Fixed point methods in nonlinear controllability problems

Controllability is one of the fundamental concepts in mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. 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 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 one hand on a class of dynamical control systems and on the other hand on the form of admissible controls. In recent years various controllability problems for different types of linear semilinear and nonlinear dynamical systems have been considered in many publications and monographs. There are, however, numerous open controllability problems for special types of nonlinear and semilinear dynamical systems, both finite-dimensional and infinite-dimensional. Controllability problems for different types of dynamical systems require 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. The main purpose of the paper is to formulate new controllability criteria for wide class of nonlinear and semilinear dynamical systems using fixed-point methods and covering nonlinear operators, which are known in nonlinear functional analysis. Moreover, in the paper we will use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability.