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

Year 2020, Volume: 49 Issue: 1, 68 - 77, 06.02.2020

Jie Bai Ling Bai Zhijun Zeng

https://doi.org/10.15672/HJMS.2018.653

Abstract

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.

Keywords

time scales, the Euler-Lagrange equation, calculus of variations, conditional extremum, $\nabla$-derivatives of exponential function

References

• [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.