Controllability of M-D linear systems

Controllability has played an instrumental role in the development of control theory during the past three decades and is now a fundamental concept in mathematical systems theory. It plays a central role in control synthesis questions, related to the very possibility of exerting effective control. As such, it enters as a crucial existence condition in many engineering-type questions, such as stabilization and optimal control. The notion of controllability is usually introduced for state space representations, where it refers to the possibility of transferring the state from an initial to a terminal value. Recently, a notion of controllability was introduced for two-dimensional discrete systems, so called 2-D systems and next extended for multidimensional discrete systems, so called M-D systems. In the present paper special case of linear M-D discrete systems with constant coefficients is investigated. This system is transformed into equivalent discrete dynamical system with infinite-dimensional state space. Fundamental definitions of approximate and exact controllability are given. Using methods of functional analysis necessary and sufficient conditions for different types of controllability are formulated. Finally, so called minimum energy control problem in infinite-dimensional state space is solved.