The Stability Of A Class Of Linear Benjamin-bona-mahony Equation And Control Issues

In this paper, we study the stability and control problems for a linearizedBenjamin-Bona-Mahony equation. Under suitable conditions the strong stabilityof this equation is established by means of the spectral analysis result in [4]. Weshow further that, however, the decay speed of its solutions may be slower thanany prescribed rate. We then construct a subspace, dense in the underlying statespace, for which the corresponding solutions decay polynomially whenever theinitial states belong to it. However, we also show that a similar construction does notlead to exponential decay. As for the control problem, we obtained a negative resultfor exact controllability and also a ?nite dimensional controllability result, whichextend the main results in [3] to general cases by means of different approaches.