Exponential Decay of Energy of Evolution Equations with Locally Distributed Damping

Abstract

Consider an evolution equation with energy dissipation, \[ \frac{\partial }{{\partial t}}y( {x,t} ) = Ay( {x,t} ) + By( {x,t} ) \] on a bounded x-domain $\Omega $, where $By$ signifies a dissipative perturbation to an otherwise energy-conserving system. This dissipation may be due to medium impurities, viscous effects, or artificially imposed dampers and stabilizers. It is distributed over only part of the domain $\Omega $. The question of when the dissipation is effective enough to cause uniform exponential decay of energy is examined. Because of the locally distributed nature of energy dissipation, the problem lacks coercivity and is not directly solvable by energy identities. Thus, to get conditions sufficient for uniform exponential decay, a different approach needs to be taken. Provided here is a set of tight sufficient conditions in terms of the influence of the dissipative operatorBon the separated eigenmodes or clustered eigenmodes ofA. The main theorems are general enough to treat the wave and beam equations in one space dimension and the Schrödinger equation on a two-dimensional rectangle or disk. In particular, for the Schrödinger equation, it shown how such phenomena as the “whispering gallery”and “bouncing ball” eigenmodes dictate the supports of effective damping functions. Thus, the theory presented is also capable of providing valuable information about the placement and design of actuators and sensors in modern distributed parameter control theory.