# American Institute of Mathematical Sciences

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

 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.
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 & 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, (1991).   Google Scholar [2] N. U. Ahmed and X. Xiang, Differential inclusions on banach spaces and their optimal control,, Nonlinear Funct. Anal.& Appl., 8 (2003), 461.   Google Scholar [3] N. U. Ahmed, Optimal relaxed controls for systems governed by impulsive differential inclusions,, Nonlinear Funct. Anal.& Appl., 10 (2005), 427.   Google Scholar [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, 19 (2007), 65.  doi: 10.1007/s00498-006-0009-x.  Google Scholar [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.   Google Scholar [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.   Google Scholar [7] N. U. Ahmed and K. L. Teo, "Optimal Control of Distributed Parameter Systems,", North Holland, (1981).   Google Scholar [8] J. P. Aubin and H. Frankowska, "Set-Valued Analysis,", Berkhauser, (1990).   Google Scholar [9] L. Cesari, "Optimization Theory and Applications,", Springer-Verlag, (1983).  doi: 10.1007/978-1-4613-8165-5.  Google Scholar [10] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar [11] H. O. Fattorini, "Infinite Dimensional optimization and Control Theory,", Encyclopedia of mathematics and its applications, (1999).   Google Scholar [12] S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis,", Kluwer Academic Publishers, (1997).   Google Scholar [13] F. Mayoral, Compact sets of compact operators in absence of $l_1$,, Proc. Am. Math. Soc., 129 (2001), 79.  doi: 10.1090/S0002-9939-00-06007-X.  Google Scholar [14] E. Serrano, C. Pineiro and J. M. Delgado, Equicompact sets of operators defined on Banach spaces,, Proc. Am. Math. Soc., 134 (2005), 689.  doi: 10.1090/S0002-9939-05-08338-3.  Google Scholar [15] E. Zeidler, "Nonlinear Functional Analysis and its Applications,", Fixed Point Theorems, (1986).   Google Scholar

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##### References:
 [1] N. U. Ahmed, "Semigroup Theory with Applications to Systems and Control,", Pitman Research Notes in Mathematics Series, (1991).   Google Scholar [2] N. U. Ahmed and X. Xiang, Differential inclusions on banach spaces and their optimal control,, Nonlinear Funct. Anal.& Appl., 8 (2003), 461.   Google Scholar [3] N. U. Ahmed, Optimal relaxed controls for systems governed by impulsive differential inclusions,, Nonlinear Funct. Anal.& Appl., 10 (2005), 427.   Google Scholar [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, 19 (2007), 65.  doi: 10.1007/s00498-006-0009-x.  Google Scholar [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.   Google Scholar [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.   Google Scholar [7] N. U. Ahmed and K. L. Teo, "Optimal Control of Distributed Parameter Systems,", North Holland, (1981).   Google Scholar [8] J. P. Aubin and H. Frankowska, "Set-Valued Analysis,", Berkhauser, (1990).   Google Scholar [9] L. Cesari, "Optimization Theory and Applications,", Springer-Verlag, (1983).  doi: 10.1007/978-1-4613-8165-5.  Google Scholar [10] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar [11] H. O. Fattorini, "Infinite Dimensional optimization and Control Theory,", Encyclopedia of mathematics and its applications, (1999).   Google Scholar [12] S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis,", Kluwer Academic Publishers, (1997).   Google Scholar [13] F. Mayoral, Compact sets of compact operators in absence of $l_1$,, Proc. Am. Math. Soc., 129 (2001), 79.  doi: 10.1090/S0002-9939-00-06007-X.  Google Scholar [14] E. Serrano, C. Pineiro and J. M. Delgado, Equicompact sets of operators defined on Banach spaces,, Proc. Am. Math. Soc., 134 (2005), 689.  doi: 10.1090/S0002-9939-05-08338-3.  Google Scholar [15] E. Zeidler, "Nonlinear Functional Analysis and its Applications,", Fixed Point Theorems, (1986).   Google Scholar
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