Stabilization of second order evolution equations by unbounded nonlinear feedback

For an abstract evolution equation of the form utt + Au + y (ut) ' 0, general conditions on the "unbounded" feedback are given, that ensure strong asymptotic stability. Essentially the directions determined by the convex of the minima of the functional y should not intersect the eigenspaces of A Equivalently , the feedback on the velocity must dissipate enough energy, in the sense that the kernel of the form is not larger than the kernel of a "strategic" observation operateur, for the uncotrolled system. The particular case where the control operator is the dual of the corresponds to more classical rank conditions on the observation operator. The present framework applies to boundary or interior , distributed or pointwise, controls. The analysis is also able to handle "unilateral controls". Several examples, including wave, beam and plate equations, possibly with interior control on thin sets, are considered