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

Abstract

Treatises on vibrations devote large space to study the dynamical behavior of an elastic system subject to known external tractions. In fact, usually a "system" is not an isolated body but it is part of a chain of mechanisms which disturb the "system" for example due to the periodic rotation of shafts. This kind of problem has been rarely studied in control theory. In the specific case we shall study, the case of a viscoelastic string, the effect of such external action is on the horizontal component of the traction, and so it affects the coefficients of the corresponding wave type equation, which will be time dependent. The usual methods used in controllability are not naturally adapted to this case. For example at first sight it might seem that moment methods can only be used in case of coefficients which are constant in time. Instead, we shall see that moment methods can be extended to study controllability of a viscoelastic string subject to external traction and in particular we shall study a controllability problem which is encountered in the solution of the inverse problem consisting in the identification of a distributed disturbance source.

Controllability of Evolution Equations with Memory

“…When a = 1, (1.1) is a controlled heat equation with a parabolic memory kernel. In this case, as it has been shown in recent years (See [11,13,23,31]), the null controllability may fail whenever the memory kernel b(·, ·) is a non-trivial constant and the control region ω is fixed, independent of time. Nevertheless, the approximate controllability property is still possible for the same equation, at least for some special cases (See [2,31]).…”

Null-controllability properties of a fractional wave equation with a memory term

“…Controllability problems for evolution equations with memory terms have been extensively studied in the past. Among other contributions, we mention [11,13,14,15,16,20,21,24] which, as in our case, deal with hyperbolic type equations. Nevertheless, in the majority of these works the issue has been addressed focusing only on the steering of the state of the system to zero at time T , without considering that the presence of the memory introduces additional effects that makes the classical controllability notion not suitable in this context.…”

Distributed Systems with Persistent Memory

“…The case K (t, s) = K (t − s) is by far the most important in applications, but also the case P(t)K (t −s) is important, since it corresponds to bodies which, being part of a chain of mechanisms, are subject to external tractions. See [79] for a moment approach to this case. dim H = +∞ (in spite of this negative result the special set K in Problem 6.8 is relatively compact, i.e., any sequence has convergent subsequences in C([0, T ]; L 2 (Ω)), since its elements take value in H 1 (Ω), which is compactly embedded in L 2 (Ω), see [23, p. 266]).…”