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Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations
Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application
School of Mathematics, Sichuan University, Chengdu 610064, China |
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.
References:
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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. |
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K. Du and Q. Meng,
A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362.
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M. Fuhrman, Y. Hu and G. Tessitore,
Stochastic maximum principle for optimal control of SPDEs, Appl. Math. Optim., 68 (2013), 181-217.
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G. Guatteri and G. Tessitore,
On the backward stochastic Riccati equation in infinite dimensions, SIAM J. Control Optim., 44 (2005), 159-194.
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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.
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K. Itô, Introduction to Probability Theory, Cambridge University Press, Cambridge, 1984. |
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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. |
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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. |
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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. |
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