Strict Lyapunov Functions and Energy Decay in Hamiltonian Chains with Degenerate Damping

We consider a Hamiltonian chain of rotators (in general nonlinear) in which the first rotator is damped. Being motivated by problems of nonequilibrium statistical mechanics of crystals, we construct a strict Lyapunov function that allows us to find a lower bound for the total energy dissipation rate when the energy and time are large. Our construction is explicit and its analysis is rather straightforward. We rely on a method going back to Matrosov, Malisoff, and Mazenc, which we review in our paper. The method is rather universal, and we show that it is applicable to a chain of oscillators as well.