Uniform exponential stability of semi-discrete scheme for Euler–Bernoulli beam equation with non-collocated feedback

In this paper, we study the uniform exponential stability of a semi-discrete scheme for an Euler–Bernoulli beam equation under an observer-based output stabilizing feedback control studied in Guo et al. (2008), where the Riesz basis approach was employed. However, it is crucial to note that the Riesz basis approach falls short when applied to the uniform exponential stability of discrete schemes. Since the original system and observer together constitutes a coupled system described by partial differential equations (PDEs), this paper innovatively constructs a Lyapunov function specifically tailored for this coupled PDEs, which gives a much direct, simple alternative approach to exponential stability. In addition, this methodology can be seamlessly applied to assess the uniform exponential stability of a semi-discretized finite difference scheme corresponding to this coupled PDE. Although the semi-discretization process is still an order reduction approach studied in our previous works, it is novel in the sense that the order of the derivatives with respect to the spatial variable has been reduced to the first order, which not only eliminates the effect of the low order derivative on the boundary in previous study but also simplifies significantly the proof of the uniform exponential stability of the semi-discrete scheme.