• Previous Article
    Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay
  • DCDS-B Home
  • This Issue
  • Next Article
    A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions
August  2013, 18(6): 1651-1661. doi: 10.3934/dcdsb.2013.18.1651

Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays

1. 

Division of Statistics and Probability, Department of Mathematical Sciences, The University of Liverpool, Peach Street, Liverpool, L69 7ZL, United Kingdom

Received  September 2011 Revised  January 2012 Published  March 2013

A class of stochastic optimal control problems of infinite dimensional Ornstein-Uhlenbeck processes of neutral type are considered. One special feature of the system under investigation is that time delays are present in the control. An equivalent formulation between an adjoint stochastic controlled delay differential equation and its lifted control system (without delays) is developed. As a consequence, the finite time quadratic regulator problem governed by this formulation is solved based on a direct solution of some associated Riccati equation.
Citation: Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651
References:
[1]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, "Representation and Control of Infinite Dimensional Systems," Second Edition, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007.

[2]

R. F. Curtain and H. J. Zwart., "An Introduction to Infinite Dimensional Linear Systems Theory," Texts in Applied Math., 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[3]

G. Da Prato and J. Zabczyk, "Second Order Partial Differential Equations in Hilbert Spaces," London Math. Soc. LNS, 293, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511543210.

[4]

J. P. Dauer and N. I. Mahmudov, Controllability of stochastic semilinear functional differential equations in Hilbert spaces, J. Math. Anal. Appl., 290 (2004), 373-394. doi: 10.1016/j.jmaa.2003.09.069.

[5]

F. Flandoli, Solution and control of a bilinear stochastic delay equation, SIAM J. Control Optim., 28 (1990), 936-949. doi: 10.1137/0328052.

[6]

M. Fuhrman and G. Tessitore, Nonolinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), 1397-1465. doi: 10.1214/aop/1029867132.

[7]

F. Gozzi and C. Marinelli, Stochastic optimal control of delay equations arising in advertising models, in "Stochastic Partial Differential Equations - VII," Lecture Notes Pure Appl. Math., 245, Chapman & Hall/CRC, Boca Raton, FL, (2006), 133-148. doi: 10.1201/9781420028720.ch13.

[8]

F. Gozzi, C. Marinelli and S. Savin, On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects, J. Optim. Theory Appl., 142 (2009), 291-321. doi: 10.1007/s10957-009-9524-5.

[9]

J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Math. Sci., 99, Springer-Verlag, New York, 1993.

[10]

A. Ichikawa, Dynamic programming approach to stochastic evolution equations, SIAM J. Control Optim., 17 (1979), 152-174. doi: 10.1137/0317012.

[11]

X. J. Li and J. M. Yong, "Optimal Control Theory for Infinite-Dimensional Systems," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.

[12]

K. Liu, The fundamental solution and its role in the optimal control of infinite dimensional neutral systems, Applied Math. Optim., 60 (2009), 1-38. doi: 10.1007/s00245-009-9065-1.

[13]

K. Liu, Finite pole assignment of linear neutral systems in infinite dimensions, in "Proceedings of the Second International Conference on Modelling and Simulation (ICMS2009)" (eds. Y. Jiang and X. G. Chen), Manchester, (2009), 1-11.

[14]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622. doi: 10.1137/S0363012901391688.

[15]

S. Nakagiri, Optimal control of linear retarded systems in Banach spaces, J. Math. Anal. Appl., 120 (1986), 169-210. doi: 10.1016/0022-247X(86)90210-6.

[16]

D. Salamon, "Control and Observation of Neutral Systems," Research Notes in Math., 91, Pitman (Advanced Publishing Program), Boston, MA, 1984.

[17]

R. Vinter and R. Kwong, The infinite time quadratic control problem for linear system with state and control delays: An evolution equation approach, SIAM J. Control Optim., 19 (1981), 139-153. doi: 10.1137/0319011.

[18]

K. Yosida, "Functional Analysis," Sixth edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123, Springer-Verlag, Berlin-New York, 1980.

[19]

J. Zabczyk, "Mathematical Control Theory: An Introduction," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992.

show all references

References:
[1]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, "Representation and Control of Infinite Dimensional Systems," Second Edition, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007.

[2]

R. F. Curtain and H. J. Zwart., "An Introduction to Infinite Dimensional Linear Systems Theory," Texts in Applied Math., 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[3]

G. Da Prato and J. Zabczyk, "Second Order Partial Differential Equations in Hilbert Spaces," London Math. Soc. LNS, 293, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511543210.

[4]

J. P. Dauer and N. I. Mahmudov, Controllability of stochastic semilinear functional differential equations in Hilbert spaces, J. Math. Anal. Appl., 290 (2004), 373-394. doi: 10.1016/j.jmaa.2003.09.069.

[5]

F. Flandoli, Solution and control of a bilinear stochastic delay equation, SIAM J. Control Optim., 28 (1990), 936-949. doi: 10.1137/0328052.

[6]

M. Fuhrman and G. Tessitore, Nonolinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), 1397-1465. doi: 10.1214/aop/1029867132.

[7]

F. Gozzi and C. Marinelli, Stochastic optimal control of delay equations arising in advertising models, in "Stochastic Partial Differential Equations - VII," Lecture Notes Pure Appl. Math., 245, Chapman & Hall/CRC, Boca Raton, FL, (2006), 133-148. doi: 10.1201/9781420028720.ch13.

[8]

F. Gozzi, C. Marinelli and S. Savin, On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects, J. Optim. Theory Appl., 142 (2009), 291-321. doi: 10.1007/s10957-009-9524-5.

[9]

J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Math. Sci., 99, Springer-Verlag, New York, 1993.

[10]

A. Ichikawa, Dynamic programming approach to stochastic evolution equations, SIAM J. Control Optim., 17 (1979), 152-174. doi: 10.1137/0317012.

[11]

X. J. Li and J. M. Yong, "Optimal Control Theory for Infinite-Dimensional Systems," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.

[12]

K. Liu, The fundamental solution and its role in the optimal control of infinite dimensional neutral systems, Applied Math. Optim., 60 (2009), 1-38. doi: 10.1007/s00245-009-9065-1.

[13]

K. Liu, Finite pole assignment of linear neutral systems in infinite dimensions, in "Proceedings of the Second International Conference on Modelling and Simulation (ICMS2009)" (eds. Y. Jiang and X. G. Chen), Manchester, (2009), 1-11.

[14]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622. doi: 10.1137/S0363012901391688.

[15]

S. Nakagiri, Optimal control of linear retarded systems in Banach spaces, J. Math. Anal. Appl., 120 (1986), 169-210. doi: 10.1016/0022-247X(86)90210-6.

[16]

D. Salamon, "Control and Observation of Neutral Systems," Research Notes in Math., 91, Pitman (Advanced Publishing Program), Boston, MA, 1984.

[17]

R. Vinter and R. Kwong, The infinite time quadratic control problem for linear system with state and control delays: An evolution equation approach, SIAM J. Control Optim., 19 (1981), 139-153. doi: 10.1137/0319011.

[18]

K. Yosida, "Functional Analysis," Sixth edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123, Springer-Verlag, Berlin-New York, 1980.

[19]

J. Zabczyk, "Mathematical Control Theory: An Introduction," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992.

[1]

Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial and Management Optimization, 2022, 18 (1) : 75-93. doi: 10.3934/jimo.2020143

[2]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic and Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013

[3]

Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142

[4]

Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871

[5]

Virginia Giorno, Serena Spina. On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 285-302. doi: 10.3934/mbe.2014.11.285

[6]

Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451

[7]

Antonio Avantaggiati, Paola Loreti. Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II). Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 525-545. doi: 10.3934/dcdss.2009.2.525

[8]

Samuel Herrmann, Nicolas Massin. Exit problem for Ornstein-Uhlenbeck processes: A random walk approach. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3199-3215. doi: 10.3934/dcdsb.2020058

[9]

Mondher Damak, Brice Franke, Nejib Yaakoubi. Accelerating planar Ornstein-Uhlenbeck diffusion with suitable drift. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4093-4112. doi: 10.3934/dcds.2020173

[10]

Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure and Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049

[11]

Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649

[12]

Alessia E. Kogoj. A Zaremba-type criterion for hypoelliptic degenerate Ornstein–Uhlenbeck operators. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3491-3494. doi: 10.3934/dcdss.2020112

[13]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[14]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control and Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[15]

Delio Mugnolo, Abdelaziz Rhandi. Ornstein–Uhlenbeck semigroups on star graphs. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022030

[16]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709

[17]

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

[18]

Qi Lü, Tianxiao Wang, Xu Zhang. Characterization of optimal feedback for stochastic linear quadratic control problems. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 11-. doi: 10.1186/s41546-017-0022-7

[19]

Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial and Management Optimization, 2007, 3 (3) : 585-596. doi: 10.3934/jimo.2007.3.585

[20]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (87)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]