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1

Stochastic controllability of systems with multiple delays in control

Klamka J.

International Journal of Applied Mathematics and Computer Science

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2009

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Vol. 19, no 1

39-47

EN

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.

2

Stochastic controllability of systems with variable delay in control

Klamka J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2008

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Vol. 56, nr 3

279-284

EN

In the paper finite-dimensional time-variable dynamical control systems described by linear stochastic ordinary differential state equations with single time-variable point delay in the control are considered. Using notations, theorems and methods taken directly from deterministic controllability problems necessary and sufficient conditions for different 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 a deterministic linear associated dynamical system is equivalent to stochastic relative exact controllability and stochastic relative approximate controllability of the original linear stochastic dynamical system. Some remarks and comments on the existing results for stochastic controllability of linear dynamical systems are also presented.

3

Stochastic controllability of linear systems with delay in control

Klamka J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2007

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Vol. 55, nr 1

23-29

EN

In the paper finite--dimensional stationary dynamical control systems described by linear stochastic ordinary differential state equations with single point delay in the control are considered. Using notations, theorems and methods taken directly from deterministic controlla-bility problems, necessary and sufficient conditions for different kinds of stochastic relative controllability are formulated and proved. It will be proved that under suitable assumptions relative controllability of a deterministic linear associated dynamical system is equivalent to stochastic relative exact controllability and stochastic relative approximate controllability of the original linear stochastic dynamical system. Same remarks and comments on the existing results for stochastic controllability of linear dynamical systems with delays are also presented. Finally, minimum energy control problem for stochastic dynamical system is formulated and solved.

4

Controllability of 2D continuous-discrete systems with delays in control

Klamka J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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1998

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Vol. 46, nr 3

363-373

EN

Recently, a new class of linear continuous-discrete 2D dynamical systems i.e., dynamical systems described by the indexed set of linear ordinary differential equations with appropriate initial and boundary conditions has been introduced. However, it should be mentioned that only a few papers deal with the 2D continuous-discrete linear dynamical systems with delays. In linear 2D continuous-discrete dynamical systems one independent variable is continuous and the second one is discrete. Such continuous-discrete models appear, for example, in the iterative learning control systems. Another example of such systems are the repetitive processes. In the present paper linear 2D continuous-discrete systems with multiple constant delays in the control and constant coefficients are considered. The general response formula for such 2D systems is derived. Fundamental definition of relative controllability in a given fixed rectangle is presented. Using some theorems taken from the theory of linear dynamical systems with delays in the state variable necessary and sufficient pure algebraic rank condition for relative controllability is formulated and proved. Moreover, remarks and comments on controllability of 2D linear continuous-discrete dynamical systems are given.