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.

The paper contains systems descriptions and fundamental results concerning the solution of the most popular linear continuous-time control models with constant coefficients. First, different kinds of stability are discussed. Next fundamental definitions of controllability both for finite-dimensional and infinite-dimensional systems are recalled and necessary and sufficient conditions for different kinds of controllability are formulated. Moreover, fundamental definitions of controllability both for finite-dimensional and infinite-dimensional control systems are presented and necessary and sufficient conditions for different kinds of controllability are given. Finally, concluding remarks and comments concerning possible extensions are presented.

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In this paper infinite-dimensional dynamical systems described by nonlinear abstract differential equations are considered. Using the generalized open mapping theorem sufficient conditions for constrained exact local controllability are formulated and proved. It is generally assumed that the values of controls are in a convex and closed cone with vertex at zero. As an illustrative example a constrained exact local controllability problem for a nonlinear delayed dynamical system is solved in detail. Some remarks and comments on the existing results for controllability of nonlinear dynamical systems are also presented. ᮊ

In the paper, finite-dimensional dynamical control systems described by semilinear ordinary differential state equations with multiple point timevariable delays in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local relative controllability are formulated and proved. It is generally assumed that the values of admissible controls are in a convex and closed cone with the vertex at zero. The special case of constant multiple point delays is also discussed. Moreover, some remarks and comments on the existing results for controllability of nonlinear and semilinear dynamical systems are also presented.