Abstract
We prove a sharp decay rate for the total energy of two classes of systems of weakly coupled hyperbolic equations. We show that we can stabilize the full system through a single damping term, in feedback form, acting on one component only of the system (\emph{indirect stabilization}). The energy estimate is achieved by means of suitable estimates of the resolvent operator norm. We apply this technique to a wave-wave system and to a wave-Petrovsky system.
Highlights
The issue of stabilizing a system of partial differential equations through suitable damping terms on each component of the system has its origin in the pioneering works of Lagnese and Lions [9] and Russell [11]
Russell [11] addressed the indirect stabilization problem, that occurs when the damping acts on a reduced number of equations of the system. In this situation the exponential decay rate usually cannot be achieved, but weaker decay rates might hold. This is the case in [2] and [3], where polynomial stability for the whole system is showed, under a suitable compatibility condition on the operators involved in the system, by means of multipliers properly adapted to the peculiar structure of the system under investigation
It turns out that different compatibility condition and multipliers are required to cope with systems with boundary conditions of similar type on each component [2] or mixed [3]
Summary
The issue of stabilizing a system of partial differential equations through suitable damping terms (possibly in feedback form) on each component of the system has its origin in the pioneering works of Lagnese and Lions [9] and Russell [11] In their approach, the multiplier method is the main tool to reach the desired estimates on the energy of each component of the system. The aim of the present paper is to derive the optimal decay rate of the total energy E(t); we will show that it decays polynomially in time with a decay rate 1/2 for initial condition in D(A), that is, for every U0 = (u0, u1, v0, v1) ∈ D(A) and for some C > 0 (see equation (31)) In this way, we succeed to improve the decay rate of an exponential factor 2.
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