Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

Abstract

Abstract. We study controllability for a nonhomogeneous string and ring under an axial stretching tension that varies with time. We consider the boundary control for a string and distributed control for a ring. For a string, we are looking for a control f (t) ∈ L 2 (0, T ) that drives the state solution to rest. We show that for a ring, two forces are required to achieve controllability. The controllability problem is reduced to a moment problem for the control. We describe the set of initial data which may be driven to rest by the control. The proof is based on an auxiliary basis property result.

On controllability of a non-homogeneous elastic string with memory

“…We proceed in two steps, which are similar but not identical to ones in [5,35] (see also the discussion in Introduction).…”

“…The proof of Riesz basis property of a family of the time-dependent functions in [1] and [4] is based on the assumption that the tension varies "slowly enough" with time. This restriction is removed in [5]. To the best of our knowledge, the papers [1], [4], [5] represent the first attempt to apply the method of moments to equations with time dependent coefficients.…”

“…This restriction is removed in [5]. To the best of our knowledge, the papers [1], [4], [5] represent the first attempt to apply the method of moments to equations with time dependent coefficients. The new difficulty in this case is the absence of an explicit representation for a family of functions arising in the moment problem.…”

Boundary controllability and source reconstruction in a viscoelastic string under external traction

“…that P does not depend on time. Control problems for the string equations under external traction have been rarely studied, see [1,2,3]. In fact, in general the system is part of a chain of mechanisms which produce "perturbations" which influence the horizontal traction in the string, often in a known way, for example due to the rotation of shafts.…”

Inverse dynamic problem for the wave equation with periodic boundary conditions

“…In the present paper, we set up and study the inverse dynamic problem for a wave equation with a potential on an interval with periodic boundary conditions. The control problems for dynamical systems for wave equation with periodic boundary conditions (the density allows certain dependence on time) were considered in [3,4]. The spectral problem for a Schrödinger operator on an interval with periodic and anti-periodic boundary conditions are used for treating the spectral problem for a Schrödinger operator with periodic potential on R, see [5].…”