Optimal control for controllable stochastic linear systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 98.

This paper is concerned with a constrained stochastic linear-quadratic optimal control problem, in which the terminal state is fixed and the initial state is constrained to lie in a stochastic linear manifold. The controllability of stochastic linear systems is studied. Then the optimal control is explicitly obtained by considering a parameterized unconstrained backward LQ problem and an optimal parameter selection problem. A notable feature of our results is that, instead of solving an equation involving derivatives with respect to the parameter, the optimal parameter is characterized by a matrix equation.

DOI : 10.1051/cocv/2020027
Classification : 49N10, 93E20, 93E24, 93B05
Mots-clés : Linear-quadratic, optimal control, controllability, controllability Gramian, Lagrange multiplier, optimal parameter, Riccati equation 
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     author = {Bi, Xiuchun and Sun, Jingrui and Xiong, Jie},
     title = {Optimal control for controllable stochastic linear systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
     doi = {10.1051/cocv/2020027},
     mrnumber = {4185058},
     zbl = {1459.49019},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2020027/}
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Bi, Xiuchun; Sun, Jingrui; Xiong, Jie. Optimal control for controllable stochastic linear systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 98. doi : 10.1051/cocv/2020027. http://www.numdam.org/articles/10.1051/cocv/2020027/

[1] M. Ait Rami, J.B. Moore and X.Y. Zhou, Indefinite stochastic linear quadratic control and generalized differential Riccati equation. SIAM J. Control Optim. 40 (2002) 1296–1311. | DOI | MR | Zbl

[2] A. Bensoussan, Lectures on stochastic control, in Nonlinear Filtering and Stochastic Control, Vol. 972 of Lecture Notes in Mathematics. Springer-Verlag, New York (1982) 1–62. | DOI | MR

[3] J.M. Bismut, Contrôle des systems linéaires quadratiques: applications de l’intégrale stochastique, in Séminaire de Probabilités XII, edited dy C. Dellacherie, P.A. Meyer and M. Weil. Vol. 649 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1978) 180–264. | DOI | Numdam | MR | Zbl

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

[5] B. Gashi, Stochastic minimum-energy control. Syst. Control Lett. 85 (2015) 70–76. | DOI | MR

[6] X. Li, J. Sun and J. Xiong, Linear quadratic optimal control problems for mean-field backward stochastic differential equations. Appl. Math. Optim. 80 (2019) 223–250. | DOI | MR | Zbl

[7] A.E.B. Lim and X.Y. Zhou, Linear-quadratic control of backward stochastic differential equations. SIAM J. Control Optim. 40 (2001) 450–474. | DOI | MR | Zbl

[8] F. Liu and S. Peng, On controllability for stochastic control systems when the coefficient is time-variant. J. Syst. Sci. Complex 23 (2010) 270–278. | DOI | MR | Zbl

[9] Q. Lü, T. Wang and X. Zhang, Characterization of optimal feedback for stochastic linear quadratic control problems. Probab. Uncertain. Quant. Risk 2 (2017) 11. | DOI | MR | Zbl

[10] S. Peng, Backward stochastic differential equation and exact controllability of stochastic control systems. Prog. Nat. Sci. 4 (1994) 274–284. | MR

[11] J. Sun, X. Li and J. Yong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems. SIAM J. Control Optim. 54 (2016) 2274–2308. | DOI | MR | Zbl

[12] J. Sun, J. Xiong and J. Yong, Indefnite stochastic linear-quadratic optimal control problems with random coefcients: Closed-loop representation of open-loop optimal controls. (2018). | arXiv

[13] 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. | DOI | MR | Zbl

[14] H. Wang, J. Sun and J. Yong, Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete Contin. Dyn. Syst. Ser. A 39 (2019) 2785–2805. | DOI | MR | Zbl

[15] Y. Wang, D. Yang, J. Yong and Z. Yu, Exact controllability of linear stochastic differential equations and related problems. Math. Control Relat. Fields 7 (2017) 305–345. | DOI | MR | Zbl

[16] Y. Wang and C. Zhang, The norm optimal control problem for stochastic linear control systems. ESAIM: COCV 21 (2015) 399–413. | Numdam | MR | Zbl

[17] W.M. Wonham, On a matrix Riccati equation of stochastic control. SIAM J. Control 6 (1968) 681–697. | DOI | MR | Zbl

[18] J. Yong and X.Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999). | DOI | MR | Zbl

[19] Z. Yu, Equivalent cost functionals and stochastic linear quadratic optimal control problems. ESAIM: COCV 19 (2013) 78–90. | Numdam | MR | Zbl

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