# American Institute of Mathematical Sciences

• Previous Article
Portfolio procurement policies for budget-constrained supply chains with option contracts and external financing
• JIMO Home
• This Issue
• Next Article
A variational inequality approach for constrained multifacility Weber problem under gauge
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  August 2017 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, 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. [13] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty Springer-Verlag, Heidelberg, 2015. [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. [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. [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. [13] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty Springer-Verlag, Heidelberg, 2015. [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. [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. [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.
 [1] Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2016016 [2] Elena K. Kostousova. On control synthesis for uncertain dynamical discrete-time systems through polyhedral techniques. Conference Publications, doi: 10.3934/proc.2015.0723 [3] Martino Bardi, Shigeaki Koike, Pierpaolo Soravia. Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2000.6.361 [4] Aleksandar Zatezalo, Dušan M. Stipanović. Control of dynamical systems with discrete and uncertain observations. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2015.35.4665 [5] Chuandong Li, Fali Ma, Tingwen Huang. 2-D analysis based iterative learning control for linear discrete-time systems with time delay. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2011.7.175 [6] Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, doi: 10.3934/proc.2013.2013.393 [7] Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2017011 [8] Mikhail Gusev. On reachability analysis for nonlinear control systems with state constraints. Conference Publications, doi: 10.3934/proc.2015.0579 [9] Alexander J. Zaslavski. The turnpike property of discrete-time control problems arising in economic dynamics. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2005.5.861 [10] Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, doi: 10.3934/proc.2003.2003.734 [11] Sofian De Clercq, Koen De Turck, Bart Steyaert, Herwig Bruneel. Frame-bound priority scheduling in discrete-time queueing systems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2011.7.767 [12] Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2017010 [13] Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2015.35.4553 [14] Christian Clason, Barbara Kaltenbacher. Avoiding degeneracy in the Westervelt equation by state constrained optimal control. Evolution Equations & Control Theory, doi: 10.3934/eect.2013.2.281 [15] H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.1995.1.77 [16] Piernicola Bettiol, Hélène Frankowska. Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2012.32.1 [17] Lars Grüne, Hasnaa Zidani. Zubov's equation for state-constrained perturbed nonlinear systems. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2015.5.55 [18] Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2012.2.207 [19] Sie Long Kek, Mohd Ismail Abd Aziz. Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2015.5.275 [20] Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo, Rohanin Ahmad. An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2013.3.109

2016 Impact Factor: 0.994