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Necessary conditions of optimality for a class of optimal control problems on time scales. (English) Zbl 1189.34172

Summary: A class of optimal control problems of a system governed by linear dynamic equations on time scales with quadratic cost functional is considered. By the Lebesgue \(\Delta \)-integral theory and the Sobolev-type space \(H^{1}\) on time scales the weak solution of linear dynamic equations on time scales for both initial value problem and backward problem are introduced, therefore the necessary conditions of optimality are presented. Some typical examples are given for demonstration.

MSC:

34N05 Dynamic equations on time scales or measure chains
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[1] S. Hilger, Ein maokettenkalkl mit anwendung auf zentrumsmannig faltigkeiten, Ph.D. Thesis, Universitat Wurzburg, 1988; S. Hilger, Ein maokettenkalkl mit anwendung auf zentrumsmannig faltigkeiten, Ph.D. Thesis, Universitat Wurzburg, 1988
[2] Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales (2003), Birkhäuser: Birkhäuser Boston · Zbl 1025.34001
[3] Brandt, A., Multiscale computation: From fast solvers to systematic upscaling, Computational Fluid and Solid Mechanics 2003, 1871-1873 (2003)
[4] Lakshmikantham, V.; Sivasundaram, S., Stability of moving invariant sets and uncertain dynamic systems on time scales, Computers and Mathematics with Applications, 36, 339-346 (1998) · Zbl 0933.93058
[5] Agarwal, R. P.; O’Regan, D., Triple solutions to boundary value problems on time scales, Applied Mathematics Letters, 13, 7-11 (2000) · Zbl 0958.34021
[6] Benchohra, M.; Henderson, J.; Ntouyas, S., Impulsive Differential Equations and Inclusions (2006), Hindawi Publishing Corporation: Hindawi Publishing Corporation New York · Zbl 1130.34003
[7] Darbha, Swaroop; Rajagopal, K. R., Aggregation of a class of interconnected, linear dynamical systems, Systems and Control Letters, 43, 387-401 (2001) · Zbl 0974.93009
[8] Hilger, S., Analysis on measure chains-a unified approach to continuous and discrete calculus, Results in Mathematics, 18, 18-56 (1990) · Zbl 0722.39001
[9] Liu, Hongbo; Xiang, X., A class of the first order impulsive dynamic equations on time scales, Nonlinear AnalysisTMA, 69, 2803-2811 (2008) · Zbl 1159.34005
[10] Wang, Da-Bin, Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales, Computers and Mathematics with Applications, 56, 1496-1504 (2008) · Zbl 1155.34313
[11] Zafer, A.; Kaymakcalan, B.; Ozgun, S. A., Asymptotic behavior of higher-order nonlinear equations on time scales, Computers and Mathematics with Applications, 36, 299-306 (1998) · Zbl 0933.39037
[12] Gong, Yurong; Xiang, X., A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales, Journal of Industrial and Management Optimization, 5, 1-13 (2009) · Zbl 1158.39300
[13] Cabada, A.; Vivero, D. R., Expression of the Lebesgue \(\Delta \)-integral on time scales as a usual Lebesgue integral; Application to the calculus of \(\Delta \)-antiderivatives, Mathematical and Computer Modelling, 43, 194-207 (2006) · Zbl 1092.39017
[14] Guseinov, G. S., Integration on time scales, Journal of Mathematical Analysis and Applications, 285, 107-127 (2003) · Zbl 1039.26007
[15] Rynne, Bryan P., \(L^2\) spaces and boundary value problems on time scales, Journal of Mathematical Analysis and Applications, 328, 1217-1236 (2007) · Zbl 1116.34021
[16] Ahmed, N. U., Elements of Finite Dimensional Systems and Control Theory (1988), Longman Scientific and Techical: Longman Scientific and Techical New York · Zbl 0658.93002
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