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July 2018, 14(3): 913-930. doi: 10.3934/jimo.2017082

## 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

* Corresponding author: ygzhu@njust.edu.cn

Received  February 2016 Revised  August 2017 Published  September 2017

Fund Project: This work is supported by the National Natural Science Foundation of China (No.61673011), the Nanhu Scholars Program for Young Scholars of XYNU and the Key Scientific Research Project for Colleges and Universities of Henan Province (No.17A120013)

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.

Citation: Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082
##### 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.

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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