On uniform exponential stability of semi-discrete scheme for 1-D wave equation with a tip mass
Most of the infinite-dimensional systems are described by partial differential equations(PDEs).For PDEs,discretization is most often necessarily for numerical simulation and applications.This paper considers the uniform ex-ponential 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 ex-ponential 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 spacial 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 simula-tions illustrate why the classical semi-discrete scheme does not preserve the uniformly exponential stability while the order reduction semi-discrete scheme does.
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