The method of adapted energies for second order evolution equations with dissipation

Abstract

This survey paper emphasizes the method of adapted energy functions to obtain sharp information on various stability questions: exponential stability, strong stability with explicit decay rates, ultimate bounds of solutions and so on. The efficiency of the method, basically related to Lyapunov’s and Barbashin–Krasovskii–LaSalle’s approaches, very often allows one to obtain close-to-optimal estimates, as demonstrated on a variety of examples connected to mechanics via differential equations of second order with respect to the time variable.