# Stochastic controllability of linear systems with state delays

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

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## Abstract

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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.

## How to cite

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Klamka, 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

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