# 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.

show all references

##### 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.
 [1] Chiun-Chuan Chen, Li-Chang Hung, Chen-Chih Lai. An N-barrier maximum principle for autonomous systems of $n$ species and its application to problems arising from population dynamics. Communications on Pure & Applied Analysis, 2019, 18 (1) : 33-50. doi: 10.3934/cpaa.2019003 [2] Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $p$-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070 [3] Qi Lü, Xu Zhang. Transposition method for backward stochastic evolution equations revisited, and its application. Mathematical Control & Related Fields, 2015, 5 (3) : 529-555. doi: 10.3934/mcrf.2015.5.529 [4] Koya Nishimura. Global existence for the Boltzmann equation in $L^r_v L^\infty_t L^\infty_x$ spaces. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1769-1782. doi: 10.3934/cpaa.2019083 [5] Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $L_K$. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086 [6] Theodore Tachim Medjo. Pullback $\mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2141-2169. doi: 10.3934/dcds.2018088 [7] Minjia Shi, Yaqi Lu. Cyclic DNA codes over $\mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$. Advances in Mathematics of Communications, 2019, 13 (1) : 157-164. doi: 10.3934/amc.2019009 [8] Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159 [9] Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2695-2708. doi: 10.3934/dcdsb.2018087 [10] Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-15. doi: 10.3934/dcdsb.2018325 [11] Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2085-2095. doi: 10.3934/dcdss.2019134 [12] Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124 [13] Imed Bachar, Habib Mâagli. Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 171-188. doi: 10.3934/dcdss.2019012 [14] Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058 [15] Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control & Related Fields, 2019, 9 (1) : 1-38. doi: 10.3934/mcrf.2019001 [16] Gyu Eun Lee. Local wellposedness for the critical nonlinear Schrödinger equation on $\mathbb{T}^3$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2763-2783. doi: 10.3934/dcds.2019116 [17] Jaime Angulo Pava, César A. Hernández Melo. On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2093-2116. doi: 10.3934/cpaa.2019094 [18] Linlin Fu, Jiahao Xu. A new proof of continuity of Lyapunov exponents for a class of $C^2$ quasiperiodic Schrödinger cocycles without LDT. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2915-2931. doi: 10.3934/dcds.2019121 [19] Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007 [20] Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129

2017 Impact Factor: 0.631