The well-posedness and stability of a beam equation with conjugate variables assigned at the same boundary point

• Published in: IEEE transactions on automatic control Vol. 50; no. 12; pp. 2087 - 2093
• Main Authors: GUO, Bao-Zhu, WANG, Jun-Min
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.12.2005; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Automatic control; Boundaries; Boundary control; Computer science; control theory; systems; Conjugates; Control system synthesis; Control systems; Control theory. Systems; Euler-Bernoulli beam; Euler-Bernoulli beams; Exact sciences and technology; Feedback; Fundamental areas of phenomenology (including applications); Genetic algorithms; Mathematical analysis; Mathematical models; MIMO; Optimal control; Optimization methods; Physics; Riccati equations; Riesz basis; Solid mechanics; Stability; Structural and continuum mechanics; Sufficient conditions; Vibration; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); C0-semigroup; Feedback regulation; Control synthesis; State space; Semigroup; Bernoulli Euler model; State space method; Modeling; Beam(mechanics); Relaxation; Feedback; Applied mathematics; Boundary control; Vibration control; Riesz basis; stability; Euler-Bernoulli beam
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2005.860275

A Euler-Bernoulli beam equation subject to a special boundary feedback is considered. The well-posedness problem of the system proposed by G. Chen is studied. This problem is in sharp contrast to the general principle in applied mathematics that the conjugate variables cannot be assigned simultaneously at the same boundary point. We use the Riesz basis approach in our investigation. It is shown that the system is well-posed in the usual energy state space and that the state trajectories approach the zero eigenspace of the system as time goes to infinity. The relaxation of the applied mathematics principle gives more freedom in the design of boundary control for suppression of vibrations of flexible structures.