Multi-Dimension Uncertain Linear Quadratic Optimal Control with Cross Term
Yuefen Chen*,** and Bo Li*
*Department of Applied Mathematics, Nanjing University of Science and Technology
Nanjing 210094, China
**College of Mathematics and Information Science, Xinyang Normal University
Xinyang 464000, China
In this paper, we consider a multi-dimension uncertain linear quadratic (LQ) optimal control with cross term. With the aid of the equation of optimality of a general multi-dimension uncertain optimal control, we present a necessary and sufficient condition for the existence of optimal linear feedback optimal control which is associated with a Riccati differential equation. Moreover, some properties of the solution for the Riccati differential equation are discussed. Furthermore, the uniqueness of the feedback optimal control for the uncertain linear quadratic optimal control with cross term is proved. Finally, as an application, an example is presented to illustrate the theory obtained.
-  R. Kalman, “Contributions to the theory of optimal control,” Boletin de la Sociedad Matematica Mexicana, Vol.5, No.2, pp. 102-119, 1960.
-  W. Wonham, “On a matrix Riccati equation of stochastic control,” SIAM J. on Control and Optimization, Vol.6, No.4, pp. 681-697, 1968.
-  M. Athans, “Special issues on linear-quadratic-Gaussian problem,” IEEE Trans. on Automatic Control, Vol.16, No.6, pp. 527-869, 1971.
-  W. Fleming and R. Rishel, “Deterministic and Stochastic Optimal Control,” Springer-Verlag Press, New York, 1986.
-  F. Carravetta and G. Mavelli, “Suboptimal stochastic linear feedback control of linear systems with state-and control-dependent noise: the incomplete information case,” Automatica, Vol.43, No.5, pp. 751-757, 2007.
-  B. Liu, “Uncertainty Theory (2nd ed.),” Springer-Verlag Press, Berlin, 2007.
-  B. Liu, “Fuzzy process, hybrid process and uncertain process,” J. of Uncertain Systems, Vol.2, No.1, pp. 3-16, 2008.
-  B. Liu, “Some research problems in uncertainy theory,” J. of Uncertain Systems, Vol.3, No.1, pp. 3-10, 2009.
-  Y. Zhu, “Uncertain optimal control with application to a portfolio selection model,” Cybernetics and Systems, Vol.41, No.7, pp. 535-547, 2010.
-  X. Xu and Y. Zhu, “Uncertain bang-bang control for continuous time model,” Cybernetics and Systems, Vol.43, pp. 515-527, 2012.
-  L. Deng and Y. Zhu, “Uncertain optimal control of linear quadratic models with jump,” Mathematical and Computer Modelling, Vol.57, No.9-10, pp. 2432-2441, 2013.
-  L. Sheng and Y. Zhu, “Optimistic value model of uncertain optimal control,” Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.21, Suppl.1, pp. 75-87, 2013.
-  H. Yan and Y. Zhu, “Bang-bang control model for uncertain switched systems,” Applied Mathematical Modelling, Vol.39, No.10-11, pp. 2994-3002, 2015.
-  X. Chen and B. Liu, “Existence and uniqueness theorem for uncertain differential equations,” Fuzzy Optimization and Decision Making, Vol.9, No.1, pp. 69-81, 2010.