Year 2020, Volume 49 , Issue 1, Pages 68 - 77 2020-02-06

Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function

Jie BAİ [1] , Ling BAİ [2] , Zhijun ZENG [3]


In this paper, we study the calculus of variations of the nabla notion on time scales including $\nabla$-derivative, $\nabla$-integral, and $\nabla$-derivatives of exponential function. The Euler-Lagrange equations of the first-order both single-variable problem and multivariable problem with nabla derivatives of exponential function on time scales are obtained. In particular, we show that the calculus of variations with multiple variables could solve the problem of conditional extreme value. Moreover, we verify the solution to the multivariable problem is exactly the extremum pair. As applications of these results, an example of conditional extremum is provided.
time scales, the Euler-Lagrange equation, calculus of variations, conditional extremum, $\nabla$-derivatives of exponential function
  • [1] R. Agarwal, M. Bohner, D. O’Regan and A. Peterson, Dynamic equations on time scales: a survey, J. Comput. Appl. Math. 141 (1), 1–26, 2002.
  • [2] C.D. Ahlbrandt, M. Bohner and J. Ridenhour, Hamiltonian systems on time scales, J. Math. Anal. Appl. 250 (2), 561–578, 2000.
  • [3] R. Almeida and D.F. Torres, Isoperimetric problems on time scales with nabla deriva- tives, J. Vib. Control, 15 (6), 951–958, 2009.
  • [4] F.M. Atici, D.C. Biles and A. Lebedinsky, An application of time scales to economics, Math. Comput. Model. 43 (7-8), 718–726, 2006.
  • [5] Z. Bartosiewicz and E. Pawluszewicz, Realizations of nonlinear control systems on time scales, IEEE Trans. Autom. Control, 53 (2), 571–575, 2008.
  • [6] Z. Bartosiewicz and E. Pawluszewicz, Dynamic feedback equivalence of time-variant control systems on homogeneous time scales, Int. J. Math. Stat. 5 (A09), 11–20, 2009.
  • [7] Z. Bartosiewicz, N. Martins and D.F. Torres, The second euler-lagrange equation of variational calculus on time scales, Eur. J. Control, 17 (1), 9–18, 2011.
  • [8] Z. Bartosiewicz, E. Piotrowska and M. Wyrwas, Stability, stabilization and observers of linear control systems on time scales, Proc. IEEE Conf. on Decision and Control, New Orleans, LA, USA, 2803–2808, 2007.
  • [9] Z. Bartosiewicz, U. Kotta, E. Pawluszewicz and M. Wyrwas, Algebraic formalism of differ- ential one-forms for nonlinear control systems on time scales, Proc. Est. Acad. Sci. 56 (3), 264–282, 2007.
  • [10] M. Bohner, Calculus of variations on time scales, Dynam. Syst. Appl. 13 (3-4), 339– 349, 2004.
  • [11] M. Bohner and A. Peterson, Dynamic equations on time scales: An introduction with applications, Birkhauser Boston, Boston, MA, 2001
  • [12] J.J. DaCunha, Stability for time varying linear dynamic systems on time scales, J. Comput. Appl. Math. 176 (2), 381–410, 2005.
  • [13] R. David, Advanced macroeconomics, McGraw-Hill, Irwin, 2011.
  • [14] R.A. Ferreira and D.F. Torres, Remarks on the calculus of variations on time scales, Int. J. Ecol. Econ. Stat. 9 (F07), 65–73, 2007.
  • [15] R.A. Ferreira and D.F. Torres, Higher-order calculus of variations on time scales, in: Mathematical Control Theory and Finance, Springer, Berlin, 136 (1), 149–159, 2008.
  • [16] M. Guzowska, A.B. Malinowska and M.R.S. Ammi, Calculus of variations on time scales: applications to economic models, Adv. Differ. Equ. 2015 (1), 203, 2015.
  • [17] R. Hilscher and V. Zeidan, Calculus of variations on time scales: weak local piecewise crd1 solutions with variable endpoints, J. Math. Anal. Appl. 289 (1), 143–166, 2004.
  • [18] R. Hilscher and V. Zeidan,Weak maximum principle and accessory problem for control problems on time scales, Nonlinear Anal. TMA, 70 (9), 3209–3226, 2009.
  • [19] A.B. Malinowska and D.F. Torres, Necessary and sufficient conditions for local pareto optimality on time scales, J. Math. Sci. (NY), 161 (6), 803–810, 2009.
  • [20] A.B. Malinowska and D.F. Torres, Strong minimizers of the calculus of variations on time scales and the weierstrass condition, P. Est. Acad. Sci. 58 (4), 205–212, 2009.
  • [21] N. Martins and D.F. Torres, Calculus of variations on time scales with nabla deriva- tives, Nonlinear Anal. TMA, 71 (12), 763–773, 2009.
  • [22] D. Mozyrska and Z. Bartosiewicz, Observability of a class of linear dynamic infinite systems on time scales, Proc. Est. Acad. Sci. 56 (4), 347–358, 2007.
  • [23] J. Seiffertt, S. Sanyal and D.C. Wunsch, Hamilton-Jacobi-Bellman equations and approximate dynamic programming on time scales, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 38 (4), 918–923, 2008.
  • [24] Z. Zhan, W. Wei and H. Xu, Hamilton-Jacobi-Bellman equations on time scales, Math. Comput. Model. 49 (9-10), 2019–2028, 2009.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0002-0127-4623
Author: Jie BAİ
Institution: Liaoning University
Country: China


Orcid: 0000-0001-6837-9983
Author: Ling BAİ
Institution: Beijing Municipal Commission Of Health And Family Planning Information Center
Country: China


Orcid: 0000-0002-7114-0559
Author: Zhijun ZENG (Primary Author)
Institution: Northeast Normal University
Country: China


Dates

Publication Date : February 6, 2020

Bibtex @research article { hujms504587, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {68 - 77}, doi = {10.15672/HJMS.2018.653}, title = {Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function}, key = {cite}, author = {BAİ, Jie and BAİ, Ling and ZENG, Zhijun} }
APA BAİ, J , BAİ, L , ZENG, Z . (2020). Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function. Hacettepe Journal of Mathematics and Statistics , 49 (1) , 68-77 . DOI: 10.15672/HJMS.2018.653
MLA BAİ, J , BAİ, L , ZENG, Z . "Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 68-77 <https://dergipark.org.tr/en/pub/hujms/issue/52287/504587>
Chicago BAİ, J , BAİ, L , ZENG, Z . "Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 68-77
RIS TY - JOUR T1 - Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function AU - Jie BAİ , Ling BAİ , Zhijun ZENG Y1 - 2020 PY - 2020 N1 - doi: 10.15672/HJMS.2018.653 DO - 10.15672/HJMS.2018.653 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 68 EP - 77 VL - 49 IS - 1 SN - 2651-477X-2651-477X M3 - doi: 10.15672/HJMS.2018.653 UR - https://doi.org/10.15672/HJMS.2018.653 Y2 - 2018 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function %A Jie BAİ , Ling BAİ , Zhijun ZENG %T Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 1 %R doi: 10.15672/HJMS.2018.653 %U 10.15672/HJMS.2018.653
ISNAD BAİ, Jie , BAİ, Ling , ZENG, Zhijun . "Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function". Hacettepe Journal of Mathematics and Statistics 49 / 1 (February 2020): 68-77 . https://doi.org/10.15672/HJMS.2018.653
AMA BAİ J , BAİ L , ZENG Z . Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 68-77.
Vancouver BAİ J , BAİ L , ZENG Z . Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 77-68.