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

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

Qi Lü and  Xu Zhang

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.

Keywords: Operator-valued backward stochastic Lyapunov equation, $ V $-transposition solution, relaxed transposition solution, Pontryagin-type maximum principle.
Mathematics Subject Classification: Primary: 93E20; Secondary: 60G05, 60H15, 60G07.
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. Google Scholar

[2]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  Google Scholar

[3]

K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362.   Google Scholar

[4]

M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, Appl. Math. Optim., 68 (2013), 181-217.   Google Scholar

[5]

G. Guatteri and G. Tessitore, On the backward stochastic Riccati equation in infinite dimensions, SIAM J. Control Optim., 44 (2005), 159-194.   Google Scholar

[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.   Google Scholar

[7]

K. Itô, Introduction to Probability Theory, Cambridge University Press, Cambridge, 1984.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

Q. Lü, J. Yong and X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals, J. Eur. Math. Soc, 14 (2012), 1795-1823.   Google Scholar

[11]

Q. Lü, H. Zhang and X. Zhang, Second order optimality conditions for optimal control problems of stochastic evolution equations, Preprint. Google Scholar

[12]

Q. Lü and X. Zhang, Well-posedness of backward stochastic differential equations with general filtration, J. Differential Equations, 254 (2013), 3200-3227.   Google Scholar

[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)  Google Scholar

[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.   Google Scholar

[15]

Q. Lü and X. Zhang, Optimal feedback for stochastic linear quadratic control and backward stochastic Riccati equations in infinite simensions, Preprint. Google Scholar

[16]

V. A. Rohlin, On the fundamental ideas of measure theory, in Amer. Math. Soc. Translation, 1952 (1952), 55 pp.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. M. Bruckner, J. B. Bruckner and B. S. Thomson, Real Analysis, Prentice Hall (Pearson), Upper Saddle River, 1997. Google Scholar

[2]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  Google Scholar

[3]

K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362.   Google Scholar

[4]

M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, Appl. Math. Optim., 68 (2013), 181-217.   Google Scholar

[5]

G. Guatteri and G. Tessitore, On the backward stochastic Riccati equation in infinite dimensions, SIAM J. Control Optim., 44 (2005), 159-194.   Google Scholar

[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.   Google Scholar

[7]

K. Itô, Introduction to Probability Theory, Cambridge University Press, Cambridge, 1984.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

Q. Lü, J. Yong and X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals, J. Eur. Math. Soc, 14 (2012), 1795-1823.   Google Scholar

[11]

Q. Lü, H. Zhang and X. Zhang, Second order optimality conditions for optimal control problems of stochastic evolution equations, Preprint. Google Scholar

[12]

Q. Lü and X. Zhang, Well-posedness of backward stochastic differential equations with general filtration, J. Differential Equations, 254 (2013), 3200-3227.   Google Scholar

[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)  Google Scholar

[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.   Google Scholar

[15]

Q. Lü and X. Zhang, Optimal feedback for stochastic linear quadratic control and backward stochastic Riccati equations in infinite simensions, Preprint. Google Scholar

[16]

V. A. Rohlin, On the fundamental ideas of measure theory, in Amer. Math. Soc. Translation, 1952 (1952), 55 pp.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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