February  2018, 11(1): 143-154. doi: 10.3934/dcdss.2018009

Fractional Herglotz variational problems of variable order

1. 

ESECS, Polytechnic Institute of Leiria, 2411–901 Leiria, Portugal

2. 

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

* Corresponding author: Dina Tavares

Received  July 2016 Revised  February 2017 Published  January 2018

Fund Project: This work is part of first author's Ph.D., which is carried out at the University of Aveiro under the Doctoral Programme Mathematics and Applications of Universities of Aveiro and Minho. It was supported by Portuguese funds through CIDMA and The Portuguese Foundation for Science and Technology (FCT), within project UID/MAT/04106/2013. Tavares was also supported by FCT through the Ph.D. fellowship SFRH/BD/42557/2007

We study fractional variational problems of Herglotz type of variable order. Necessary optimality conditions, described by fractional differential equations depending on a combined Caputo fractional derivative of variable order, are proved. Two different cases are considered: the fundamental problem, with one independent variable, and the general case, with several independent variables. We end with some illustrative examples of the results of the paper.

Citation: Dina Tavares, Ricardo Almeida, Delfim F. M. Torres. Fractional Herglotz variational problems of variable order. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 143-154. doi: 10.3934/dcdss.2018009
References:
[1]

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

[2]

F. M. CoimbraC. M. Soon and M. H. Kobayashi, The variable viscoelasticity operator, Annalender Physik, 14 (2005), 378-389. Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

A. B. Malinowska and D. F. M. Torres, Fractional calculus of variations for a combined Caputo derivative, Fract. Calc. Appl. Anal., 14 (2011), 523-537. doi: 10.2478/s13540-011-0032-6. Google Scholar

[9]

T. OdzijewiczA. B. Malinowska and D. F. M. Torres, Fractional variational calculus with classical and combined Caputo derivatives, Nonlinear Anal., 75 (2012), 1507-1515. doi: 10.1016/j.na.2011.01.010. Google Scholar

[10]

T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Fractional variational calculus of variable order, In: Advances in Harmonic Analysis and Operator Theory, 291–301, Oper. Theory Adv. Appl. , 229, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0516-2_16. Google Scholar

[11]

T. OdzijewiczA. B. Malinowska and D. F. M. Torres, Noether's theorem for fractional variational problems of variable order, Cent. Eur. J. Phys., 11 (2013), 691-701. doi: 10.2478/s11534-013-0208-2. Google Scholar

[12]

S. G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transform. Spec. Funct., 1 (1993), 277-300. doi: 10.1080/10652469308819027. Google Scholar

[13]

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

[14]

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

[15]

S. P. S. Santos, N. Martins and D. F. M. Torres, An optimal control approach to Herglotz variational problems, in Optimization in the Natural Sciences (eds. A. Plakhov, T. Tchemisova and A. Freitas), Communications in Computer and Information Science, Vol. 499, Springer, 2015,107–117. doi: 10.1007/978-3-319-20352-2_7. Google Scholar

[16]

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

[17]

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

[18]

H. Sun, S. Hu, Y. Chen, W. Chen and Z. Yu, A dynamic-order fractional dynamic system Chinese Phys. Lett. , 30 (2013), 046601, 4 pp. doi: 10.1088/0256-307X/30/4/046601. Google Scholar

[19]

D. TavaresR. Almeida and D. F. M. Torres, Optimality conditions for fractional variational problems with dependence on a combined caputo derivative of variable order, Optim., 64 (2015), 1381-1391. doi: 10.1080/02331934.2015.1010088. Google Scholar

show all references

References:
[1]

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

[2]

F. M. CoimbraC. M. Soon and M. H. Kobayashi, The variable viscoelasticity operator, Annalender Physik, 14 (2005), 378-389. Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

A. B. Malinowska and D. F. M. Torres, Fractional calculus of variations for a combined Caputo derivative, Fract. Calc. Appl. Anal., 14 (2011), 523-537. doi: 10.2478/s13540-011-0032-6. Google Scholar

[9]

T. OdzijewiczA. B. Malinowska and D. F. M. Torres, Fractional variational calculus with classical and combined Caputo derivatives, Nonlinear Anal., 75 (2012), 1507-1515. doi: 10.1016/j.na.2011.01.010. Google Scholar

[10]

T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Fractional variational calculus of variable order, In: Advances in Harmonic Analysis and Operator Theory, 291–301, Oper. Theory Adv. Appl. , 229, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0516-2_16. Google Scholar

[11]

T. OdzijewiczA. B. Malinowska and D. F. M. Torres, Noether's theorem for fractional variational problems of variable order, Cent. Eur. J. Phys., 11 (2013), 691-701. doi: 10.2478/s11534-013-0208-2. Google Scholar

[12]

S. G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transform. Spec. Funct., 1 (1993), 277-300. doi: 10.1080/10652469308819027. Google Scholar

[13]

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

[14]

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

[15]

S. P. S. Santos, N. Martins and D. F. M. Torres, An optimal control approach to Herglotz variational problems, in Optimization in the Natural Sciences (eds. A. Plakhov, T. Tchemisova and A. Freitas), Communications in Computer and Information Science, Vol. 499, Springer, 2015,107–117. doi: 10.1007/978-3-319-20352-2_7. Google Scholar

[16]

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

[17]

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

[18]

H. Sun, S. Hu, Y. Chen, W. Chen and Z. Yu, A dynamic-order fractional dynamic system Chinese Phys. Lett. , 30 (2013), 046601, 4 pp. doi: 10.1088/0256-307X/30/4/046601. Google Scholar

[19]

D. TavaresR. Almeida and D. F. M. Torres, Optimality conditions for fractional variational problems with dependence on a combined caputo derivative of variable order, Optim., 64 (2015), 1381-1391. doi: 10.1080/02331934.2015.1010088. Google Scholar

Figure 1.  Graphics of function $z(\overline x, t)$
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