Equivalent cost functionals and stochastic linear quadratic optimal control problems

Yu, Zhiyong

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 78-90.

This paper is concerned with the stochastic linear quadratic optimal control problems (LQ problems, for short) for which the coefficients are allowed to be random and the cost functionals are allowed to have negative weights on the square of control variables. We propose a new method, the equivalent cost functional method, to deal with the LQ problems. Comparing to the classical methods, the new method is simple, flexible and non-abstract. The new method can also be applied to deal with nonlinear optimization problems.

[1] B.D.O. Anderson and J.B. Moore, Optimal control-Linear quadratic methods. Prentice-Hall, New York (1989). | Zbl

[2] A. Bensoussan, Lecture on stochastic cntrol, Part I, in Nonlinear Filtering and Stochastic Control, Lecture Notes in Math. 972. Springer-Verlag, Berlin (1983) 1-39. | MR | Zbl

[3] J.M. Bismut, Controle des systems linears quadratiques : applications de l'integrale stochastique, in Séminaire de Probabilités XII, Lecture Notes in Math. 649, edited by C. Dellacherie, P.A. Meyer and M. Weil. Springer-Verlag, Berlin (1978) 180-264. | Numdam | MR | Zbl

[4] S. Chen and J. Yong, Stochastic linear quadratic optimal control problems. Appl. Math. Optim. 43 (2001) 21-45. | MR | Zbl

[5] S. Chen and Z. Zhou, Stochastic linaer quadratic regulators with indefinite control weight costs. II. SIAM J. Control Optim. 39 (2000) 1065-1081. | MR | Zbl

[6] S. Chen, X. Li and X. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J. Control Optim. 36 (1998) 1685-1702. | MR | Zbl

[7] M.H.A. Davis, Linear estimation and stochastic control. Chapman and Hall, London (1977). | MR | Zbl

[8] Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations. Prob. Theory Relat. Fields 103 (1995) 273-283. | MR | Zbl

[9] M. Jeanblanc and Z. Yu , Optimal investment problems with uncertain time horizon. Working paper.

[10] R.E. Kalman, Contributions to the theory of optimal control. Bol. Soc. Math. Mexicana 5 (1960) 102-119. | MR | Zbl

[11] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, Lecture Notes in Math. 1702. Springer-Verlag, New York (1999). | MR | Zbl

[12] S. Peng, New development in stochastic maximum principle and related backward stochastic differential equations, in proceedings of 31st CDC Conference. Tucson (1992).

[13] S. Peng, Open problems on backward stochastic differential equations, in Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998). edited by S. Chen et al., Kluwer Academic Publishers, Boston (1999) 966-979. | MR | Zbl

[14] S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equation and applications to optimal control. SIAM J. Control Optim. 37 (1999) 825-843. | MR | Zbl

[15] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, New Jersey (1970). | MR | Zbl

[16] S. Tang, General linear quadratic optimal stochastic control problems with random coefficients : linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim. 42 (2003) 53-75. | MR | Zbl

[17] W.M. Wonham, On a matrix Riccati equation of stochastic control. SIAM J. Control Optim. 6 (1968) 312-326 . | MR | Zbl

[18] Z. Wu, Forward-backward stochastic differential equations, linear quadratic stochastic optimal control and nonzero sum differential games. Journal of Systems Science and Complexity 18 (2005) 179-192. | MR | Zbl

[19] J. Yong and X. Zhou, Stochastic controls : Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999). | MR | Zbl