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On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity
Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems
1. | School of Mathematics and Statistics, Xinyang Normal University, Xinyang, Henan 464000, China |
2. | School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China |
Uncertainty theory is a branch of axiomatic mathematics that deals with human uncertainty. Based on uncertainty theory, this paper discusses linear quadratic (LQ) optimal control with process state inequality constraints for discrete-time uncertain systems, where the weighting matrices in the cost function are assumed to be indefinite. By means of the maximum principle with mixed inequality constraints, we present a necessary condition for the existence of optimal state feedback control that involves a constrained difference equation. Moreover, the existence of a solution to the constrained difference equation is equivalent to the solvability of the indefinite LQ problem. Furthermore, the well-posedness of the indefinite LQ problem is proved. Finally, an example is provided to demonstrate the effectiveness of our theoretical results.
References:
[1] |
M. Athans,
The matrix minimum principle, Information and Control, 11 (1967), 592-606.
doi: 10.1016/S0019-9958(67)90803-0. |
[2] |
K. Bahlali, B. Djehiche and B. Mezerdi,
On the stochastic maximum principle in optimal control of degenerate diffusions with Lipschitz coefficients, Applied Mathematics and Optimization, 56 (2007), 364-378.
doi: 10.1007/s00245-007-9017-6. |
[3] |
A. Bensoussan, S. P. Sethi, R. G. Vickson and N. Derzko,
Stochastic production planning with production constraints: A summary, SIAM Journal on Control and Optimization, 22 (1984), 920-935.
doi: 10.1137/0322060. |
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D. P. Bertsekas,
Dynamic Programming and Stochastic Control, Mathematics in Science and Engineering, 125. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. |
[5] |
S. P. Chen, X. J. Li and X. Y. Zhou,
Stochastic linear quadratic regulators with indefinite control weight costs, SIAM Journal on Control and Optimization, 36 (1998), 1685-1702.
doi: 10.1137/S0363012996310478. |
[6] |
X. Chen, Y. Liu and D. A. Ralescu,
Uncertain stock model with periodic dividends, Fuzzy Optimization and Decision Making, 12 (2013), 111-123.
doi: 10.1007/s10700-012-9141-x. |
[7] |
Y. Gao,
Uncertain models for single facility location problems on networks, Applied Mathematical Modelling, 36 (2012), 2592-2599.
doi: 10.1016/j.apm.2011.09.042. |
[8] |
M. R. Hestenes,
Calculus of Variations and Optimal Control Theory Wiley, New York, 1966. |
[9] |
Y. Hu and X. Y. Zhou,
Constrained stochastic LQ control with random coefficients, and application to portfolio selection, SIAM Journal on Control and Optimization, 44 (2005), 444-466.
doi: 10.1137/S0363012904441969. |
[10] |
D. Kahneman and A. Tversky,
Prospect theory: an analysis of decision under risk, Econometrica, 47 (1979), 263-292.
|
[11] |
X. Li and X. Y. Zhou,
Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon, Communications on Information and Systems, 2 (2002), 265-282.
doi: 10.4310/CIS.2002.v2.n3.a4. |
[12] |
B. Liu,
Uncertainty Theory 2nd edition, Springer-Verlag, Berlin, 2004.
doi: 10.1007/978-3-540-39987-2. |
[13] |
B. Liu,
Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty Springer-Verlag, Heidelberg, 2015.
doi: 10.1007/978-3-662-44354-5. |
[14] |
B. Liu,
Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10.
|
[15] |
X. Liu, Y. Li and W. Zhang,
Stochastic linear quadratic optimal control with constraint for discrete-time systems, Applied Mathematics and Computation, 228 (2014), 264-270.
doi: 10.1016/j.amc.2013.09.036. |
[16] |
B. Liu and K. Yao,
Uncertain multilevel programming: Algorithm and applications, Computers and Industrial Engineering, 89 (2014), 235-240.
doi: 10.1016/j.cie.2014.09.029. |
[17] |
R. Penrose,
A generalized inverse of matrices, Mathematical Proceedings of the Cambridge Philosophical Society, 51 (1955), 406-413.
doi: 10.1017/S0305004100030401. |
[18] |
L. Sheng and Y. Zhu,
Optimistic value model of uncertain optimal control, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 21 (2013), 75-87.
doi: 10.1142/S0218488513400060. |
[19] |
Y. Shu and Y. Zhu,
Stability and optimal control for uncertain continuous-time singular systems, European Journal of Control, 34 (2017), 16-23.
doi: 10.1016/j.ejcon.2016.12.003. |
[20] |
V. K. Socgnia and O. Menoukeu-Pamen,
An infinite horizon stochastic maximum principle for discounted control problem with Lipschitz coefficients, Journal of Mathematical Analysis and Applications, 422 (2015), 684-711.
doi: 10.1016/j.jmaa.2014.09.010. |
[21] |
Z. Wang, J. Guo, M. Zheng and Y. Yang,
A new approach for uncertain multiobjective programming problem based on $\mathcal{P}_E$ principle, Journal of Industrial and Management Optimization, 11 (2015), 13-26.
doi: 10.3934/jimo.2015.11.13. |
[22] |
W. M. Wonham,
On a matrix Riccati equation of stochastic control, SIAM Journal on Control and Optimization, 6 (1968), 681-697.
doi: 10.1137/0306044. |
[23] |
H. Yan, Y. Sun and Y. Zhu,
A linear-quadratic control problem of uncertain discrete-time switched systems, Journal of Industrial and Management Optimization, 13 (2017), 267-282.
doi: 10.3934/jimo.2016016. |
[24] |
J. Yong and X. Y. Zhou,
Stochastic Controls: Hamiltonian Systems and HJB Equations Springer, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
[25] |
W. Zhang, H. Zhang and B. S. Chen,
Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion, IEEE Transactions on Automatic Control, 53 (2008), 1630-1642.
doi: 10.1109/TAC.2008.929368. |
[26] |
W. Zhang and B. S. Chen,
On stabilizability and exact observability of stochastic systems with their applications, Automatica, 40 (2004), 87-94.
doi: 10.1016/j.automatica.2003.07.002. |
[27] |
W. Zhang and G. Li, Discrete-time indefinite stochastic linear quadratic optimal control with second moment constraints Mathematical Problems in Engineering 2014 (2014), Art. ID 278142, 9 pp.
doi: 10.1155/2014/278142. |
[28] |
X. Y. Zhou and D. Li,
Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.
doi: 10.1007/s002450010003. |
[29] |
Y. Zhu,
Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems: An International Journal, 41 (2010), 535-547.
doi: 10.1080/01969722.2010.511552. |
[30] |
Y. Zhu,
Functions of uncertain variables and uncertain programming, Journal of Uncertain Systems, 6 (2012), 278-288.
|
show all references
References:
[1] |
M. Athans,
The matrix minimum principle, Information and Control, 11 (1967), 592-606.
doi: 10.1016/S0019-9958(67)90803-0. |
[2] |
K. Bahlali, B. Djehiche and B. Mezerdi,
On the stochastic maximum principle in optimal control of degenerate diffusions with Lipschitz coefficients, Applied Mathematics and Optimization, 56 (2007), 364-378.
doi: 10.1007/s00245-007-9017-6. |
[3] |
A. Bensoussan, S. P. Sethi, R. G. Vickson and N. Derzko,
Stochastic production planning with production constraints: A summary, SIAM Journal on Control and Optimization, 22 (1984), 920-935.
doi: 10.1137/0322060. |
[4] |
D. P. Bertsekas,
Dynamic Programming and Stochastic Control, Mathematics in Science and Engineering, 125. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. |
[5] |
S. P. Chen, X. J. Li and X. Y. Zhou,
Stochastic linear quadratic regulators with indefinite control weight costs, SIAM Journal on Control and Optimization, 36 (1998), 1685-1702.
doi: 10.1137/S0363012996310478. |
[6] |
X. Chen, Y. Liu and D. A. Ralescu,
Uncertain stock model with periodic dividends, Fuzzy Optimization and Decision Making, 12 (2013), 111-123.
doi: 10.1007/s10700-012-9141-x. |
[7] |
Y. Gao,
Uncertain models for single facility location problems on networks, Applied Mathematical Modelling, 36 (2012), 2592-2599.
doi: 10.1016/j.apm.2011.09.042. |
[8] |
M. R. Hestenes,
Calculus of Variations and Optimal Control Theory Wiley, New York, 1966. |
[9] |
Y. Hu and X. Y. Zhou,
Constrained stochastic LQ control with random coefficients, and application to portfolio selection, SIAM Journal on Control and Optimization, 44 (2005), 444-466.
doi: 10.1137/S0363012904441969. |
[10] |
D. Kahneman and A. Tversky,
Prospect theory: an analysis of decision under risk, Econometrica, 47 (1979), 263-292.
|
[11] |
X. Li and X. Y. Zhou,
Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon, Communications on Information and Systems, 2 (2002), 265-282.
doi: 10.4310/CIS.2002.v2.n3.a4. |
[12] |
B. Liu,
Uncertainty Theory 2nd edition, Springer-Verlag, Berlin, 2004.
doi: 10.1007/978-3-540-39987-2. |
[13] |
B. Liu,
Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty Springer-Verlag, Heidelberg, 2015.
doi: 10.1007/978-3-662-44354-5. |
[14] |
B. Liu,
Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10.
|
[15] |
X. Liu, Y. Li and W. Zhang,
Stochastic linear quadratic optimal control with constraint for discrete-time systems, Applied Mathematics and Computation, 228 (2014), 264-270.
doi: 10.1016/j.amc.2013.09.036. |
[16] |
B. Liu and K. Yao,
Uncertain multilevel programming: Algorithm and applications, Computers and Industrial Engineering, 89 (2014), 235-240.
doi: 10.1016/j.cie.2014.09.029. |
[17] |
R. Penrose,
A generalized inverse of matrices, Mathematical Proceedings of the Cambridge Philosophical Society, 51 (1955), 406-413.
doi: 10.1017/S0305004100030401. |
[18] |
L. Sheng and Y. Zhu,
Optimistic value model of uncertain optimal control, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 21 (2013), 75-87.
doi: 10.1142/S0218488513400060. |
[19] |
Y. Shu and Y. Zhu,
Stability and optimal control for uncertain continuous-time singular systems, European Journal of Control, 34 (2017), 16-23.
doi: 10.1016/j.ejcon.2016.12.003. |
[20] |
V. K. Socgnia and O. Menoukeu-Pamen,
An infinite horizon stochastic maximum principle for discounted control problem with Lipschitz coefficients, Journal of Mathematical Analysis and Applications, 422 (2015), 684-711.
doi: 10.1016/j.jmaa.2014.09.010. |
[21] |
Z. Wang, J. Guo, M. Zheng and Y. Yang,
A new approach for uncertain multiobjective programming problem based on $\mathcal{P}_E$ principle, Journal of Industrial and Management Optimization, 11 (2015), 13-26.
doi: 10.3934/jimo.2015.11.13. |
[22] |
W. M. Wonham,
On a matrix Riccati equation of stochastic control, SIAM Journal on Control and Optimization, 6 (1968), 681-697.
doi: 10.1137/0306044. |
[23] |
H. Yan, Y. Sun and Y. Zhu,
A linear-quadratic control problem of uncertain discrete-time switched systems, Journal of Industrial and Management Optimization, 13 (2017), 267-282.
doi: 10.3934/jimo.2016016. |
[24] |
J. Yong and X. Y. Zhou,
Stochastic Controls: Hamiltonian Systems and HJB Equations Springer, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
[25] |
W. Zhang, H. Zhang and B. S. Chen,
Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion, IEEE Transactions on Automatic Control, 53 (2008), 1630-1642.
doi: 10.1109/TAC.2008.929368. |
[26] |
W. Zhang and B. S. Chen,
On stabilizability and exact observability of stochastic systems with their applications, Automatica, 40 (2004), 87-94.
doi: 10.1016/j.automatica.2003.07.002. |
[27] |
W. Zhang and G. Li, Discrete-time indefinite stochastic linear quadratic optimal control with second moment constraints Mathematical Problems in Engineering 2014 (2014), Art. ID 278142, 9 pp.
doi: 10.1155/2014/278142. |
[28] |
X. Y. Zhou and D. Li,
Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.
doi: 10.1007/s002450010003. |
[29] |
Y. Zhu,
Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems: An International Journal, 41 (2010), 535-547.
doi: 10.1080/01969722.2010.511552. |
[30] |
Y. Zhu,
Functions of uncertain variables and uncertain programming, Journal of Uncertain Systems, 6 (2012), 278-288.
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