Fast control of a laminated Timoshenko beam system with only boundary inputs

In this paper, we investigates the fast control problem of double-layer Timoshenko composite beam under unique boundary conditions characterized by anti-damping and anti-stiffness terms. Building upon our previous findings concerning single-layer Timoshenko beam, we explore novel aspects including the achievement of an arbitrarily swift decay rate for the double-layer Timoshenko composite beam and the impact of boundary characteristics that potentially lead to rapid divergence. Our methodology revolves around a transformative approach: proposing a Riemann-like transformation to convert the state variables of double-layer Timoshenko beams into a 1-D hyperbolic system integrating partial integro-differential equations (PIDEs) and ordinary differential equations (ODEs). Then we harness PDE backstepping techniques to design a control law ensuring closed-loop stability at the origin in the $L^2$ norm. The adjustable control parameters facilitates the attainment of arbitrarily rapid convergence rates. Moreover, we extend our control approach to accommodate longitudinal dynamics, yielding comparable outcomes of significant merit. Finally, the numerical simulation verifies our theoretical result.