Sharp stability estimates for quasi-autonomous evolution equations of hyperbolic type

We study the energy decay of the difference of two solutions for dissipativ e evolution problems of the type: u " + Lu + g(u') = h(t), t greater than or equal to 0, including wave and plate equations and ordinary differential equations. In the general case, when the damping term g behaves like a power of the veloc ity u', the energy decreases like a negative power of time, multiplied by a constant depending on the initial energies. We provide estimates on these constants and prove their optimality. In the special case of the ordinary d ifferential equation with periodic forcing, we establish, relying on a cont rollability-like technique, that the decay is in fact exponential, even und er very weak damping.