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Vietnam Journal of Mathematics 36:4(2008) 463-472  Distance from an Exactly Controllable System to Not Approximately Controllable Systems B. T. Anh$^1$, D. C. Khanh$^2$ and D. D. X. Thanh$^3$ Abstract.  Given an exactly controllable time-invariant linear control system on a Hilbert space, the distance from the given system to the set of not approximately controllable systems is the norm of the smallest perturbation that makes the given system not approximately controllable. In this paper, the distances when both or only one of the system operators is perturbed are formulated in terms of optimization problems depending on a complex variable. In some cases, these optimization problems can be reduced to depend on one real variable, as well as the real and complex radii are shown to be equal. The obtained results in this paper also generalize the recent work of [6,12].   2000 Mathematics Subject Classification: 93B05, 93C25, 93C05. Keywords: Controllability, time-invariant linear systems, distance, Hilbert spaces.

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