Optimal control of ∞-dimensional stochastic systems via generalized solutions of HJB equations

N.U. Ahmed

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2001)

  • Volume: 21, Issue: 1, page 97-126
  • ISSN: 1509-9407

Abstract

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In this paper, we consider optimal feedback control for stochastc infinite dimensional systems. We present some new results on the solution of associated HJB equations in infinite dimensional Hilbert spaces. In the process, we have also developed some new mathematical tools involving distributions on Hilbert spaces which may have many other interesting applications in other fields. We conclude with an application to optimal stationary feedback control.

How to cite

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N.U. Ahmed. "Optimal control of ∞-dimensional stochastic systems via generalized solutions of HJB equations." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 21.1 (2001): 97-126. <http://eudml.org/doc/271549>.

@article{N2001,
abstract = {In this paper, we consider optimal feedback control for stochastc infinite dimensional systems. We present some new results on the solution of associated HJB equations in infinite dimensional Hilbert spaces. In the process, we have also developed some new mathematical tools involving distributions on Hilbert spaces which may have many other interesting applications in other fields. We conclude with an application to optimal stationary feedback control.},
author = {N.U. Ahmed},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {optimal control; stochastic systems; infinite dimension; HJB equation; stationary feedback control; optimal feedback control; stochastic infinite-dimensional systems; HJB equations; distributions on Hilbert spaces},
language = {eng},
number = {1},
pages = {97-126},
title = {Optimal control of ∞-dimensional stochastic systems via generalized solutions of HJB equations},
url = {http://eudml.org/doc/271549},
volume = {21},
year = {2001},
}

TY - JOUR
AU - N.U. Ahmed
TI - Optimal control of ∞-dimensional stochastic systems via generalized solutions of HJB equations
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2001
VL - 21
IS - 1
SP - 97
EP - 126
AB - In this paper, we consider optimal feedback control for stochastc infinite dimensional systems. We present some new results on the solution of associated HJB equations in infinite dimensional Hilbert spaces. In the process, we have also developed some new mathematical tools involving distributions on Hilbert spaces which may have many other interesting applications in other fields. We conclude with an application to optimal stationary feedback control.
LA - eng
KW - optimal control; stochastic systems; infinite dimension; HJB equation; stationary feedback control; optimal feedback control; stochastic infinite-dimensional systems; HJB equations; distributions on Hilbert spaces
UR - http://eudml.org/doc/271549
ER -

References

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  1. [1] F. Gozzi and E. Rouy, Regular Solutions of Second-Order Stationary Hamilton-Jacobi Equations, Journal of Differential Equations 130 (1996), 210-234. Zbl0864.34058
  2. [2] G. Da Prato, Perturbations of Ornstein-Uhlenbeck Transition Semigroups by a Subquadratic Potential, Communications in Applied Analysis 2 (3) (1998), 431-444. Zbl0895.93040
  3. [3] B. Goldys and B. Maslowski, Ergodic Control of Semilinear Stochastic Equations and Hamilton-Jacobi Equations, preprint, (1998). Zbl0939.93043
  4. [4] G. Da Prato and J. Zabczyk, Regular Densities of Invariant Measures in Hilbert Spaces, Journal of Functional Analysis 130 (1995), 427-449. Zbl0832.60069
  5. [5] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Note, 229, Cambridge University Press 1996. Zbl0849.60052
  6. [6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, Encyclopedia of Mathematics and its Applications series, 44, Cambridge University Press 1992. Zbl0761.60052
  7. [7] S. Cerrai, A Hille-Yosida Theorem for Weakly Continuous Semigroups, Semigroup Forum 49 (1994), 349-367. Zbl0817.47048
  8. [8] N.U. Ahmed and J. Zabczyk, Nonlinear Filtering for Semilinear Stochastic Differential Equations on Hilbert Spaces, Preprint 522, Inst. Math. Polish Academy of Sciences, Warsaw, Poland. 
  9. [9] N.U. Ahmed, Relaxed Controls for Stochastic Boundary Value Problems in Infinite Dimension, Lect. Notes in Contr. and Inf. Sciences 149 (1990), 1-10. 
  10. [10] N.U. Ahmed, Optimal Relaxed Controls for Nonlinear Infinite Dimensional Stochastic Differential Inclusions, International Symposium on Optimal Control of infinite Dimensional Systems, (ed. N.H. Pavel), Lect. Notes in Pure and Applied Math, Marcel Dekker, New York and Basel 160 (1994), 1-19. 
  11. [11] N.U. Ahmed, Optimal Relaxed Controls for Infinite Dimensional Stochastic Systems of Zakai Type, SIAM J. Control and Optimization 34 (5) (1996). Zbl0861.93030
  12. [12] N.U. Ahmed, M. Fuhrman and J. Zabczyk, On Filtering Equations in Infinite Dimensions, Journal of Functional Analysis 143 (1), 1997. Zbl0880.60043
  13. [13] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol 1, Theory, Kluwer Academic Publishers, Dordrecht, Boston, London 1997. Zbl0887.47001

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