Well-Posedness of Systems of Linear Elasticity with Dirichlet Boundary Control and Observation

Abstract

An open-loop system of linear elasticity with Dirichlet boundary control and collocated observation is considered. The main result we obtained states that this system is well-posed in the sense of Salamon. The result deduces the exponential stability of the closed-loop system under proportional output feedback. Hence it answers positively an open question proposed by Liu and Krstić [IMA J. Appl. Math., 65 (2000), pp. 109–121]. Moreover, the well-posedness, together with the regularity of the open-loop system in the sense of Weiss that was obtained in our separate paper [S. G. Chai and B. Z. Guo, Feedthrough Operator for Linear Elasticity System with Boundary Control and Observation, Preprint, School of Computational and Applied Mathematics, University of the Witwatersrand, South Africa, 2008], makes this infinite-dimensional system parallel in many ways to a linear finite-dimensional system in the framework of well-posed and regular linear infinite-dimensional systems.