Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems.

*(English)*Zbl 1412.49068Summary: 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.

##### MSC:

49N10 | Linear-quadratic optimal control problems |

49L20 | Dynamic programming in optimal control and differential games |

65K05 | Numerical mathematical programming methods |

##### Keywords:

indefinite LQ control; process state inequality constraints; discrete-time uncertain systems; constrained difference equation
PDF
BibTeX
XML
Cite

\textit{Y. Chen} and \textit{Y. Zhu}, J. Ind. Manag. Optim. 14, No. 3, 913--930 (2018; Zbl 1412.49068)

Full Text:
DOI

##### References:

[1] | M. Athans, The matrix minimum principle, Information and Control, 11, 592, (1967) · Zbl 0176.07301 |

[2] | K. Bahlali; B. Djehiche; B. Mezerdi, On the stochastic maximum principle in optimal control of degenerate diffusions with Lipschitz coefficients, Applied Mathematics and Optimization, 56, 364, (2007) · Zbl 1135.60323 |

[3] | A. Bensoussan; S. P. Sethi; R. G. Vickson; N. Derzko, Stochastic production planning with production constraints: A summary, SIAM Journal on Control and Optimization, 22, 920, (1984) · Zbl 0561.90044 |

[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. · Zbl 0549.93064 |

[5] | S. P. Chen; X. J. Li; X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, SIAM Journal on Control and Optimization, 36, 1685, (1998) · Zbl 0916.93084 |

[6] | X. Chen; Y. Liu; D. A. Ralescu, Uncertain stock model with periodic dividends, Fuzzy Optimization and Decision Making, 12, 111, (2013) · Zbl 1411.91550 |

[7] | Y. Gao, Uncertain models for single facility location problems on networks, Applied Mathematical Modelling, 36, 2592, (2012) · Zbl 1246.90083 |

[8] | M. R. Hestenes, Calculus of Variations and Optimal Control Theory Wiley, New York, 1966. |

[9] | Y. Hu; X. Y. Zhou, Constrained stochastic LQ control with random coefficients, and application to portfolio selection, SIAM Journal on Control and Optimization, 44, 444, (2005) · Zbl 1210.93082 |

[10] | D. Kahneman; A. Tversky, Prospect theory: an analysis of decision under risk, Econometrica, 47, 263, (1979) · Zbl 0411.90012 |

[11] | X. Li; X. Y. Zhou, Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon, Communications on Information and Systems, 2, 265, (2002) · Zbl 1119.93418 |

[12] | B. Liu, Uncertainty Theory 2\^{}{nd} 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, 3, (2009) |

[15] | X. Liu; Y. Li; W. Zhang, Stochastic linear quadratic optimal control with constraint for discrete-time systems, Applied Mathematics and Computation, 228, 264, (2014) · Zbl 1364.49032 |

[16] | B. Liu; K. Yao, Uncertain multilevel programming: algorithm and applications, Computers and Industrial Engineering, 89, 235, (2014) |

[17] | R. Penrose, A generalized inverse of matrices, Mathematical Proceedings of the Cambridge Philosophical Society, 51, 406, (1955) · Zbl 0065.24603 |

[18] | L. Sheng; Y. Zhu, Optimistic value model of uncertain optimal control, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 21, 75, (2013) · Zbl 1322.93064 |

[19] | Y. Shu; Y. Zhu, Stability and optimal control for uncertain continuous-time singular systems, European Journal of Control, 34, 16, (2017) · Zbl 1358.93154 |

[20] | V. K. Socgnia; O. Menoukeu-Pamen, An infinite horizon stochastic maximum principle for discounted control problem with Lipschitz coefficients, Journal of Mathematical Analysis and Applications, 422, 684, (2015) · Zbl 1341.49031 |

[21] | Z. Wang; J. Guo; M. Zheng; Y. Yang, A new approach for uncertain multiobjective programming problem based on \(\mathcal{P}_E\) principle, Journal of Industrial and Management Optimization, 11, 13, (2015) · Zbl 1304.90190 |

[22] | W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM Journal on Control and Optimization, 6, 681, (1968) · Zbl 0182.20803 |

[23] | H. Yan; Y. Sun; Y. Zhu, A linear-quadratic control problem of uncertain discrete-time switched systems, Journal of Industrial and Management Optimization, 13, 267, (2017) · Zbl 1368.49041 |

[24] | J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations Springer, New York, 1999. · Zbl 0943.93002 |

[25] | W. Zhang; H. Zhang; B. S. Chen, Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion, IEEE Transactions on Automatic Control, 53, 1630, (2008) · Zbl 1367.93549 |

[26] | W. Zhang; B. S. Chen, On stabilizability and exact observability of stochastic systems with their applications, Automatica, 40, 87, (2004) · Zbl 1043.93009 |

[27] | W. Zhang and G. Li, Discrete-time indefinite stochastic linear quadratic optimal control with second moment constraints Mathematical Problems in Engineering2014 (2014), Art. ID 278142, 9 pp. |

[28] | X. Y. Zhou; D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42, 19, (2000) · Zbl 0998.91023 |

[29] | Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems: An International Journal, 41, 535, (2010) · Zbl 1225.93121 |

[30] | Y. Zhu, Functions of uncertain variables and uncertain programming, Journal of Uncertain Systems, 6, 278, (2012) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.