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

Qi Lü; Xu Zhang

doi:10.3934/mcrf.2018014

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2018, Volume 8, Issue 1: 337-381. Doi: 10.3934/mcrf.2018014

This issue Previous Article Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations Next Article

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 93E20; Secondary: 60G05, 60H15, 60G07.

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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.