# Stochastic controllability of linear systems with state delays

International Journal of Applied Mathematics and Computer Science (2007)

- Volume: 17, Issue: 1, page 5-13
- ISSN: 1641-876X

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topKlamka, Jerzy. "Stochastic controllability of linear systems with state delays." International Journal of Applied Mathematics and Computer Science 17.1 (2007): 5-13. <http://eudml.org/doc/207822>.

@article{Klamka2007,

abstract = {A class of finite-dimensional stationary dynamic control systems described by linear stochastic ordinary differential state equations with a single point delay in the state variables is considered. Using a theorem and methods adopted directly from deterministic controllability problems, necessary and sufficient conditions for various kinds of stochastic relative controllability are formulated and proved. It will be demonstrated that under suitable assumptions the relative controllability of an associated deterministic linear dynamic system is equivalent to the stochastic relative exact controllability and the stochastic relative approximate controllability of the original linear stochastic dynamic system. Some remarks and comments on the existing results for the controllability of linear dynamic systems with delays are also presented. Finally, a minimum energy control problem for a stochastic dynamic system is formulated and solved.},

author = {Klamka, Jerzy},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {linear control systems; minimum energy control; controllability; stochastic control systems; delayed state variables},

language = {eng},

number = {1},

pages = {5-13},

title = {Stochastic controllability of linear systems with state delays},

url = {http://eudml.org/doc/207822},

volume = {17},

year = {2007},

}

TY - JOUR

AU - Klamka, Jerzy

TI - Stochastic controllability of linear systems with state delays

JO - International Journal of Applied Mathematics and Computer Science

PY - 2007

VL - 17

IS - 1

SP - 5

EP - 13

AB - A class of finite-dimensional stationary dynamic control systems described by linear stochastic ordinary differential state equations with a single point delay in the state variables is considered. Using a theorem and methods adopted directly from deterministic controllability problems, necessary and sufficient conditions for various kinds of stochastic relative controllability are formulated and proved. It will be demonstrated that under suitable assumptions the relative controllability of an associated deterministic linear dynamic system is equivalent to the stochastic relative exact controllability and the stochastic relative approximate controllability of the original linear stochastic dynamic system. Some remarks and comments on the existing results for the controllability of linear dynamic systems with delays are also presented. Finally, a minimum energy control problem for a stochastic dynamic system is formulated and solved.

LA - eng

KW - linear control systems; minimum energy control; controllability; stochastic control systems; delayed state variables

UR - http://eudml.org/doc/207822

ER -

## References

top- Arapostathis A., George R.K., Ghosh M.K. (2001): On the controllability of a class of nonlinear stochastic systems. - Syst. Contr. Lett., Vol.44, No.1, pp.25-34. Zbl0986.93007
- Balasubramaniam P. and Dauer J.P. (2001): Controllability of semilinear stochastic evolution equations in Hilbert spaces. - J. Appl. Math. Stoch.Anal., Vol.14, No.4, pp.329-339. Zbl1031.93040
- Bashirov A.E., and Kerimov K.R. (1997): On controllability conception for stochastic systems. - SIAM J. Contr. Optim., Vol.35, No.2, pp.348-398. Zbl0873.93076
- Bashirov A.E. and Mahmudov N.I. (1999): On concepts of controllability for deterministic and stochastic systems. - SIAM J. Contr. Optim., Vol.37, No.6, pp.1808-1821. Zbl0940.93013
- Ehrhard M. and Kliemann W. (1982): Controllability of stochastic linear systems. - Syst. Contr.Lett., Vol.2, No.2, pp.145-153. Zbl0493.93009
- Fernandez-Cara E., Garrido-Atienza M.J. and Real J. (1999): On the approximate controllability of a stochastic parabolic equation with multiplicative noise. - C.R. Acad. Sci. Paris, t.328, Sèrie1, pp.675-680.
- Kim Jong Uhn (2004): Approximate controllability of a stochastic wave equation. - Appl. Math.Optim., Vol.49, No.1, pp.81-98. Zbl1059.93019
- Klamka J. (1991): Controllability of Dynamical Systems. - Dordrecht: Kluwer Academic.
- Klamka J. (1993): Controllability of dynamical systems-A survey. - Arch. Contr. Sci., Vol.2, No.3/4, pp.281-307. Zbl0818.93002
- Klamka J. (1996): Constrained controllability of nonlinear systems. - J. Math. Anal. Applic., Vol.201, No.2, pp.365-374. Zbl0858.93014
- Klamka J. (2000): Schauder's fixed point theorem in nonlinear controllability problems. - Contr.Cybern., Vol.29, No.3, pp.377-393. Zbl1011.93001
- Klamka J. and Socha L. (1977): Some remarks about stochastic controllability. - IEEE Trans.Automat. Contr., Vol.AC-22, No.5, pp.880-881. Zbl0363.93048
- Klamka J. and Socha L. (1980): Some remarks about stochastic controllability for delayed linear systems. - Int. J. Contr., Vol.32, No.3, pp.561-566. Zbl0443.93011
- Mahmudov N.I. (2001): Controllability of linear stochastic systems. - IEEE Trans. Automat. Contr., Vol.AC-46, No.4, pp.724-731. Zbl1031.93034
- Mahmudov N.I. (2001): Controllability of linear stochastic systems in Hilbert spaces. - J. Math.Anal. Applic., Vol.259, No.1, pp.64-82. Zbl1031.93032
- Mahmudov N.I. (2002): On controllability of semilinear stochastic systems in Hilbert spaces. - IMA J. Mathemat. Contr. Inf., Vol.19, No.2, pp.363-376. Zbl1138.93313
- Mahmudov N.I. (2003): Controllability and observability of linear stochastic systems in Hilbert spaces. - Progress in Probability, Vol.53, No.1, pp.151-167. Zbl1175.93210
- Mahmudov N.I. (2003): Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. - SIAM J. Contr.Optim., Vol.42, No.5, pp.1604-1622. Zbl1084.93006
- Mahmudov N.I. and Denker A. (2000): On controllability of linear stochastic systems. - Int. J.Contr., Vol.73, No.2, pp.144-151. Zbl1031.93033
- Mahmudov N.I. and Zorlu S. (2003): Controllability of nonlinear stochastic systems. - Int. J.Contr., Vol.76, No.2, pp.95-104. Zbl1111.93301
- Subramaniam R. and Balachandran K. (2002): Controllability of stochastic Volterra integrodifferential systems. - Korean J. Comput. Appl. Math., Vol.9, No.2, pp.583-589. Zbl1031.93039
- Sunahara Y., Kabeuchi T., Asada S., Aihara S. and Kishino K. (1974): On stochastic controllability for nonlinear systems. - IEEE Trans. Automat.Contr., Vol.AC-19, No.1, pp.49-54. Zbl0276.93011
- Sunahara Y., Aihara S. and Kishino K. (1975): On the stochastic observability and controllability for nonlinear systems. - Int. J. Contr., Vol.22, No.1, pp.65-82. Zbl0315.93021
- Zabczyk J. (1991): Controllability of stochastic linear systems. - Syst. Contr. Lett., Vol.1, No.1, pp.25-31 Zbl0481.93054

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