Lyapunov Based Boundary Control of Strain Gradient Microscale Beams With Exponential Decay Rate

Abstract

In nonclassical microbeams, the governing partial differential equation (PDE) of the system and corresponding boundary conditions are obtained based on the nonclassical continuum mechanics. In this study, exponential decay rate of a vibrating nonclassical microscale Euler–Bernoulli beam is investigated using a linear boundary control law and by implementing a proper Lyapunov functional. To illustrate the performance of the designed controllers, the closedloop PDE model of the system is simulated via finite element method (FEM). To this end, new nonclassical beam element stiffness and mass matrices are developed based on the strain gradient theory and verification of this new beam element is accomplished in this work.