Indirect Boundary Stabilization of Weakly Coupled Hyperbolic Systems

This work is concerned with the boundary stabilization of an abstract system of two coupled second order evolution equations wherein only one of the equations is stabilized (indirect damping; see, e.g., J. Math. Anal. Appl., 173 (1993), pp. 339--358). We show that, under a condition on the operators of each equation and on the boundary feedback operator, the energy of smooth solutions of this system decays polynomially at ∞. We then apply this abstract result to several systems of partial differential equations (wave-wave systems, Kirchhoff--Petrowsky systems, and wave-Petrowsky systems).