Uniformly Exponential Convergence Of Semi-Discretization Scheme For One-dimensional Wave Equation With A Tip Mass Under Boundary Non-collocated Control

For most control systems described by partial differential equations(PDEs),if they are to be applied to engineering,a numerical discretization process is essential.One would rather discretize the system into an ordinary differential equations(ODEs)system for which most of engineers are familiar with by keeping the time continuous and the spatial variable to be discretized.This process is called the semi-discretization scheme.In the last two decades,there has been a great discovery in the numerical implementation of feedback control for systems described by PDEs:for exponentially stable PDEs,the semi-discrete system is not uniformly decaying for the step size after discretization of the spatial variables.This obviously hinders the application of feedback control of PDEs in engineering,especially in computer digitization.Therefore,solving the non-uniformly exponential stability after discretization of PDEs is an important work of this paper.This paper considers the uniformly exponential stability of a semi-discrete model for a 1-d wave equation with tip mass under boundary feedback control.The original closed-loop system is transformed firstly into a low-order equivalent system by order reduction method and the exponential stability of the transformed system by an indirect Lyapunov method is established.The equivalent system is then discretized into a series of semi-discrete systems in spatial variable.Parallel to the continuous system,the semi-discrete systems are proved to be uniformly exponentially stable by means of the indirect Lyapunov method.Numerical simulations illustrate why the classical semi-discrete scheme does not preserve the uniformly exponential stability while the order reduction semi-discrete scheme does.Then,we consider stability of the one-dimensional wave equation with tip mass under non-collocated control.First,an observer-based output feedback control is designed to obtain a closed-loop system consisting of the coupled same type of PDEs and ODEs.The exponential stability of the closed-loop system is proved by constructing a global Lyapunov function and combining the properties of the C0-semigroup.The semi-discrete finite difference scheme for the closedloop system is constructed by the reduced-order finite difference method.The proof procedure based on the continuous system proves the semi-discrete scheme to be uniformly exponentially stable.