# American Institute of Mathematical Sciences

March 2018, 8(1): 337-381. doi: 10.3934/mcrf.2018014

## Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application

 School of Mathematics, Sichuan University, Chengdu 610064, China

Received  May 2017 Revised  October 2017 Published  January 2018

We establish the well-posedness of operator-valued backward stochastic Lyapunov equations in infinite dimensions, in the sense of $V$-transposition solution and of relaxed transposition solution. As an application, we obtain a Pontryagin-type maximum principle for the optimal control of controlled stochastic evolution equations.

Citation: Qi Lü, Xu Zhang. Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application. Mathematical Control & Related Fields, 2018, 8 (1) : 337-381. doi: 10.3934/mcrf.2018014
##### References:
 [1] A. M. Bruckner, J. B. Bruckner and B. S. Thomson, Real Analysis, Prentice Hall (Pearson), Upper Saddle River, 1997. [2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [3] K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362. [4] M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, Appl. Math. Optim., 68 (2013), 181-217. [5] G. Guatteri and G. Tessitore, On the backward stochastic Riccati equation in infinite dimensions, SIAM J. Control Optim., 44 (2005), 159-194. [6] G. Guatteri and G. Tessitore, Well posedness of operator valued backward stochastic Riccati equations in infinite dimensional spaces, SIAM J. Control Optim., 52 (2014), 3776-3806. [7] K. Itô, Introduction to Probability Theory, Cambridge University Press, Cambridge, 1984. [8] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I. Elementary Theory, Graduate Studies in Mathematics, 15. American Mathematical Society, Providence, RI, 1997. [9] Q. Lü, Second order necessary conditions for optimal control problems of stochastic evolution equations, Proceedings of the 35th Chinese Control Conference, Chengdu, China, (2016), 2620-2625. [10] Q. Lü, J. Yong and X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals, J. Eur. Math. Soc, 14 (2012), 1795-1823. [11] Q. Lü, H. Zhang and X. Zhang, Second order optimality conditions for optimal control problems of stochastic evolution equations, Preprint. [12] Q. Lü and X. Zhang, Well-posedness of backward stochastic differential equations with general filtration, J. Differential Equations, 254 (2013), 3200-3227. [13] Q. Lü and X. Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, Springer Briefs in Mathematics, Springer, New York, 2014. (See also http://arXiv.org/abs/1204.3275) [14] Q. Lü and X. Zhang, Transposition method for backward stochastic evolution equations revisited, and its application, Math. Control Relat. Fields., 5 (2015), 529-555. [15] Q. Lü and X. Zhang, Optimal feedback for stochastic linear quadratic control and backward stochastic Riccati equations in infinite simensions, Preprint. [16] V. A. Rohlin, On the fundamental ideas of measure theory, in Amer. Math. Soc. Translation, 1952 (1952), 55 pp. [17] J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab., 35 (2007), 1438-1478. [18] J. M. A. M. van Neerven, γ-radonifying operators—a survey, The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, 1-61. Proc. Centre Math. Appl. Austral. Nat. Univ., 44, Austral. Nat. Univ., Canberra, 2010.

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##### References:
 [1] A. M. Bruckner, J. B. Bruckner and B. S. Thomson, Real Analysis, Prentice Hall (Pearson), Upper Saddle River, 1997. [2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [3] K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362. [4] M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, Appl. Math. Optim., 68 (2013), 181-217. [5] G. Guatteri and G. Tessitore, On the backward stochastic Riccati equation in infinite dimensions, SIAM J. Control Optim., 44 (2005), 159-194. [6] G. Guatteri and G. Tessitore, Well posedness of operator valued backward stochastic Riccati equations in infinite dimensional spaces, SIAM J. Control Optim., 52 (2014), 3776-3806. [7] K. Itô, Introduction to Probability Theory, Cambridge University Press, Cambridge, 1984. [8] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I. Elementary Theory, Graduate Studies in Mathematics, 15. American Mathematical Society, Providence, RI, 1997. [9] Q. Lü, Second order necessary conditions for optimal control problems of stochastic evolution equations, Proceedings of the 35th Chinese Control Conference, Chengdu, China, (2016), 2620-2625. [10] Q. Lü, J. Yong and X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals, J. Eur. Math. Soc, 14 (2012), 1795-1823. [11] Q. Lü, H. Zhang and X. Zhang, Second order optimality conditions for optimal control problems of stochastic evolution equations, Preprint. [12] Q. Lü and X. Zhang, Well-posedness of backward stochastic differential equations with general filtration, J. Differential Equations, 254 (2013), 3200-3227. [13] Q. Lü and X. Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, Springer Briefs in Mathematics, Springer, New York, 2014. (See also http://arXiv.org/abs/1204.3275) [14] Q. Lü and X. Zhang, Transposition method for backward stochastic evolution equations revisited, and its application, Math. Control Relat. Fields., 5 (2015), 529-555. [15] Q. Lü and X. Zhang, Optimal feedback for stochastic linear quadratic control and backward stochastic Riccati equations in infinite simensions, Preprint. [16] V. A. Rohlin, On the fundamental ideas of measure theory, in Amer. Math. Soc. Translation, 1952 (1952), 55 pp. [17] J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab., 35 (2007), 1438-1478. [18] J. M. A. M. van Neerven, γ-radonifying operators—a survey, The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, 1-61. Proc. Centre Math. Appl. Austral. Nat. Univ., 44, Austral. Nat. Univ., Canberra, 2010.
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