Observability and stabilization of the vibrating string equipped with bouncing point sensors and actuators

Abstract

SUMMARYThe issues of observability and stabilization are analysed for a vibrating string equipped, respectively, either with a point sensor measuring its displacement or a vertical tie-down viscous damper. It is well known that the aforementioned properties hold for the stationary devices of these types only when they are placed at the 'irrational points' of the string. The latter creates obvious di culties in applications. In this article we discuss how mobile point sensors and dampers, bouncing in a rather simple fashion between any two points on the string, can be employed instead in a more 'feasible and robust' way.

Observability of a 1D Schrödinger equation with time-varying boundaries

“…It is remakable that the lower estimte cannot be true when ε = 0 on any rational point a ; the fact that the considered domains extend however, seem to 'middle out' this obstacle. Closely related to this observation are works of Castro and Khapalov [6,14,13] where on a fixed domain Ω a moving point observer is considered, with similar conclusions. We also mention results from Moyano [28,29] where in a two-dimensional circle the radius ℓ(t) is used as a control parameter.…”

Exact controllability of the 1-d wave equation from a moving interior point

“…This problem has been previously studied in [8] and [9] where some sufficient conditions on the curve are given. Here we introduce a new approach that provides the exact controllability property for a larger class of curves.…”

Exact Observability of a 1-Dimensional Wave Equation on a Noncylindrical Domain

“…One way to counter this problem is to obtain observability results for the average of |u| 2 in a small neighbourhood of a fixed internal point a, see [15]. It is also well known that another way to counter this problem is to consider a moving interior point, see for example [7,20,19]. We follow in this article the idea that fixed domain with moving observers should somehow behave similar to moving domains with fixed observers.…”