# Nonlinear dynamic systems and optimal control problems on time scales

Yunfei Peng; Xiaoling Xiang; Yang Jiang

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 3, page 654-681
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topPeng, Yunfei, Xiang, Xiaoling, and Jiang, Yang. "Nonlinear dynamic systems and optimal control problems on time scales." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 654-681. <http://eudml.org/doc/272772>.

@article{Peng2011,

abstract = {This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula and Hamiltonian function on time scales, the necessary conditions of optimality are derived respectively. Some examples on continuous optimal control problems, discrete optimal control problems, mathematical programming and variational problems are also presented for demonstration.},

author = {Peng, Yunfei, Xiang, Xiaoling, Jiang, Yang},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {time scale; weak solution; optimal control; subdifferentials; existence; necessary conditions of optimality},

language = {eng},

number = {3},

pages = {654-681},

publisher = {EDP-Sciences},

title = {Nonlinear dynamic systems and optimal control problems on time scales},

url = {http://eudml.org/doc/272772},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Peng, Yunfei

AU - Xiang, Xiaoling

AU - Jiang, Yang

TI - Nonlinear dynamic systems and optimal control problems on time scales

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2011

PB - EDP-Sciences

VL - 17

IS - 3

SP - 654

EP - 681

AB - This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula and Hamiltonian function on time scales, the necessary conditions of optimality are derived respectively. Some examples on continuous optimal control problems, discrete optimal control problems, mathematical programming and variational problems are also presented for demonstration.

LA - eng

KW - time scale; weak solution; optimal control; subdifferentials; existence; necessary conditions of optimality

UR - http://eudml.org/doc/272772

ER -

## References

top- [1] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusion. Hindawi Publishing Corporation, New York (2006). Zbl1130.34003MR2322133
- [2] R.A.C. Ferreira and D.F.M. Torres, Higher-order calculus of variations on time scales, in Mathematical control theory and finance, Springer, Berlin (2008) 149–159. Zbl1191.49017MR2484109
- [3] Y. Gong and X. Xiang, A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales. J. Ind. Manag. Opt.5 (2009) 1–13. Zbl1158.39300MR2470701
- [4] G.S. Guseinov, Integration on time scales. J. Math. Anal. Appl.285 (2003) 107–127. Zbl1039.26007MR2000143
- [5] R. Hilscher and V. Zeidan, Weak maximum principle and accessory problem for control problems on time scales. Nonlinear Anal.70 (2009) 3209–3226. Zbl1157.49030MR2503067
- [6] S. Hu and N.S. Papageoriou, Handbook of Multivalued Analysis. Kluwer Academic Publishers, Dordrecht (1997). Zbl0887.47001
- [7] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamical Systems on Measure Chains. Kluwer Acadamic Publishers, Dordrecht (1996). Zbl0869.34039MR1419803
- [8] H. Liu and X. Xiang, A class of the first order impulsive dynamic equations on time scales. Nonlinear Anal.69 (2008) 2803–2811. Zbl1159.34005MR2452091
- [9] A.B. Malinowska and D.F.M. Torres, Strong minimizers of the calculus of variations on time scales and the Weierstrass condition, in Proceedings of the Estonian Academy of Sciences58 (2009) 205–212. Zbl1179.49025MR2604248
- [10] Y. Peng and X. Xiang, Necessary conditions of optimality for a class of optimal control problem on time scales. Comp. Math. Appl.58 (2009) 2035–2045. Zbl1189.34172MR2557525
- [11] B.P. Rynne, L2 spaces and boundary value problems on time-scales. J. Math. Anal. Appl.328 (2007) 1217–1236. Zbl1116.34021MR2290047
- [12] S.I. Suslov, Semicontinuouity of an integral functional in Banach space. Sib. Math. J.38 (1997) 350–359. Zbl0868.49009MR1457790
- [13] C.C. Tisdell and A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling. Nonlinear Anal.68 (2008) 3504–3524. Zbl1151.34005MR2401364
- [14] D.-B. Wang, Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales. Comp. Math. Appl.56 (2008) 1496–1504. Zbl1155.34313MR2440567
- [15] E. Zeidler, Nonlinear Functional Analysis and its Applications III. Springer-Verlag, New York (1985). Zbl0583.47051MR768749
- [16] Z. Zhan and W. Wei, Necessary conditions for a class of optimal control problems on time scales. Abstr. Appl. Anal. 2009 (2009) e1–e14. Zbl1163.49013MR2506995
- [17] Z. Zhan and W. Wei, On existence of optimal control governed by a class of the first-order linear dynamic systems on time scales. Appl. Math. Comput.215 (2009) 2070–2081. Zbl1183.49005MR2557091
- [18] Z. Zhan, W. Wei and H. Xu, Hamilton-Jacobi-Bellman equations on time scales. Math. Comp. Model.49 (2009) 2019–2028. Zbl1171.39302MR2532106

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.