American Institute of Mathematical Sciences

January  2015, 11(1): 329-343. doi: 10.3934/jimo.2015.11.329

Second order sufficient optimality conditions for hybrid control problems with state jump

 1 School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China, China 2 Business School, University of Shanghai for Science and Technology, Shanghai, 200093

Received  May 2013 Revised  February 2014 Published  May 2014

In this paper, an optimal control problem for a class of hybrid systems is considered. By introducing a new time variable and transforming the hybrid optimal control problem into an equivalent problem, second order sufficient optimality conditions for this hybrid problem are derived. It is shown that sufficient optimality conditions can be verified by checking the Legendre-Clebsch condition and solving some Riccati equations with certain boundary and jump conditions. An example is given to show the effectiveness of the main results.
Citation: Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial & Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329
References:
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References:
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Duan, Control parameterization enhancing transform for optimal control of switched systems,, Mathematical and Computer Modelling, 43 (2006), 1393.  doi: 10.1016/j.mcm.2005.08.012.  Google Scholar [10] Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems,, Dynamics of Continuous, 18 (2011), 59.   Google Scholar [11] C. Y. Liu, Z. H. Guan, E. M. Feng and H. C. Yin, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch fermentation,, Journal of Industrial and Management Optimization, 5 (2009), 835.   Google Scholar [12] R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica, 45 (2009), 2250.  doi: 10.1016/j.automatica.2009.05.029.  Google Scholar [13] S. F. 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Rosendahl, Sufficient Optimality Conditions for Nonsmooth Optimal Control Problems,, Ph.D thesis, (2009).   Google Scholar [22] M. Rungger and O. Stursberg, A numerical method for hybrid optimal control based on dynamic programming,, Nonlinear Analysis: Hybrid Systems, 5 (2011), 254.  doi: 10.1016/j.nahs.2010.09.002.  Google Scholar [23] M. S. Shaikh and P. E. Caines, On the hybird optimal control problem: Theory and algorithms,, IEEE Transactions on Automatic Control, 52 (2007), 1587.  doi: 10.1109/TAC.2007.904451.  Google Scholar [24] H. J. Sussman, A maximum principle for hybrid optimization,, in Proceedings of IEEE Conference on Decision and Control, (1999), 425.   Google Scholar [25] X. P. Xu and J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants,, IEEE Transactions on Automatic Control, 49 (2004), 2.  doi: 10.1109/TAC.2003.821417.  Google Scholar [26] V. Zeidan, The riccati equation for optimal control problems with mixed state-control constraints: necessary and sufficiency,, SIAM Journal on Control and Optimization, 32 (1994), 1297.  doi: 10.1137/S0363012992233640.  Google Scholar
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