# Stochastic controllability of systems with multiple delays in control

• Volume: 19, Issue: 1, page 39-47
• ISSN: 1641-876X

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## Abstract

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Finite-dimensional stationary dynamic control systems described by linear stochastic ordinary differential state equations with multiple point delays in control are considered. Using the notation, theorems and methods used for deterministic controllability problems for linear dynamic systems with delays in control as well as necessary and sufficient conditions for various kinds of stochastic relative controllability in a given time interval are formulated and proved. It will be proved that, under suitable assumptions, relative controllability of an associated deterministic linear dynamic system is equivalent to stochastic relative exact controllability and stochastic relative approximate controllability of the original linear stochastic dynamic system. As a special case, relative stochastic controllability of dynamic systems with a single point delay is also considered. Some remarks and comments on the existing results for stochastic controllability of linear dynamic systems are also presented.

## How to cite

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Jerzy Klamka. "Stochastic controllability of systems with multiple delays in control." International Journal of Applied Mathematics and Computer Science 19.1 (2009): 39-47. <http://eudml.org/doc/207919>.

@article{JerzyKlamka2009,
abstract = {Finite-dimensional stationary dynamic control systems described by linear stochastic ordinary differential state equations with multiple point delays in control are considered. Using the notation, theorems and methods used for deterministic controllability problems for linear dynamic systems with delays in control as well as necessary and sufficient conditions for various kinds of stochastic relative controllability in a given time interval are formulated and proved. It will be proved that, under suitable assumptions, relative controllability of an associated deterministic linear dynamic system is equivalent to stochastic relative exact controllability and stochastic relative approximate controllability of the original linear stochastic dynamic system. As a special case, relative stochastic controllability of dynamic systems with a single point delay is also considered. Some remarks and comments on the existing results for stochastic controllability of linear dynamic systems are also presented.},
author = {Jerzy Klamka},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {controllability; linear control systems; stochastic control systems; delayed controls; multiple delays},
language = {eng},
number = {1},
pages = {39-47},
title = {Stochastic controllability of systems with multiple delays in control},
url = {http://eudml.org/doc/207919},
volume = {19},
year = {2009},
}

TY - JOUR
AU - Jerzy Klamka
TI - Stochastic controllability of systems with multiple delays in control
JO - International Journal of Applied Mathematics and Computer Science
PY - 2009
VL - 19
IS - 1
SP - 39
EP - 47
AB - Finite-dimensional stationary dynamic control systems described by linear stochastic ordinary differential state equations with multiple point delays in control are considered. Using the notation, theorems and methods used for deterministic controllability problems for linear dynamic systems with delays in control as well as necessary and sufficient conditions for various kinds of stochastic relative controllability in a given time interval are formulated and proved. It will be proved that, under suitable assumptions, relative controllability of an associated deterministic linear dynamic system is equivalent to stochastic relative exact controllability and stochastic relative approximate controllability of the original linear stochastic dynamic system. As a special case, relative stochastic controllability of dynamic systems with a single point delay is also considered. Some remarks and comments on the existing results for stochastic controllability of linear dynamic systems are also presented.
LA - eng
KW - controllability; linear control systems; stochastic control systems; delayed controls; multiple delays
UR - http://eudml.org/doc/207919
ER -

## References

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1. Arapostathis, A., George, R.K. and Ghosh, M.K. (2001). On the controllability of a class of nonlinear stochastic systems, Systems and Control Letters 44(1): 25-34. Zbl0986.93007
2. Bashirov, A.E. and Kerimov, K.R. (1997). On controllability conception for stochastic systems, SIAM Journal on Control and Optimization 35(3): 348-398. Zbl0873.93076
3. Bashirov, A.E. and Mahmudov, N.I. (1999). On concepts of controllability for deterministic and stochastic systems, SIAM Journal on Control and Optimization 37(6): 1808-1821. Zbl0940.93013
4. Ehrhard, M. and Kliemann, W. (1982). Controllability of stochastic linear systems, Systems and Control Letters 2(2): 145-153. Zbl0493.93009
5. Fernandez-Cara, E., Garrido-Atienza, M.J., and Real, J. (1999).On the approximate controllability of a stochastic parabolic equation with multiplicative noise, Comptes Rendus de l'Académie des Sciences, Paris 328(1): 675-680. Zbl0932.93007
6. Kim, J. U. (2004). Approximate controllability of a stochastic wave equation, Applied Mathematics and Optimization 49(1): 81-98. Zbl1059.93019
7. Klamka, J. (1991). Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht. Zbl0732.93008
8. Klamka, J. (1993). Controllability of dynamical systems-A survey, Archives of Control Sciences 2(3/4): 281-307. Zbl0818.93002
9. Klamka, J. (1996). Constrained controllability of nonlinear systems, Journal of Mathematical Analysis and Applications 201(2): 365-374. Zbl0858.93014
10. Klamka, J. (2000). Schauder's fixed point theorem in nonlinear controllability problems, Control and Cybernetics 29(3): 377-393. Zbl1011.93001
11. Klamka, J. and Socha, L. (1977). Some remarks about stochastic controllability, IEEE Transactions on Automatic Control 22(6): 880-881. Zbl0363.93048
12. Klamka, J. and Socha, L. (1980). Some remarks about stochastic controllability for delayed linear systems, International Journal of Control 32(5): 561-566. Zbl0443.93011
13. Mahmudov, N.I. (2001a). Controllability of linear stochastic systems, IEEE Transactions on Automatic Control 46(5): 724-731. Zbl1031.93034
14. Mahmudov, N.I. (2001b). Controllability of linear stochastic systems in Hilbert spaces, Journal of Mathematical Analysis and Applications 259(1): 64-82. Zbl1031.93032
15. Mahmudov, N.I. (2002). On controllability of semilinear stochastic systems in Hilbert spaces, IMA Journal of Mathematical Control and Information 19(2): 363-376. Zbl1138.93313
16. Mahmudov, N.I. (2003a). Controllability and observability of linear stochastic systems in Hilbert spaces, Progress in Probability 53(2): 151-167. Zbl1175.93210
17. Mahmudov, N. I. (2003b). Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM Journal on Control and Optimization 42(5): 1604-1622. Zbl1084.93006
18. Mahmudov, N.I. and Denker, A. (2000). On controllability of linear stochastic systems, International Journal of Control 73(2): 144-151. Zbl1031.93033
19. Mahmudov, N.I. and Zorlu, S. (2003). Controllability of nonlinear stochastic systems, International Journal of Control 76(2): 95-104. Zbl1111.93301
20. Subramaniam, R. and Balachandran, K. (2002). Controllability of stochastic Volterra integrodifferential systems, Korean Journal on Computing and Applied Mathematics 9(2): 583-589. Zbl1031.93039
21. Sunahara, Y., Kabeuchi, T., Asada, S., Aihara, S. and Kishino, K. (1974). On stochastic controllability for nonlinear systems, IEEE Transactions on Automatic Control 19(1): 49-54. Zbl0276.93011
22. Sunahara, Y., Aihara, S. and Kishino, K. (1975). On the stochastic observability and controllability for nonlinear systems, International Journal of Control 22(1): 65-82. Zbl0315.93021
23. Zabczyk, J. (1991). Controllability of stochastic linear systems, Systems and Control Letters 1(1): 25-31. Zbl0481.93054

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