Discrete-time controllability for distributed parameter systems

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Abstract
The aim of this paper is to study controllability for systems described by first-order evolution equations and for systems described by second-order evolution equations in the case of discrete-time controls. For systems with finite-dimensional controls we present necessary and sufficient conditions for controllability. We show that distributed parameter systems are never finite-step controllable. We also investigate the relations between systems with discrete-time controls and systems with continuous -time controls from the view-point of controllability. Moreover we see the essential difference between the parabolic case and the hyperbolic case.