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November  2011, 16(4): 1137-1155. doi: 10.3934/dcdsb.2011.16.1137

## A class of nonlinear impulsive differential equation and optimal controls on time scales

 1 Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025 2 Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025

Received  October 2010 Revised  March 2011 Published  August 2011

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear impulsive differential equation on time scale. The reasonable weak solution of nonlinear impulsive differential equation on time scale is introduced and the existence and uniqueness of the weak solution and its properties are presented. By $L^{1}-$strong$-$weak lower semicontinuity of integral functional on time scale, we give the existence of optimal controls. Using integration by parts formula on time scale, the necessary conditions of optimality are derived. An example on mathematical programming is also presented for demonstration.
Citation: Yunfei Peng, X. Xiang. A class of nonlinear impulsive differential equation and optimal controls on time scales. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1137-1155. doi: 10.3934/dcdsb.2011.16.1137
##### References:
 [1] M. U. Akhmet and M. Turan, The differential equations on time scales through impulsive differential equations, Nonlinear Analysis, 65 (2006), 2043-2060. doi: 10.1016/j.na.2005.12.042. [2] M. Benchohra, J. Henderson and S. Ntouyas, "Impulsive Differential Equations and Inclusion," Contemporary Mathematics and its Applications, 2, Hindawi Publishing Corporation, New York, 2006. [3] Rui A. C. Ferreira and Delfim F. M. Torres, Higher-order calculus of variations on time scales, in "Mathematical Control Theory and Finance," Springer, Berlin, (2008), 149-159. [4] P. Gajardo, H. Ramirez and A. Rapaport, Minimal time sequential Banach reactors with bounded and impulse controls for one or more species, SIAM J. Control Optim., 47 (2008), 2827-2856. doi: 10.1137/070695204. [5] 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. Industrial and Management Optimization, 5 (2009), 1-10. [6] S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis. Vol. I. Theory," Mathematics and its Applications, 419, Kluwer Academic Publishers, Dordrecht, 1997. [7] Roman Hilscher and Vera Zeidan, Weak maximum principle and accessory problem for control problems on time scales, Nonlinear Analysis, 70 (2009), 3209-3226. doi: 10.1016/j.na.2008.04.025. [8] V. Lakshmikantham and S. Sivasundaram, B. Kaymakcalan, "Dynamical Systems on Measure Chains,'' Kluwer Acadamic Publishers, Dordrecht, 1996. [9] G. Liu, X. Xiang and Y. Peng, Nonlinear integro-differential equation and optimal controls on time scales, Computers and Mathematics with Applications, 61 (2011), 155-169. doi: 10.1016/j.camwa.2010.10.013. [10] H. Liu and X. Xiang, A class of the first order impulsive dynamic equations on time scales, Nonlinear Analysis, 69 (2008), 2803-2811. doi: 10.1016/j.na.2007.08.052. [11] Yajun Ma and Jitao Sun, Uniform eventual Lipschitz stability of impulsive systems on time scales, Applied Mathematics and Computation, 211 (2009), 246-250. doi: 10.1016/j.amc.2009.01.033. [12] Agnieszka B. Malinowska and Delfim F. M. Torres, Strong minimizers of the calculus of variations on time scales and the Weierstrass condition, Proceedings of the Estonian Academy of Sciences, 58 (2009), 205-212. doi: 10.3176/proc.2009.4.02. [13] Agnieszka B. Malinowska and Delfim F. M. Torres, Leitmann's direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales, Applied Mathematics and Computation, 217 (2010), 1158-1162. doi: 10.1016/j.amc.2010.01.015. [14] Y. Peng and X. Xiang, Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls, J. Industrial and Management Optimization, 4 (2008), 17-32. [15] Y. Peng, X. Xiang, Y. Gong and G. Liu, Necessary conditions of optimality for a class of optimal control problems on time scales, Computers and Mathematics with Applications, 58 (2009), 2035-2045. doi: 10.1016/j.camwa.2009.08.032. [16] Y. Peng, X. Xiang and Yang Jiang, Nonliear dynaminc systems and optimal control problems on time scales, ESAIM Control, Optimization and Calculus of Variations, 17(2011), 654-681. doi: 10.1051/cocv/2010022. [17] E. Zeidler, "Nonlinear Functional Analysis and Its Applications III, Variational Methods and Optimization," Springer-Verlag, New York, 1985. [18] Z. Zhan and W. Wei, On existence of optimal control governed by a class of the first-order linear dynamic systems on time scales, Applied Mathematics and Computation, 215 (2009), 2070-2081. doi: 10.1016/j.amc.2009.08.009. [19] Z. Zhan and W. Wei, Necessary conditions for a class of optimal control problems on time scales, Abstract and Applied Analysis, 2009, Article ID 974394, 14 pp.

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##### References:
 [1] M. U. Akhmet and M. Turan, The differential equations on time scales through impulsive differential equations, Nonlinear Analysis, 65 (2006), 2043-2060. doi: 10.1016/j.na.2005.12.042. [2] M. Benchohra, J. Henderson and S. Ntouyas, "Impulsive Differential Equations and Inclusion," Contemporary Mathematics and its Applications, 2, Hindawi Publishing Corporation, New York, 2006. [3] Rui A. C. Ferreira and Delfim F. M. Torres, Higher-order calculus of variations on time scales, in "Mathematical Control Theory and Finance," Springer, Berlin, (2008), 149-159. [4] P. Gajardo, H. Ramirez and A. Rapaport, Minimal time sequential Banach reactors with bounded and impulse controls for one or more species, SIAM J. Control Optim., 47 (2008), 2827-2856. doi: 10.1137/070695204. [5] 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. Industrial and Management Optimization, 5 (2009), 1-10. [6] S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis. Vol. I. Theory," Mathematics and its Applications, 419, Kluwer Academic Publishers, Dordrecht, 1997. [7] Roman Hilscher and Vera Zeidan, Weak maximum principle and accessory problem for control problems on time scales, Nonlinear Analysis, 70 (2009), 3209-3226. doi: 10.1016/j.na.2008.04.025. [8] V. Lakshmikantham and S. Sivasundaram, B. Kaymakcalan, "Dynamical Systems on Measure Chains,'' Kluwer Acadamic Publishers, Dordrecht, 1996. [9] G. Liu, X. Xiang and Y. Peng, Nonlinear integro-differential equation and optimal controls on time scales, Computers and Mathematics with Applications, 61 (2011), 155-169. doi: 10.1016/j.camwa.2010.10.013. [10] H. Liu and X. Xiang, A class of the first order impulsive dynamic equations on time scales, Nonlinear Analysis, 69 (2008), 2803-2811. doi: 10.1016/j.na.2007.08.052. [11] Yajun Ma and Jitao Sun, Uniform eventual Lipschitz stability of impulsive systems on time scales, Applied Mathematics and Computation, 211 (2009), 246-250. doi: 10.1016/j.amc.2009.01.033. [12] Agnieszka B. Malinowska and Delfim F. M. Torres, Strong minimizers of the calculus of variations on time scales and the Weierstrass condition, Proceedings of the Estonian Academy of Sciences, 58 (2009), 205-212. doi: 10.3176/proc.2009.4.02. [13] Agnieszka B. Malinowska and Delfim F. M. Torres, Leitmann's direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales, Applied Mathematics and Computation, 217 (2010), 1158-1162. doi: 10.1016/j.amc.2010.01.015. [14] Y. Peng and X. Xiang, Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls, J. Industrial and Management Optimization, 4 (2008), 17-32. [15] Y. Peng, X. Xiang, Y. Gong and G. Liu, Necessary conditions of optimality for a class of optimal control problems on time scales, Computers and Mathematics with Applications, 58 (2009), 2035-2045. doi: 10.1016/j.camwa.2009.08.032. [16] Y. Peng, X. Xiang and Yang Jiang, Nonliear dynaminc systems and optimal control problems on time scales, ESAIM Control, Optimization and Calculus of Variations, 17(2011), 654-681. doi: 10.1051/cocv/2010022. [17] E. Zeidler, "Nonlinear Functional Analysis and Its Applications III, Variational Methods and Optimization," Springer-Verlag, New York, 1985. [18] Z. Zhan and W. Wei, On existence of optimal control governed by a class of the first-order linear dynamic systems on time scales, Applied Mathematics and Computation, 215 (2009), 2070-2081. doi: 10.1016/j.amc.2009.08.009. [19] Z. Zhan and W. Wei, Necessary conditions for a class of optimal control problems on time scales, Abstract and Applied Analysis, 2009, Article ID 974394, 14 pp.
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