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December  2012, 1(2): 235-250. doi: 10.3934/eect.2012.1.235

Existence of optimal output feedback control law for a class of uncertain infinite dimensional stochastic systems: A direct approach

N. U. Ahmed 1,

1. 

EECS, University of Ottawa, Ottawa, Ontario, Canada

Received  May 2012 Revised  June 2012 Published  October 2012

In this paper we consider a class of partially observed semilinear stochastic evolution equations on infinite dimensional Hilbert spaces subject to measurement uncertainty. We prove the existence of optimal feedback control law from a class of operator valued functions furnished with the Tychonoff product topology. This is an extension of our previous results for uncertain systems governed by deterministic differential equations on Banach spaces. Also we present a result on existence of optimal feedback control law for a class of uncertain stochastic systems modeled by differential inclusions.
Keywords: feedback operators, Partially observed uncertain systems, operator valued functions, existence of saddle points., differential inclusions.
Mathematics Subject Classification: Primary: 49J24, 49J27, 93E20, 93B52, 35R60, 35R70, 34G25, 34H05, 47A6.
Citation: N. U. Ahmed. Existence of optimal output feedback control law for a class of uncertain infinite dimensional stochastic systems: A direct approach. Evolution Equations and Control Theory, 2012, 1 (2) : 235-250. doi: 10.3934/eect.2012.1.235
References:
[1]

N. U. Ahmed, "Semigroup Theory with Applications to Systems and Control," Pitman Research Notes in Mathematics Series, 246, Longman Scientific and Technical, U. K; Co-published with John-Wiely & Sons, Inc., New York, 1991.

[2]

N. U. Ahmed and X. Xiang, Differential inclusions on banach spaces and their optimal control, Nonlinear Funct. Anal.& Appl., 8 (2003), 461-488.

[3]

N. U. Ahmed, Optimal relaxed controls for systems governed by impulsive differential inclusions, Nonlinear Funct. Anal.& Appl., 10 (2005), 427-460.

[4]

N. U. Ahmed and C. D. Charalambous, Minimax games for stochastic systems subject to relative entropy uncertainty: Applications to SDE's on Hilbert spaces, J. Mathematics of Control, Signals and Systems, 19 (2007), 65-91. doi: 10.1007/s00498-006-0009-x.

[5]

N. U. Ahmed, Optimal output feedback boundary control for systems governed by semilinear parabolic inclusions: uncertain systems, Advances in Nonlinear Variational Inequalities, 11 (2008), 61-79.

[6]

N. U. Ahmed and Suruz Miah, Optimal feedback control law for a class of partially observed dynamic systems: a min-max problem, Dynamic Systems and Applications, 20 (2011), 149-168.

[7]

N. U. Ahmed and K. L. Teo, "Optimal Control of Distributed Parameter Systems," North Holland, New York, Oxford, 1981.

[8]

J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Berkhauser, Boston, 1990.

[9]

L. Cesari, "Optimization Theory and Applications," Springer-Verlag, 1983. doi: 10.1007/978-1-4613-8165-5.

[10]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.

[11]

H. O. Fattorini, "Infinite Dimensional optimization and Control Theory," Encyclopedia of mathematics and its applications, 62, Cambridge University Press, 1999.

[12]

S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis," Kluwer Academic Publishers, Dordrecht, Boston, London, vol 1, 1997.

[13]

F. Mayoral, Compact sets of compact operators in absence of $l_1$, Proc. Am. Math. Soc., 129 (2001),79-82. doi: 10.1090/S0002-9939-00-06007-X.

[14]

E. Serrano, C. Pineiro and J. M. Delgado, Equicompact sets of operators defined on Banach spaces, Proc. Am. Math. Soc., 134 (2005), 689-695. doi: 10.1090/S0002-9939-05-08338-3.

[15]

E. Zeidler, "Nonlinear Functional Analysis and its Applications," Fixed Point Theorems, Springer-Verlag, New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest, Vol. 1, 1986.

show all references

References:
[1]

N. U. Ahmed, "Semigroup Theory with Applications to Systems and Control," Pitman Research Notes in Mathematics Series, 246, Longman Scientific and Technical, U. K; Co-published with John-Wiely & Sons, Inc., New York, 1991.

[2]

N. U. Ahmed and X. Xiang, Differential inclusions on banach spaces and their optimal control, Nonlinear Funct. Anal.& Appl., 8 (2003), 461-488.

[3]

N. U. Ahmed, Optimal relaxed controls for systems governed by impulsive differential inclusions, Nonlinear Funct. Anal.& Appl., 10 (2005), 427-460.

[4]

N. U. Ahmed and C. D. Charalambous, Minimax games for stochastic systems subject to relative entropy uncertainty: Applications to SDE's on Hilbert spaces, J. Mathematics of Control, Signals and Systems, 19 (2007), 65-91. doi: 10.1007/s00498-006-0009-x.

[5]

N. U. Ahmed, Optimal output feedback boundary control for systems governed by semilinear parabolic inclusions: uncertain systems, Advances in Nonlinear Variational Inequalities, 11 (2008), 61-79.

[6]

N. U. Ahmed and Suruz Miah, Optimal feedback control law for a class of partially observed dynamic systems: a min-max problem, Dynamic Systems and Applications, 20 (2011), 149-168.

[7]

N. U. Ahmed and K. L. Teo, "Optimal Control of Distributed Parameter Systems," North Holland, New York, Oxford, 1981.

[8]

J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Berkhauser, Boston, 1990.

[9]

L. Cesari, "Optimization Theory and Applications," Springer-Verlag, 1983. doi: 10.1007/978-1-4613-8165-5.

[10]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.

[11]

H. O. Fattorini, "Infinite Dimensional optimization and Control Theory," Encyclopedia of mathematics and its applications, 62, Cambridge University Press, 1999.

[12]

S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis," Kluwer Academic Publishers, Dordrecht, Boston, London, vol 1, 1997.

[13]

F. Mayoral, Compact sets of compact operators in absence of $l_1$, Proc. Am. Math. Soc., 129 (2001),79-82. doi: 10.1090/S0002-9939-00-06007-X.

[14]

E. Serrano, C. Pineiro and J. M. Delgado, Equicompact sets of operators defined on Banach spaces, Proc. Am. Math. Soc., 134 (2005), 689-695. doi: 10.1090/S0002-9939-05-08338-3.

[15]

E. Zeidler, "Nonlinear Functional Analysis and its Applications," Fixed Point Theorems, Springer-Verlag, New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest, Vol. 1, 1986.

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