Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality

N.U. Ahmed

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2014)

  • Volume: 34, Issue: 1, page 105-129
  • ISSN: 1509-9407

Abstract

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In this paper we consider the question of optimal control for a class of stochastic evolution equations on infinite dimensional Hilbert spaces with controls appearing in both the drift and the diffusion operators. We consider relaxed controls (measure valued random processes) and briefly present some results on the question of existence of mild solutions including their regularity followed by a result on existence of partially observed optimal relaxed controls. Then we develop the necessary conditions of optimality for partially observed relaxed controls. This is the main topic of this paper. Further we present an algorithm for computation of optimal policies followed by a brief discussion on regular versus relaxed controls. The paper is concluded by an example of a non-convex problem which is readily solvable by our approach.

How to cite

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N.U. Ahmed. "Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 34.1 (2014): 105-129. <http://eudml.org/doc/270523>.

@article{N2014,
abstract = {In this paper we consider the question of optimal control for a class of stochastic evolution equations on infinite dimensional Hilbert spaces with controls appearing in both the drift and the diffusion operators. We consider relaxed controls (measure valued random processes) and briefly present some results on the question of existence of mild solutions including their regularity followed by a result on existence of partially observed optimal relaxed controls. Then we develop the necessary conditions of optimality for partially observed relaxed controls. This is the main topic of this paper. Further we present an algorithm for computation of optimal policies followed by a brief discussion on regular versus relaxed controls. The paper is concluded by an example of a non-convex problem which is readily solvable by our approach.},
author = {N.U. Ahmed},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {differential equations; Hilbert spaces; relaxed controls; optimal control; necessary conditions of optimality; stochastic evolution equations; necessary optimality conditions},
language = {eng},
number = {1},
pages = {105-129},
title = {Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality},
url = {http://eudml.org/doc/270523},
volume = {34},
year = {2014},
}

TY - JOUR
AU - N.U. Ahmed
TI - Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2014
VL - 34
IS - 1
SP - 105
EP - 129
AB - In this paper we consider the question of optimal control for a class of stochastic evolution equations on infinite dimensional Hilbert spaces with controls appearing in both the drift and the diffusion operators. We consider relaxed controls (measure valued random processes) and briefly present some results on the question of existence of mild solutions including their regularity followed by a result on existence of partially observed optimal relaxed controls. Then we develop the necessary conditions of optimality for partially observed relaxed controls. This is the main topic of this paper. Further we present an algorithm for computation of optimal policies followed by a brief discussion on regular versus relaxed controls. The paper is concluded by an example of a non-convex problem which is readily solvable by our approach.
LA - eng
KW - differential equations; Hilbert spaces; relaxed controls; optimal control; necessary conditions of optimality; stochastic evolution equations; necessary optimality conditions
UR - http://eudml.org/doc/270523
ER -

References

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  1. [1] N.U. Ahmed, Deterministic and stochastic neutral systems on Banach spaces and their optimal fedback controls, J. Nonlin. Syst. Appl. (2009) 151-160. 
  2. [2] N.U. Ahmed, Measure valued solutions for systems governed by neutral differential equations on Banach spaces and their optimal control, Discuss. Math. DICO 33 (2013) 89-109. doi: 10.7151/dmdico.1142. Zbl1297.49009
  3. [3] N.U. Ahmed, Relaxed solutions for stochastic evolution equations on Hilbert space with polynomial growth, Publ. Math. Debrechen 54 (1-2) (1999) 75-101. 
  4. [4] N.U. Ahmed, Stochastic neutral evolution equations on Hilbert spaces with partially observed relaxed control and their necessary conditions of optimality, Nonlin. Anal. TMA 101 (2014) 66-79. Zbl1285.49017
  5. [5] N.U. Ahmed, Some Recent Developments in Systems and Control Theory on Infinite Dimensional Banach Spaces, Part 1 & 2, Proceedings of the 5th International Conference on Optimization and Control with Applications, (Edited by: K.L. Teo, H. Xu and Y. Zhang), Beijing, China, 2012; Publisher: Springer-Verlag (in Print). 
  6. [6] N.U. Ahmed, Optimization and Identification of Systems Governed by Evolution Equations on Banach Spaces, Pitman research Notes in Mathematics Series, Vol. 184, Longman Scientific and Technical, U.K; Co-published with John-Wiely & Sons, Inc. New York, 1988. 
  7. [7] N.U. Ahmed, Semigroup Theory with Applications to Systema and Control, Pitman Research Notes in Mathematics Series, Vol. 246, Longman Scientific and Technical, U.K; Co-published with John-Wiely & Sons, Inc. New York, 1991. 
  8. [8] N.U. Ahmed, Dynamic Systems and Control with Applications, World Scientific (New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2006). ISBN: 981-270-053-6 
  9. [9] N.U. Ahmed and C.D. Charalambous, Stochastic minimum principle for partially observed systems subject to continuous and jump diffusion processes and drviven by relaxed controls, SIAM J. Control and Optim. 51 (4) (2013) 3235-3257. doi: 10.1137/120885656 Zbl1290.49034
  10. [10] S. Bahlali, Necessary and sufficient optimality coditions for relaxed and strict control problems, SIAM J. Control and Optim. 47 (2008) 2078-2095. doi: 10.1137/070681053 Zbl1167.49024
  11. [11] N. Dunford and J.T. Schwartz, Linear Operators, Part 1 (Interscience Publishers, Inc., New York, 1958). Zbl0084.10402
  12. [12] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions (Cambridge University Press, 1992). doi: 10.1017/CBO9780511666223 Zbl0761.60052
  13. [13] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. 1, Theory (Kluwer Academic publishers, Dordrecht/Boston/London, 1997). Zbl0887.47001
  14. [14] Y. Hu and S. Peng, Adaptive solution of a Backward semilinear stochastic evolution equation, Stoch. Anal. Appl. 9 (4) (1991) 445-459. doi: 10.1080/07362999108809250 Zbl0736.60051
  15. [15] W. Wei, Maximum principle for optimal control of neutral stochastic functional differential systems, Science China Math. (to appear) Zbl1327.93421

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