On controllability of linear stochastic systems

Stochastic Controllability of Linear Systems With State Delays

“…Controllability is one of the fundamental concepts in mathematical control theory and plays an important role in both deterministic and stochastic control theories (Klamka, 1991;Klamka, 1993;Mahmudov and Denker, 2000). Controllability is a qualitative property of dynamic control systems and is of particular importance in control theory.…”

“…Roughly speaking, controllability generally means that it is possible to steer a dynamic control system from an arbitrary initial state to an arbitrary final state using a set of admissible controls. In the literature there are many different definitions of controllability for both linear (Klamka, 1991;Klamka, 1993;Mahmudov, 2001;Mahmudov and Denker, 2000) and nonlinear dynamic systems (Klamka, 2000;Mahmudov, 2002;Mahmudov and Zorlu, 2003), which do depend on the class of dynamic control systems and the set of admissible controls † This work was supported by the Ministry of Science and Higher Education under the grant no. 8T11A 002 27.…”

“…In recent years the extensions of deterministic controllability concepts to stochastic control systems have been discussed only in a limited number of publications. In the papers (Bashirov and Kerimov, 1997;Bashirov and Mahmudov, 1999;Ehrhard and Kliemann, 1982;Mahmudov, 2001;Mahmudov and Denker, 2000;Zabczyk, 1991) different kinds of stochastic controllability were discussed for linear finite dimensional stationary and nonstation-6 J. Klamka ary control systems. The papers (Fernandez-Cara et al, 1999;Kim Jong Uhn, 2004;Mahmudov, 2001; are devoted to a systematic study of approximate and exact stochastic controllability for linear infinite dimensional control systems defined in Hilbert spaces.…”

“…The papers (Fernandez-Cara et al, 1999;Kim Jong Uhn, 2004;Mahmudov, 2001; are devoted to a systematic study of approximate and exact stochastic controllability for linear infinite dimensional control systems defined in Hilbert spaces. Stochastic controllability for finite dimensional nonlinear stochastic systems was discussed in (Arapostathis et al, 2001;Balasubramaniam and Dauer, 2001;Mahmudov and Zorlu, 2003;Sunahara et al, 1974;Sunahara et al, 1975). Using the theory of bounded nonlinear operators and linear semigroups, various different types of stochastic controllability for nonlinear infinite dimensional control systems defined in Hilbert spaces were considered in (Mahmudov, 2002;.…”

“…Specifically we shall consider stochastic relative exact and approximate controllability problems for finite-dimensional linear stationary dynamic systems with single constant point delays in the state variables described by stochastic ordinary differential state equations. More precisely, using techniques similar to those presented in (Mahmudov, 2001;Mahmudov, 2001;Mahmudov and Denker, 2000), we shall formulate and prove necessary and sufficient conditions for stochastic relative exact controllability in a prescribed time interval for linear stationary stochastic dynamic systems with one constant point delay in the state variables.…”