The second Euler-Lagrange equation of variational calculus on time scales

European Journal of Control

Volumn 17, Issue 1, 2011, Pages 9-18

  Bartosiewicz, Zbigniew ^a   Martins, Natália ^b   Torres, Delfim F M ^b  

^a BIALYSTOK UNIVERSITY OF TECHNOLOGY   (Poland)

^b UNIVERSITY OF AVEIRO   (Portugal)

Author keywords

Calculus of variations; DuBois Reymond, and second Erdmann necessary optimality conditions; Euler Lagrange; Noether's theorem; Optimal control; Time scales

Indexed keywords

CALCULATIONS; EQUATIONS OF MOTION; LAGRANGE MULTIPLIERS; TIME MEASUREMENT;

CALCULUS OF VARIATIONS; DELTA INTEGRALS; DUBOI-REYMOND, AND SECOND ERDMANN NECESSARY OPTIMALITY CONDITION; EULER-LAGRANGE; EULER-LAGRANGE EQUATIONS; MINIMISATION; NECESSARY OPTIMALITY CONDITION; NOETHER'S THEOREM; OPTIMAL CONTROLS; TIME-SCALES;

VARIATIONAL TECHNIQUES;

EID: 79951562167     PISSN: 09473580     EISSN: None     Source Type: Journal    
DOI: 10.3166/ejc.17.9-18     Document Type: Article

References (39)

1. Agarwal R, Bohner M, O'Regan D, Peterson A. Dynamic equations on time scales: a survey, J Comput Appl Math 2002; 141(1-2): 1-26.
2. Ahlbrandt CD, Bohner M, Ridenhour J. Hamiltonian systems on time scales. J Math Anal Appl 2000; 250(2): 561-578.
3. Almeida R, Torres DFM. Isoperimetric problems on time scales with nabla derivatives. J Vib Control 2009; 15(6): 951-958.
4. Atici FM, Biles DC, Lebedinsky A. An application of time scales to economics. Math Comput Modelling 2006; 43(7-8): 718-726.
5. Aulbach B, Hilger S. A unified approach to continuous and discrete dynamics. In: Qualitative Theory of Differential Equations (Szeged, 1988). Colloq. Math. Soc. J'anos Bolyai 53: 37-56.
6. Bangerezako G. Variational q-calculus. J Math Anal Appl 2004; 289(2): 650-665.
7. Bartosiewicz Z, Kotta U, Pawłuszewicz E, Wyrwas M. Algebraic formalism of differential one-forms for nonlinear control systems on time scales. Proc Estonian Acad Sci Phys Math 2007; 56(3): 264-282.
8. Bartosiewicz Z, Pawłuszewicz E. Realizations of nonlinear control systems on time scales. IEEE Trans Automat Control 2008; 53(2), 571-575.
9. Bartosiewicz Z, Pawłuszewicz E. Dynamic feedback equivalence of time-variant control systems on homogeneous time scales. Int J Math Stat 2009; 5(A09): 11-20.
10. Bartosiewicz Z, Piotrowska E, Wyrwas M. Stability, stabilization and observers of linear control systems on time scales. In Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, December 12-14, 2007; 2803-2808.
11. Bartosiewicz Z, Torres DFM. Noether's theorem on time scales. J Math Anal Appl 2008; 342(2), 1220-1226.
12. Bayen T. Analytical parameterization of rotors and proof of a Goldberg conjecture by optimal control theory. SIAM J Control Optim 2008; 47(6): 3007-3036.
13. Bohner M. Calculus of variations on time scales. Dynam Systems Appl 2004; 13(3-4), 339-349.
14. Bohner M, Peterson A. Dynamic Equations on Time Scales. Birkhäuser Boston, Boston, MA, 2001.
15. Cesari L. Optimization-Theory and Applications. Springer, New York, 1983.
16. Clarke FH. Optimization and Nonsmooth Analysis. Wiley, New York, 1983.
17. Clarke FH. Methods of Dynamic and Nonsmooth Optimization. SIAM, Philadelphia, PA, 1989.
18. DaCunha JJ. Stability for time varying linear dynamic systems on time scales. J Comput Appl Math 2005; 176(2): 381-410.
19. Ferreira RAC, Torres DFM. Remarks on the calculus of variations on time scales. Int J Ecol Econ Stat 2007; 9(F07), 65-73.
20. Ferreira RAC, Torres DFM. Higher-order calculus of variations on time scales. In Mathematical Control Theory and Finance. Springer, Berlin 2008; 149-159.
21. Frederico GSF, Torres DFM. Fractional conservation laws in optimal control theory. Nonlinear Dyn 2008; 53(3), 215-222.
22. Gouveia PDF, Torres DFM. Automatic computation of conservation laws in the calculus of variations and optimal control. Comput Methods Appl Math 2005; 5(4), 387-409.
23. Guseinov GSh, Kaymakçalan B. On a disconjugacy criterion for second order dynamic equations on time scales. J Comput Appl Math 2002; 141(1-2), 187-196.
24. Hilscher R, Zeidan V. Calculus of variations on time scales: weak local piecewise C1 rd solutions with variable endpoints. J Math Anal Appl 2004; 289(1): 143-166.
25. Hilscher R, Zeidan V. Weak maximum principle and accessory problem for control problems on time scales. Nonlinear Anal 2009; 70(9): 3209-3226.
26. Kelley WG, Peterson AC. Difference Equations. Academic Press, Boston, MA, 1991.
27. Logan JD. Applied Mathematics. Wiley, New York, 1987.
28. Malinowska AB, Torres DFM. Necessary and sufficient conditions for local Pareto optimality on time scales. J Math Sci (NY) 2009; 161(6), 803-810.
29. Malinowska AB, Torres DFM. Strong minimizers of the calculus of variations on time scales and the Weierstrass condition. Proc Est Acad Sci 2009; 58(4): 205-212.
30. Martins N, Torres DFM. Calculus of variations on time scales with nabla derivatives. Nonlinear Anal 2009; 71(12): e763-e773.
31. Mozyrska D, Bartosiewicz Z. Observability of a class of linear dynamic infinite systems on time scales. Proc Estonian Acad Sci Phys Math 2007; 56(4), 347-358.
32. Seiffertt J, Sanyal S, Wunsch DC. Hamilton-Jacobi-Bellmam equations and approximate dynamic programming on time scales. IEEE Trans Syst Man Cybern. Part B: Cybern 2008; 38(4), 918-923.
33. Torres DFM. On the Noether theorem for optimal control. Eur J Control 2002; 8(1): 56-63.
34. Torres DFM. Integrals of motion for discrete-time optimal control problems. In Control Applications of Optimisation, IFAC Workshop Series, 2003, 33-38.
35. Torres DFM. Carathéodory equivalence, Noether theorems, and Tonelli full-regularity in the calculus of variations and optimal control. J Math Sci (NY) 2004; 120(1): 1032-1050.
36. Torres DFM. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Commun Pure Appl Anal 2004; 3(3): 491-500.
37. Troutman JL. Variational Calculus and Optimal Control, 2nd edn. Springer, New York, 1996.
38. Vinter R. Optimal Control. Birkhäuser Boston, Boston, MA, 2000.
39. Zhan Z, Wei W, Xu H. Hamilton-Jacobi-Bellman equations on time scales. Math Comput Modelling 2009; 49(9-10): 2019-2028.