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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.

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##### 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.
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