January  2019, 4: 1 doi: 10.1186/s41546-018-0035-x

Mixed deterministic and random optimal control of linear stochastic systems with quadratic costs

1. Univ Rennes, CNRS, IRMAR-UMR 6625, 35000 Rennes, France;

2. Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433 China

Received  November 26, 2018 Revised  December 11, 2018

Fund Project: Hu acknowledges research supported by Lebesgue center of mathematics "Investissements d'avenir" program-ANR-11-LABX-0020-01, by CAESARS-ANR-15-CE05-0024 and by MFG-ANR-16-CE40-0015-01. Tang acknowledges research supported by National Science Foundation of China (Grant No. 11631004) and Science and Technology Commission of Shanghai Municipality (Grant No. 14XD1400400).

In this paper, we consider the mixed optimal control of a linear stochastic system with a quadratic cost functional, with two controllers-one can choose only deterministic time functions, called the deterministic controller, while the other can choose adapted random processes, called the random controller. The optimal control is shown to exist under suitable assumptions. The optimal control is characterized via a system of fully coupled forward-backward stochastic differential equations (FBSDEs) of mean-field type. We solve the FBSDEs via solutions of two (but decoupled) Riccati equations, and give the respective optimal feedback law for both deterministic and random controllers, using solutions of both Riccati equations. The optimal state satisfies a linear stochastic differential equation (SDE) of mean-field type. Both the singular and infinite time-horizonal cases are also addressed.
Citation: Ying Hu, Shanjian Tang. Mixed deterministic and random optimal control of linear stochastic systems with quadratic costs. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 1-. doi: 10.1186/s41546-018-0035-x
References:
[1]

Bensoussan, A.:Lectures on stochastic control. In:Mitter, S.K., Moro, A. (eds.) Nonlinear filtering and stochastic control, Proceedings of the 3rd 1981 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), Held at Cortona, July 1-10, 1981 pp. 1-62. Lecture Notes in Mathematics 972. Springer-Verlag, Berlin (1982),

[2]

Bismut, J.M.:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim. 14, 419-444(1976),

[3]

Buckdahn, R., Li, J., Peng, S.:Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Appl. 119, 3133-3154(2009),

[4]

Haussmann, U.G.:Optimal stationary control with state and control dependent noise. SIAM J. Control. 9, 184-198(1971),

[5]

Kohlmann, M., Tang, S.:Minimization of risk and linear quadratic optimal control theory. SIAM J. Control Optim. 42, 1118-1142(2003),

[6]

Peng, S.:Stochastic Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 30, 284-304(1992),

[7]

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

[8]

Wonham, W.M.:On a matrix Riccati equation of stochastic control. SIAM J. Control. 6, 681-697(1968),

[9]

Wu, H., Zhou, X.:Stochastic frequency characteristics. SIAM J. Control Optim. 40, 557-576(2001),

[10]

Yong, J., Zhou, X.Y.:Stochastic Controls:Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999),

show all references

References:
[1]

Bensoussan, A.:Lectures on stochastic control. In:Mitter, S.K., Moro, A. (eds.) Nonlinear filtering and stochastic control, Proceedings of the 3rd 1981 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), Held at Cortona, July 1-10, 1981 pp. 1-62. Lecture Notes in Mathematics 972. Springer-Verlag, Berlin (1982),

[2]

Bismut, J.M.:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim. 14, 419-444(1976),

[3]

Buckdahn, R., Li, J., Peng, S.:Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Appl. 119, 3133-3154(2009),

[4]

Haussmann, U.G.:Optimal stationary control with state and control dependent noise. SIAM J. Control. 9, 184-198(1971),

[5]

Kohlmann, M., Tang, S.:Minimization of risk and linear quadratic optimal control theory. SIAM J. Control Optim. 42, 1118-1142(2003),

[6]

Peng, S.:Stochastic Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 30, 284-304(1992),

[7]

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

[8]

Wonham, W.M.:On a matrix Riccati equation of stochastic control. SIAM J. Control. 6, 681-697(1968),

[9]

Wu, H., Zhou, X.:Stochastic frequency characteristics. SIAM J. Control Optim. 40, 557-576(2001),

[10]

Yong, J., Zhou, X.Y.:Stochastic Controls:Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999),

[1]

Naïla Hayek. Infinite-horizon multiobjective optimal control problems for bounded processes. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1121-1141. doi: 10.3934/dcdss.2018064

[2]

Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control & Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007

[3]

Ralf Banisch, Carsten Hartmann. Addendum to "A sparse Markov chain approximation of LQ-type stochastic control problems". Mathematical Control & Related Fields, 2017, 7 (4) : 623-623. doi: 10.3934/mcrf.2017023

[4]

Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation. Communications on Pure & Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023

[5]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97

[6]

Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235

[7]

Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming formulations of deterministic infinite horizon optimal control problems in discrete time. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3821-3838. doi: 10.3934/dcdsb.2017192

[8]

Wei-guo Wang, Wei-chao Wang, Ren-cang Li. Deflating irreducible singular M-matrix algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 491-518. doi: 10.3934/naco.2013.3.491

[9]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001

[10]

Charlotte Beauthier, Joseph J. Winkin, Denis Dochain. Input/state invariant LQ-optimal control: Application to competitive coexistence in a chemostat. Evolution Equations & Control Theory, 2015, 4 (2) : 143-158. doi: 10.3934/eect.2015.4.143

[11]

Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082

[12]

Mustapha Ait Rami, John Moore. Partial stabilizability and hidden convexity of indefinite LQ problem. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 221-239. doi: 10.3934/naco.2016009

[13]

Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020087

[14]

Fabio Bagagiolo. An infinite horizon optimal control problem for some switching systems. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 443-462. doi: 10.3934/dcdsb.2001.1.443

[15]

Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022

[16]

Fengmin Xu, Yanfei Wang. Recovery of seismic wavefields by an lq-norm constrained regularization method. Inverse Problems & Imaging, 2018, 12 (5) : 1157-1172. doi: 10.3934/ipi.2018048

[17]

Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021

[18]

Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683

[19]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129

[20]

Renato Iturriaga, Héctor Sánchez-Morgado. Limit of the infinite horizon discounted Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 623-635. doi: 10.3934/dcdsb.2011.15.623

 Impact Factor: 

Metrics

  • PDF downloads (6)
  • HTML views (17)
  • Cited by (0)

Other articles
by authors

[Back to Top]