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. 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. FuhrmanY. 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. LüJ. 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 NeervenM. 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. FuhrmanY. 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. LüJ. 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 NeervenM. 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

[1]

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

[2]

Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017

[3]

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

[4]

Fuzhi Li, Dongmei Xu, Jiali Yu. Regular measurable backward compact random attractor for $ g $-Navier-Stokes equation. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3137-3157. doi: 10.3934/cpaa.2020136

[5]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405

[6]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3351-3386. doi: 10.3934/dcdss.2020440

[7]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[8]

Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021017

[9]

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

[10]

Niklas Sapountzoglou, Aleksandra Zimmermann. Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2341-2376. doi: 10.3934/dcds.2020367

[11]

Chengxin Du, Changchun Liu. Time periodic solution to a two-species chemotaxis-Stokes system with $ p $-Laplacian diffusion. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4321-4345. doi: 10.3934/cpaa.2021162

[12]

Lingyu Diao, Jian Gao, Jiyong Lu. Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Advances in Mathematics of Communications, 2020, 14 (4) : 555-572. doi: 10.3934/amc.2020029

[13]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328

[14]

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

[15]

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

[16]

Guash Haile Taddele, Poom Kumam, Habib ur Rehman, Anteneh Getachew Gebrie. Self adaptive inertial relaxed $ CQ $ algorithms for solving split feasibility problem with multiple output sets. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021172

[17]

Magdalena Foryś-Krawiec, Jana Hantáková, Piotr Oprocha. On the structure of $ \alpha $-limit sets of backward trajectories for graph maps. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021159

[18]

Ruonan Liu, Run Xu. Hermite-Hadamard type inequalities for harmonical $ (h1,h2)- $convex interval-valued functions. Mathematical Foundations of Computing, 2021, 4 (2) : 89-103. doi: 10.3934/mfc.2021005

[19]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052

[20]

Brahim Alouini. Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 45-72. doi: 10.3934/dcdsb.2021032

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (197)
  • HTML views (291)
  • Cited by (1)

Other articles
by authors

[Back to Top]