Equivalent cost functionals and stochastic linear quadratic optimal control problems
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.

DOI : https://doi.org/10.1051/cocv/2011206
Classification : 93E20,  49N10,  60H10
Mots clés : stochastic LQ problem, stochastic hamiltonian system, forward-backward stochastic differential equation, Riccati equation, stochastic maximum principle
@article{COCV_2013__19_1_78_0,
     author = {Yu, Zhiyong},
     title = {Equivalent cost functionals and stochastic linear quadratic optimal control problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {78--90},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {1},
     year = {2013},
     doi = {10.1051/cocv/2011206},
     zbl = {1258.93129},
     mrnumber = {3023061},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2011206/}
}
TY  - JOUR
AU  - Yu, Zhiyong
TI  - Equivalent cost functionals and stochastic linear quadratic optimal control problems
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
DA  - 2013///
SP  - 78
EP  - 90
VL  - 19
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2011206/
UR  - https://zbmath.org/?q=an%3A1258.93129
UR  - https://www.ams.org/mathscinet-getitem?mr=3023061
UR  - https://doi.org/10.1051/cocv/2011206
DO  - 10.1051/cocv/2011206
LA  - en
ID  - COCV_2013__19_1_78_0
ER  - 
Yu, Zhiyong. Equivalent cost functionals and stochastic linear quadratic optimal control problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 78-90. doi : 10.1051/cocv/2011206. http://www.numdam.org/articles/10.1051/cocv/2011206/

[1] B.D.O. Anderson and J.B. Moore, Optimal control-Linear quadratic methods. Prentice-Hall, New York (1989). | Zbl 0751.49013

[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 705931 | Zbl 0505.93078

[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 520007 | Zbl 0389.93052

[4] S. Chen and J. Yong, Stochastic linear quadratic optimal control problems. Appl. Math. Optim. 43 (2001) 21-45. | MR 1804393 | Zbl 0969.93044

[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 1814267 | Zbl 1023.93072

[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 1626817 | Zbl 0916.93084

[7] M.H.A. Davis, Linear estimation and stochastic control. Chapman and Hall, London (1977). | MR 476099 | Zbl 0437.60001

[8] Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations. Prob. Theory Relat. Fields 103 (1995) 273-283. | MR 1355060 | Zbl 0831.60065

[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 127472 | Zbl 0112.06303

[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 1704232 | Zbl 0927.60004

[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 1777419 | Zbl 0981.93079

[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 1675098 | Zbl 0931.60048

[15] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, New Jersey (1970). | MR 274683 | Zbl 0932.90001

[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 1982735 | Zbl 1035.93065

[17] W.M. Wonham, On a matrix Riccati equation of stochastic control. SIAM J. Control Optim. 6 (1968) 312-326 . | MR 239161 | Zbl 0164.19101

[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 2136983 | Zbl 1156.93409

[19] J. Yong and X. Zhou, Stochastic controls : Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999). | MR 1696772 | Zbl 0943.93002

Cité par Sources :