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:
[1]

S. A. Attia, V. Azhmyakov and J. Raisch, On an optimization problem for a class of impulsive hybrid systems,, Discrete Event Dynamic Systems, 20 (2010), 215.  doi: 10.1007/s10626-009-0068-5.  Google Scholar

[2]

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[3]

A. S. Bortakovskii, Sufficient optimality conditions for continuous-discrete systems with multiple instantaneous switchings of the discrete part,, Journal of Computer and Systems Sciences International, 51 (2012), 183.  doi: 10.1134/S1064230712020049.  Google Scholar

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A. V. Dmitruk and A. M. Kaganovich, The hybrid maximum principle is a consequence of Pontryagin maximum principle,, Systems $&$ Control Letters, 57 (2008), 964.  doi: 10.1016/j.sysconle.2008.05.006.  Google Scholar

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M. Garavello and B. Piccoli, Hybrid necessary principle,, SIAM Journal on Control and Optimization, 43 (2005), 1867.  doi: 10.1137/S0363012903416219.  Google Scholar

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C. Jiang, K. L. Teo, R. Loxton and G. R. Duan, A neighboring extremal solution for an optimal switched impulsive control problem,, Journal of Industrial and Management Optimization, 8 (2012), 591.  doi: 10.3934/jimo.2012.8.591.  Google Scholar

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M. Kamgarpour and C. Tomlin, On optimal control of non-autonomousswitched systems with a fixed mode sequence,, Automatica, 48 (2012), 1177.  doi: 10.1016/j.automatica.2012.03.019.  Google Scholar

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R. Li, K. L. Teo, K. H. Wong and G. R. 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

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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

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S. F. Maharramov, Necessary optimality conditions for switching control problems,, Journal of Industrial and Management Optimization, 6 (2010), 47.  doi: 10.3934/jimo.2010.6.47.  Google Scholar

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S. F. Maharramov, Optimality condition of a nonsmooth switching control system,, Automatic Control and Computer Sciences, 42 (2008), 94.  doi: 10.3103/S0146411608020077.  Google Scholar

[15]

K. Malanowski, H. Maurer and S. Pickenhain, Second-order sufficient conditions for state-constrained optimal control problems,, Journal of Optimization Theory and Applications, 123 (2004), 595.  doi: 10.1007/s10957-004-5725-0.  Google Scholar

[16]

H. Maurer, First and second order sufficient optimality conditions in mathematical programming and optimal control,, Mathematical Programming Study, 14 (1981), 163.   Google Scholar

[17]

H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal control problems with free final time: the Riccati approach,, SIAM Journal on Control and Optimization, 41 (2002), 380.  doi: 10.1137/S0363012900377419.  Google Scholar

[18]

H. Maurer and S. Pickenhain, Second-order sufficient conditions for control problems with mixed control-state constrains,, Journal of Optimization Theory and Applications, 86 (1995), 649.  doi: 10.1007/BF02192163.  Google Scholar

[19]

H. J. Oberle and R. Rosendahl, Numerical computation of a singular-state subarc in an economic optimal control model,, Optimal Control Applications and Methods, 27 (2006), 211.  doi: 10.1002/oca.775.  Google Scholar

[20]

N. P. Osmolovskii and H. Maurer, Second-order sufficient optimality conditions for a control problem with continuous and bang-bang control components: Riccati approach,, in IFIP Conference on System Modeling and optimization, 312 (2009), 411.  doi: 10.1007/978-3-642-04802-9_24.  Google Scholar

[21]

R. 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

show all references

References:
[1]

S. A. Attia, V. Azhmyakov and J. Raisch, On an optimization problem for a class of impulsive hybrid systems,, Discrete Event Dynamic Systems, 20 (2010), 215.  doi: 10.1007/s10626-009-0068-5.  Google Scholar

[2]

D. Augustin and H. Maurer, Second order sufficient conditions and sensitivity analysis for optimal multiprocess control problems,, Control and Cybernetics, 29 (2000), 11.   Google Scholar

[3]

A. S. Bortakovskii, Sufficient optimality conditions for continuous-discrete systems with multiple instantaneous switchings of the discrete part,, Journal of Computer and Systems Sciences International, 51 (2012), 183.  doi: 10.1134/S1064230712020049.  Google Scholar

[4]

A. V. Dmitruk and A. M. Kaganovich, The hybrid maximum principle is a consequence of Pontryagin maximum principle,, Systems $&$ Control Letters, 57 (2008), 964.  doi: 10.1016/j.sysconle.2008.05.006.  Google Scholar

[5]

M. Egerstedt, Y. Wardi and H. Axelsson, Transition-time optimization for switched-mode dynamical systems,, IEEE Transactions on Automatic Control, 51 (2006), 110.  doi: 10.1109/TAC.2005.861711.  Google Scholar

[6]

M. Garavello and B. Piccoli, Hybrid necessary principle,, SIAM Journal on Control and Optimization, 43 (2005), 1867.  doi: 10.1137/S0363012903416219.  Google Scholar

[7]

C. Jiang, K. L. Teo, R. Loxton and G. R. Duan, A neighboring extremal solution for an optimal switched impulsive control problem,, Journal of Industrial and Management Optimization, 8 (2012), 591.  doi: 10.3934/jimo.2012.8.591.  Google Scholar

[8]

M. Kamgarpour and C. Tomlin, On optimal control of non-autonomousswitched systems with a fixed mode sequence,, Automatica, 48 (2012), 1177.  doi: 10.1016/j.automatica.2012.03.019.  Google Scholar

[9]

R. Li, K. L. Teo, K. H. Wong and G. R. 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. Maharramov, Necessary optimality conditions for switching control problems,, Journal of Industrial and Management Optimization, 6 (2010), 47.  doi: 10.3934/jimo.2010.6.47.  Google Scholar

[14]

S. F. Maharramov, Optimality condition of a nonsmooth switching control system,, Automatic Control and Computer Sciences, 42 (2008), 94.  doi: 10.3103/S0146411608020077.  Google Scholar

[15]

K. Malanowski, H. Maurer and S. Pickenhain, Second-order sufficient conditions for state-constrained optimal control problems,, Journal of Optimization Theory and Applications, 123 (2004), 595.  doi: 10.1007/s10957-004-5725-0.  Google Scholar

[16]

H. Maurer, First and second order sufficient optimality conditions in mathematical programming and optimal control,, Mathematical Programming Study, 14 (1981), 163.   Google Scholar

[17]

H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal control problems with free final time: the Riccati approach,, SIAM Journal on Control and Optimization, 41 (2002), 380.  doi: 10.1137/S0363012900377419.  Google Scholar

[18]

H. Maurer and S. Pickenhain, Second-order sufficient conditions for control problems with mixed control-state constrains,, Journal of Optimization Theory and Applications, 86 (1995), 649.  doi: 10.1007/BF02192163.  Google Scholar

[19]

H. J. Oberle and R. Rosendahl, Numerical computation of a singular-state subarc in an economic optimal control model,, Optimal Control Applications and Methods, 27 (2006), 211.  doi: 10.1002/oca.775.  Google Scholar

[20]

N. P. Osmolovskii and H. Maurer, Second-order sufficient optimality conditions for a control problem with continuous and bang-bang control components: Riccati approach,, in IFIP Conference on System Modeling and optimization, 312 (2009), 411.  doi: 10.1007/978-3-642-04802-9_24.  Google Scholar

[21]

R. 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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