The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

Stochastic Controllability of Linear Systems With State Delays: 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.

The main purpose of the paper is to discuss controllability problem for infinite-dimensional fractional linear discrete-time control systems. Moreover, the minimum energy control problem of infinite-dimensional fractional-discrete time linear systems is also addressed. Using theory of linear operators necessary and sufficient conditions for the exact controllability of the system are established. Sufficient conditions for the solvability of the minimum energy control of the infinite-dimensional fractional discrete-time systems are given. A procedure for computation of the optimal sequence of inputs minimizing the quadratic performance index is proposed.