Exact controllability of nonselfadjoint Euler–Bernoulli beam model via spectral decomposition method

The zero controllability problem for the hyperbolic equation, which governs the vibrations of the Euler–Bernoulli beam model of a finite length, is studied in this paper. The equation of motion is supplied with a one-parameter family of physically meaningful boundary conditions containing damping terms. The control is introduced as a separable forcing term g(x)f(t) in the right-hand side of the equation. A force profile function, g(x), is assumed to be given. To construct the control, f(t), which brings a given initial state of the system to zero on the specific time interval [0, T], the spectral decomposition method is applied. The necessary and/or sufficient conditions for the exact controllability as well as the explicit formulas for the control laws are given. Approximate controllability is also discussed. The exact controllability results are generalized to the case of multiple eigenvalues of the main dynamics generator.