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A necessary and sufficient condition for the controllability of linear systems in Hilbert spaces and applications

Abstract:

As we have announced in the title of this work, we show that a broad class of linear evolution equations is exactly controllable. This class is represented by the following infinite-dimensional linear control system: z = Az + Bu(t), t > 0, z ∈ Z, u(t) ∈ U, where Z, U are Hilbert spaces, the control function u belong to L2 (0, t1; U), t1 > 0, B ∈ L (U, Z) and A generates a strongly continuous semigroup operator T (t) according to Pazy. We give a necessary and sufficient condition for the exact controllability of this system and apply this result to a linear controlled damped wave equation. © The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.