Exact boundary controllability of two Euler-Bernoulli beams connected by a point mass

• Published in: Mathematical and computer modelling 2000-11, Vol.32 (9), p.955-969
• Main Authors: Castro, C., Zuazua, E.
• Format: Article
• Language: English
• Subjects: Calculus of variations and optimal control; Controllability; Exact sciences and technology; Flexible beams; Fourier analysis; Fourier series; Fundamental areas of phenomenology (including applications); Mathematical analysis; Mathematics; Ordinary differential equations; Physics; Point mass; Sciences and techniques of general use; Solid mechanics; Structural and continuum mechanics; Structural mechanics (beam, string...); Theory and numerical methods
• ISSN: 0895-7177; 1872-9479
• DOI: 10.1016/S0895-7177(00)00182-5

We consider a hybrid system consisting of two flexible beams connected by a point mass. In a previous work, we showed that when the constant of rotational inertia γ is positive, due to the presence of the mass, the system is well posed in asymmetric spaces, i.e., spaces with different regularity to both sides of the mass. As a consequence of this, the space of controllable data when we act on the free extreme of the system is also an asymmetric space when γ > 0. In this paper, we study the case γ = 0 in which we recover the classical Euler-Bernoulli model for the beams. We prove in this case that the system is not well posed in asymmetric spaces and then the presence of the point mass does not affect the controllability of the system. The proofs are based in the development of solutions in Fourier series and the use of nonharmonic Fourier series.