On Controllability of an Elastic Ring

Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

“…In Section 5 we sketch an extension of our results to controllability of an elastic ring. This extends the results obtained in [4] under more restrictive conditions on P (t).…”

“…The paper represents a continuation of the previous papers by first two authors [1], [2], and [4] and the papers [24]- [26] of the third author. In the papers [1], [4], the density of the string and ring correspondingly is constant but the tension is supposed to be the sum of a constant the method of moments to equations with time dependent coefficients. As we said already, the new difficulty in this case is the absence of an explicit representation for a basis.…”

“…We also note that the function Q(s) (which is called Liouville transformation of P (t)) is "small" if P (t) (or ω(t) = P (t) ) varies "slowly"; the last fact was very important for the proof in [1], [4] but is not used at all here. Applying the transformation (2.23), (2.25) to IC (1.4) yields the following initial conditions for z n (s),…”

“…In [1], [4] the basis property of the system {Γ ± n (s)} is proved with the help of the Krein theorem about a basis that is quadratically close to an orthonormal one (see [15].) It is important for the proofs in both papers that the tension P (t) changes "slowly" with time (to be precise, its Liouville transformation is supposed to be small.)…”

“…The main goal of this paper, concerning the wave equation with time varying coefficient P (t) is to remove the restriction on the tension (used in the papers [1,2,3,4]) with the help of an idea suggested in [24]- [26].…”