Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-31T05:16:06.549Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear dynamic systems and optimal control problems on time scales*

Published online by Cambridge University Press:  23 April 2010

Yunfei Peng ,
Xiaoling Xiang  and
Yang Jiang
Yunfei Peng
Affiliation:
College of Science, Guizhou University, Guiyang, Guizhou 550025, P.R. China. pengyf0803@163.com; xxl3621070@yahoo.com.cn; jiangyang415@163.com
Xiaoling Xiang
Affiliation:
College of Science, Guizhou University, Guiyang, Guizhou 550025, P.R. China. pengyf0803@163.com; xxl3621070@yahoo.com.cn; jiangyang415@163.com
Yang Jiang
Affiliation:
College of Science, Guizhou University, Guiyang, Guizhou 550025, P.R. China. pengyf0803@163.com; xxl3621070@yahoo.com.cn; jiangyang415@163.com
Get access

Abstract

This paper is mainlyconcerned with a class of optimal control problems of systemsgoverned by the nonlinear dynamic systems on time scales.Introducing the reasonable weak solution of nonlinear dynamicsystems, the existence of the weak solution for the nonlineardynamic systems on time scales and its properties are presented.Discussing L1-strong-weak lower semicontinuity of integralfunctional, we give sufficient conditions for the existence ofoptimal controls. Using integration by parts formula and Hamiltonianfunction on time scales, the necessary conditions of optimality arederived respectively. Some examples on continuous optimal controlproblems, discrete optimal control problems, mathematicalprogramming and variational problems are also presented for demonstration.

Keywords

Time scaleweak solutionoptimalcontrolsubdifferentialsexistencenecessary conditions ofoptimality

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011 , pp. 654 - 681
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusion. Hindawi Publishing Corporation, New York (2006).
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.
Gong, Y. and Xiang, X., 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.
Guseinov, G.S., Integration on time scales. J. Math. Anal. Appl. 285 (2003) 107–127. CrossRef
Hilscher, R. and Zeidan, V., Weak maximum principle and accessory problem for control problems on time scales. Nonlinear Anal. 70 (2009) 3209–3226. CrossRef
S. Hu and N.S. Papageoriou, Handbook of Multivalued Analysis. Kluwer Academic Publishers, Dordrecht (1997).
V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamical Systems on Measure Chains. Kluwer Acadamic Publishers, Dordrecht (1996).
Liu, H. and Xiang, X., A class of the first order impulsive dynamic equations on time scales. Nonlinear Anal. 69 (2008) 2803–2811. CrossRef
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 Sciences 58 (2009) 205–212. CrossRef
Peng, Y. and Xiang, X., Necessary conditions of optimality for a class of optimal control problem on time scales. Comp. Math. Appl. 58 (2009) 2035–2045. CrossRef
Rynne, B.P., L 2 spaces and boundary value problems on time-scales. J. Math. Anal. Appl. 328 (2007) 1217–1236. CrossRef
Suslov, S.I., Semicontinuouity of an integral functional in Banach space. Sib. Math. J. 38 (1997) 350–359. CrossRef
Tisdell, C.C. and Zaidi, A., 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. CrossRef
Wang, D.-B., Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales. Comp. Math. Appl. 56 (2008) 1496–1504. CrossRef
E. Zeidler, Nonlinear Functional Analysis and its Applications III. Springer-Verlag, New York (1985).
Z. Zhan and W. Wei, Necessary conditions for a class of optimal control problems on time scales. Abstr. Appl. Anal. 2009 (2009) e1–e14.
Zhan, Z. and Wei, W., 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.
Zhan, Z., Wei, W. and Hamilton-Jacobi-Bellman, H. Xu equations on time scales. Math. Comp. Model. 49 (2009) 2019–2028. CrossRef