Geodesic
Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 34 (2014) no. 1, pp. 105-129.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: differential equations, Hilbert spaces, relaxed controls, optimal control, necessary conditions of optimality 
@article{DMDICO_2014_34_1_a5,
     author = {Ahmed, N.},
     title = {Stochastic evolution equations on {Hilbert} spaces with partially observed relaxed controls and their necessary conditions of optimality},
     journal = {Discussiones Mathematicae. Differential Inclusions, Control and Optimization},
     pages = {105--129},
     publisher = {mathdoc},
     volume = {34},
     number = {1},
     year = {2014},
     zbl = {1332.49030},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a5/}
}
TY  - JOUR
AU  - Ahmed, N.
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
SP  - 105
EP  - 129
VL  - 34
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a5/
LA  - en
ID  - DMDICO_2014_34_1_a5
ER  - 
%0 Journal Article
%A Ahmed, N.
%T Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality
%J Discussiones Mathematicae. Differential Inclusions, Control and Optimization
%D 2014
%P 105-129
%V 34
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a5/
%G en
%F DMDICO_2014_34_1_a5
Ahmed, N. Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 34 (2014) no. 1, pp. 105-129. http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a5/

[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] 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.

[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] 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.

[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] 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] 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] 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] 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

[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

[11] N. Dunford and J.T. Schwartz, Linear Operators, Part 1 (Interscience Publishers, Inc., New York, 1958).

[12] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions (Cambridge University Press, 1992). doi: 10.1017/CBO9780511666223

[13] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. 1, Theory (Kluwer Academic publishers, Dordrecht/Boston/London, 1997).

[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

[15] W. Wei, Maximum principle for optimal control of neutral stochastic functional differential systems, Science China Math. (to appear)