Uniform approximation of exponential stability for an Euler–Bernoulli beam under bending moment feedback

Abstract

Uniformly approximating infinite-dimensional systems, such as those described by partial differential equations (PDEs), using infinitely many systems described by ordinary differential equations is crucial for their application in engineering. This paper examines uniform exponential stability for a semi-discrete scheme applied to a one-dimensional Euler–Bernoulli beam with bending moment feedback, which is sharp contrast to scenarios involving shear force feedback control. For decades, establishing the exponential stability of this continuous PDE has posed a significant challenge, primarily due to the absence of an appropriate time domain multiplier. This obstacle, in a way, spurred the advancement of the frequency domain multiplier method in the 1980s. Notably, the frequency multiplier utilized here is a specified exponential function of both concerned varying real numbers and spacial variable, distinguishing it from recent studies on the Schrödinger equation. Consequently, the question of uniform stability for the semi-discrete scheme has remained unresolved for an extended period. In this paper, we first establish the exponential stability of the continuous system using the frequency domain method. Subsequently, we employ an order-reduction technique to devise a spatially semi-discretized finite difference scheme for this continuous system. Ultimately, to demonstrate the uniformly exponential stability of the resulting semi-discretized system, we adopt the discrete frequency domain method, with the proof mirroring the approach used for the continuous system. The approach presented in this paper, which effectively addresses the specified exponential function involving both concerned varying real numbers and spatial variables, holds potential for application to other PDEs that encounter analogous challenges.