Stabilization and parameter estimation for an Euler-Bernoulli beam equation with uncertain harmonic disturbance under boundary output feedback control (Q1775873)

This is a lengthy article involving a clever use of functional analysis. While all problems are formulated in the \(L_2\) setting, the proof of existence of closed-loop solutions uses the \(L_\infty\) weak star topology to conclude the existence of generalized derivatives \(u_{xxxx}\). The uniqueness of classical solutions is proved by the use of a Lyapunov-type function, followed by a Galerkin approximation. In the proof of asymptotic stability, the authors first construct another Lyapunov functional, this time a fairly standard one, and estimate time derivatives along the path of the solution. Finally, they introduce the Riesz basis of eigenfunctions for a family of eigenvalues and show that they are orthogonal. This confirms that their approximations are equal to the limit of estimated parameters.