Stochastic Controllability of Systems with Multiple Delays in Control

Jerzy Klamka

doi:10.2478/v10006-009-0003-9

.blurhash-client-img { display: none !important; }

Stochastic Controllability of Systems with Multiple Delays in Control

International Journal of Applied Mathematics and Computer Science

Volume 19 (2009): Issue 1 (March 2009)

By: Jerzy Klamka

Open Access

|Apr 2009

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.

References

Arapostathis, A., George, R.K. and Ghosh, M.K. (2001). On the controllability of a class of nonlinear stochastic systems,44(1): 25-34.
Open DOI Search in Google Scholar
Bashirov, A.E. and Kerimov, K.R. (1997). On controllability conception for stochastic systems,35(3): 348-398.
Open DOI Search in Google Scholar
Bashirov, A.E. and Mahmudov, N.I. (1999). On concepts of controllability for deterministic and stochastic systems,37(6): 1808-1821.
Open DOI Search in Google Scholar
Ehrhard, M. and Kliemann, W. (1982). Controllability of stochastic linear systems,2(2): 145-153.
Open DOI Search in Google Scholar
Fernandez-Cara, E., Garrido-Atienza, M.J., and Real, J. (1999).On the approximate controllability of a stochastic parabolic equation with multiplicative noise,328(1): 675-680.
Open DOI Search in Google Scholar
Kim, J.U. (2004). Approximate controllability of a stochastic wave equation,49(1): 81-98.
Open DOI Search in Google Scholar
Klamka, J. (1991)., Kluwer Academic Publishers, Dordrecht.
Search in Google Scholar
Klamka, J. (1993). Controllability of dynamical systems—A survey,2(3/4): 281-307.
Search in Google Scholar
Klamka, J. (1996). Constrained controllability of nonlinear systems,201(2): 365-374.
Open DOI Search in Google Scholar
Klamka, J. (2000). Schauder's fixed point theorem in nonlinear controllability problems,29(3): 377-393.
Search in Google Scholar
Klamka, J. and Socha, L. (1977). Some remarks about stochastic controllability,22(6): 880-881.
Open DOI Search in Google Scholar
Klamka, J. and Socha, L. (1980). Some remarks about stochastic controllability for delayed linear systems,32(5): 561-566.
Open DOI Search in Google Scholar
Mahmudov, N.I. (2001a). Controllability of linear stochastic systems,46(5): 724-731.
Open DOI Search in Google Scholar
Mahmudov, N.I. (2001b). Controllability of linear stochastic systems in Hilbert spaces,259(1): 64-82.
Open DOI Search in Google Scholar
Mahmudov, N.I. (2002). On controllability of semilinear stochastic systems in Hilbert spaces,19(2): 363-376.
Open DOI Search in Google Scholar
Mahmudov, N.I. (2003a). Controllability and observability of linear stochastic systems in Hilbert spaces,53(2): 151-167.
Open DOI Search in Google Scholar
Mahmudov, N.I. (2003b). Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces,42(5): 1604-1622.
Open DOI Search in Google Scholar
Mahmudov, N.I. and Denker, A. (2000).On controllability of linear stochastic systems,73(2): 144-151.
Open DOI Search in Google Scholar
Mahmudov, N.I. and Zorlu, S. (2003). Controllability of nonlinear stochastic systems,76(2): 95-104.
Open DOI Search in Google Scholar
Subramaniam, R. and Balachandran, K. (2002). Controllability of stochastic Volterra integrodifferential systems,9(2): 583-589.
Open DOI Search in Google Scholar
Sunahara, Y., Kabeuchi, T., Asada, S., Aihara, S. and Kishino, K. (1974). On stochastic controllability for nonlinear systems,19(1): 49-54.
Open DOI Search in Google Scholar
Sunahara, Y., Aihara, S. and Kishino, K. (1975). On the stochastic observability and controllability for nonlinear systems,22(1): 65-82.
Open DOI Search in Google Scholar
Zabczyk, J. (1991). Controllability of stochastic linear systems,1(1): 25-31.
Open DOI Search in Google Scholar