Stabilization of an abstract second order system with application to wave equations under non-collocated control and observations

The objective of this paper is to deal with the stabilization of multi-dimensional wave equations under non-collocated control and observation with the following cases: (a) locally internal distributed control and observation; (b) internal distributed control and boundary observation; (c) boundary control and internal distributed observation; (d) locally internal distributed control and boundary observation. We first discuss the stability of an abstract second order infinite-dimensional system under the estimated state feedback control. The well-posedness, in the sense of D.Salamon, is specially emphasized. The result is then applied to the multi-dimensional wave equations with the aforementioned three different non-collocated control and observations (a)–(c). Finally, we prove the stabilizability of a wave equation with non-collocated control and observation (d) by using the same idea, although it cannot be cast into the abstract setting.