Exact Controllability for String with Attached Masses

Abstract

We consider the problem of boundary control for a vibrating string with N interior point masses. We assume the control is at the left end, and the string is fixed at the right end. Singularities in waves are "smoothed" out to one order as they cross a point mass. We characterize the reachable set for a L 2 control. The control problem is reduced to a moment problem, which is then solved using the theory of exponential divided differences in tandem with unique shape and velocity controllability results.2 Existence, uniqueness, and regularity of solutions

Riesz basis generation and boundary stabilization of two strings connected by a point mass with variable coefficients

“…Some recent progress are concerned to the boundary controllability of hybrid systems with internal point masses and variable coefficients. [12][13][14][15] In this paper, we study the boundary feedback stabilization of a one-dimensional linear hybrid system which composed by two vibrating nonhomogeneous strings connected by a point mass. We assume that the first string occupies the interval (−1, 0) and the second one occupies the interval (0, 1).…”

Controllability for a string with attached masses and Riesz bases for asymmetric spaces

“…In this paper, we consider the controllability of a vibrating string with N attached masses. The controllability of a string with a single attached mass was considered in [28], [20], [21], also [34], and more recently in [6], [18] and [19]. In proving our 454 SERGEI AVDONIN AND JULIAN EDWARD controllability results we construct Riesz bases of the associated asymmetric spaces.…”

“…One novelty of this paper and [6] is a new method to prove controllability that combines dynamical and spectral approaches. We now state one of the key results from our dynamical approach for mixed control; the analogue for Dirichlet control can be found in .…”

“…We now compare this paper to other results in the literature. This paper, along with [6], were inspired by the paper of Hansen and Zuazua, [28]. In that work, the authors consider only Dirichlet control with N = 1.…”

“…The current work is an improvement in several ways. First, in [6] we assumed ρ j = ξ j = 1 for all j, and second we only considered Dirichlet control. Third, in the present paper we prove full controllability in the optimal time T = 2 in the case N = 1 with Dirichlet control, whereas in all other cases Theorems 1.2 and 7.3 show that full controllability in the optimal time T = 2 is not generally possible.…”