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A control system of the second order in time with control$u=u(t)\in L^2([0,T];U)$ is considered. If thesystem is controllable in a strong sense anduT is the controlsteering the system to the rest at timeT,then the L2–norm of uT decreases as $1/\sqrt T$while the $L^1([0,T];U)$–norm of uT is approximately constant.The proof is based on the moment approachand properties of the relevant exponential family. Results areapplied to the wave equation with boundary or distributed controls.
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