Spectral properties of non-homogeneous Timoshenko beam and its rest to rest controllability

Abstract

The controllability of a slowly rotating non-homogeneous beam clamped to a disc is considered. It is assumed that at the beginning the beam remains at the position of rest and it is supposed to rotate by the given angle and stop. The movement is governed by the system of two differential equations with non-constant coefficients: mass density, flexural rigidity and shear stiffness. To solve the problem of controllability, the spectrum of the operator generating the dynamics of the model is studied. Then the problem of controllability is reduced to the moment problem that is, in turn, solved with the use of the asymptotics of the spectrum.

Controlling a non-homogeneous Timoshenko beam with the aid of the torque

“…One can notice that D(A) is dense in H and A is surjective (Sklyar and Szkibiel, 2008b). Also, it is proved there that the operator A is self-adjoint, positive and possesses a compact resolvent.…”

“…Sklyar and Szkibiel (2008b) that the eigenspaces of the operator A are at most of the dimension 2. It is proved (Krabs and Sklyar, 2002) that if the parameter functions , R, K and E are constant (and positive) then all the eigenvalues of A are simple.…”

“…We consider the following system of equations describing the rotating Timoshenko beam (see Sklyar and Szkibiel, 2008a;2008b):…”

Construction of an Optimal Rest-to-Rest Control of a Rotating Beam

“…The moment problem in question, (8), is generated by some special functions. We exploit the fact that the exponents √ λ n have a special asymptotic behavior, given by (7). This means that those exponential functions are in fact close to trigonometric functions-at least for large indices.…”

“…Those results were used to obtain a new controllability condition of the beam in the form of smoothness of end states [6]. A similar model of a non-homogeneous beam was considered in [7] where the minimal time for a rest-to-rest controllability was found.…”