143-154.  doi: 10.3934/dcdss.2018009.  Google Scholar

[21]

X. Tian and Y. Zhang, Noether's theorem for fractional Herglotz variational principle in phase space, Chaos Solitons and Fractals, 119 (2019), 50-54.  doi: 10.1016/j.chaos.2018.12.005.  Google Scholar

[22]

B. van Brunt, The Calculus of Variations, Universitext, Springer-Verlag, New York, 2004. doi: 10.1007/b97436.  Google Scholar

[23]

Y. Zhang and X. Tian, Conservation laws of nonconservative nonholonomic system based on Herglotz variational problem, Phys. Lett. A, 383 (2019), 691-696.  doi: 10.1016/j.physleta.2018.11.034.  Google Scholar

[24]

A. Zinober and S. Sufahani, A non-standard optimal control problem arising in an economics application, Pesqui. Oper., 33 (2013). doi: 10.1590/S0101-74382013000100004.  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]

P. A. F. CruzD. F. M. Torres and A. S. I. Zinober, A non-classical class of variational problems, Int. J. Mathematical Modelling and Numerical Optimisation, 1 (2010), 227-236.   Google Scholar

[4]

B. Georgieva, Symmetries of the Herglotz variational principle in the case of one independent variable, Annuaire Univ. Sofia Fac. Math. Inform., 100 (2010), 113-122.   Google Scholar

[5]

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

[6]

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

[7]

B. GeorgievaR. 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

[8]

R. B. Guenther and J. A. Gottsch, The Herglotz lectures on contact transformations and Hamiltonian systems, Juliusz Schauder Center for Nonlinear Studies, Nicholas Copernicus University, Torún, 1 (1996). Google Scholar

[9]

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

[10]

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

[11]

K. A. Hoffman, Stability results for constrained calculus of variations problems: An analysis of the twisted elastic loop, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 1357-1381.  doi: 10.1098/rspa.2004.1435.  Google Scholar

[12]

L. MachadoL. Abrunheiro and N. Martins, Variational and optimal control approaches for the second-order Herglotz problem on spheres, J. Optim. Theory Appl., 182 (2019), 965-983.  doi: 10.1007/s10957-018-1424-0.  Google Scholar

[13]

A. B. Malinowska and D. F. M. Torres, Natural boundary conditions in the calculus of variations, Math. Methods Appl. Sci., 33 (2010), 1712-1722.  doi: 10.1002/mma.1289.  Google Scholar

[14]

J. C. OrumR. T. HudspethW. 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

[15]

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

[16]

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

[17]

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., Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., 2015 (2015), 990-999.  doi: 10.3934/proc.2015.990.  Google Scholar

[18]

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

[19]

S. P. S. SantosN. Martins and D. F. M. Torres, Noether currents for higher-order variational problems of Herglotz type with time delay, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 91-102.  doi: 10.3934/dcdss.2018006.  Google Scholar

[20]

D. TavaresR. Almeida and D. F. M. Torres, Fractional Herglotz variational problems of variable order, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 143-154.  doi: 10.3934/dcdss.2018009.  Google Scholar

[21]

X. Tian and Y. Zhang, Noether's theorem for fractional Herglotz variational principle in phase space, Chaos Solitons and Fractals, 119 (2019), 50-54.  doi: 10.1016/j.chaos.2018.12.005.  Google Scholar

[22]

B. van Brunt, The Calculus of Variations, Universitext, Springer-Verlag, New York, 2004. doi: 10.1007/b97436.  Google Scholar

[23]

Y. Zhang and X. Tian, Conservation laws of nonconservative nonholonomic system based on Herglotz variational problem, Phys. Lett. A, 383 (2019), 691-696.  doi: 10.1016/j.physleta.2018.11.034.  Google Scholar

[24]

A. Zinober and S. Sufahani, A non-standard optimal control problem arising in an economics application, Pesqui. Oper., 33 (2013). doi: 10.1590/S0101-74382013000100004.  Google Scholar

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