American Institute of Mathematical Sciences

Advanced
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 & 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

show all references

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

[1]

Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial & Management Optimization, 2020, 16 (2) : 707-724. doi: 10.3934/jimo.2018174
[2]

Qian Liu, Xinmin Yang, Heung Wing Joseph Lee. On saddle points of a class of augmented lagrangian functions. Journal of Industrial & Management Optimization, 2007, 3 (4) : 693-700. doi: 10.3934/jimo.2007.3.693
[3]

Mariusz Michta. Stochastic inclusions with non-continuous set-valued operators. Conference Publications, 2009, 2009 (Special) : 548-557. doi: 10.3934/proc.2009.2009.548
[4]

Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations & Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028
[5]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116
[6]

Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control & Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013
[7]

Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028
[8]

Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48
[9]

Qing Yang, Shiji Song, Cheng Wu. Inventory policies for a partially observed supply capacity model. Journal of Industrial & Management Optimization, 2013, 9 (1) : 13-30. doi: 10.3934/jimo.2013.9.13
[10]

Aihua Fan, Lingmin Liao, Jacques Peyrière. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1103-1128. doi: 10.3934/dcds.2008.21.1103
[11]

Kody Law, Abhishek Shukla, Andrew Stuart. Analysis of the 3DVAR filter for the partially observed Lorenz'63 model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1061-1078. doi: 10.3934/dcds.2014.34.1061
[12]

Tomasz Kapela, Piotr Zgliczyński. A Lohner-type algorithm for control systems and ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 365-385. doi: 10.3934/dcdsb.2009.11.365
[13]

Saïd Abbas, Mouffak Benchohra, John R. Graef. Coupled systems of Hilfer fractional differential inclusions in banach spaces. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2479-2493. doi: 10.3934/cpaa.2018118
[14]

Olaf Klein. On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2591-2614. doi: 10.3934/dcds.2015.35.2591
[15]

Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017
[16]

Mariusz Michta. On solutions to stochastic differential inclusions. Conference Publications, 2003, 2003 (Special) : 618-622. doi: 10.3934/proc.2003.2003.618
[17]

Guoshan Zhang, Shiwei Wang, Yiming Wang, Wanquan Liu. LS-SVM approximate solution for affine nonlinear systems with partially unknown functions. Journal of Industrial & Management Optimization, 2014, 10 (2) : 621-636. doi: 10.3934/jimo.2014.10.621
[18]

Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 629-654. doi: 10.3934/dcdsb.2010.14.629
[19]

Yong-Kui Chang, Xiaojing Liu. Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020036
[20]

JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences