Journal of Dynamical and Control Systems, Vol. 12, No. 3, July 2006, 405–418 ( c 2006) REGULARITY OF AN EULER–BERNOULLI EQUATION WITH NEUMANN CONTROL AND COLLOCATED OBSERVATION BAO-ZHU GUO and ZHI-CHAO SHAO Abstract. This paper studies the regularity of an Euler–Bernoulli plate equation on a bounded domain of R n , n ≥ 2, with partial Neu- mann control and collocated observation. It is shown that the system is not only well posed in the sense of D. Salamon but also regular in the sense of G. Weiss. It is also shown that the corresponding feedthrough operator is zero. 1. Introduction and main result In the last fifteen years, much efforts have been concentrated on a wide class of linear infinite-dimensional systems called well-posed and regular lin- ear systems (see the surveys [5,23]). This general framework covers many partial differential equations with actuators and sensors supported at iso- lated points, on sub-domains, or on a part of the boundary of the spatial regions. The studies have shown that this class of infinite-dimensional sys- tems possesses many properties that are parallel in many ways to those of finite-dimensional systems (see, e.g., [6, 22]). The concept of regularity while very useful in this framework, rarely appears in the literature on con- trol of systems of partial differential equation [4]. In [2], the well-posedness of a wave equation with Dirichlet input and collocated output on the 2-D disk was proved by a direct method. The well-posedness of this equation on a bounded open domain of R n , n ≥ 2, with a smooth boundary was proved in [1] by the microlocal analysis. The well-posedness and regularity of a multidimensional heat equation with both Dirichlet and Neumann type boundary controls is derived in [3]. Other examples on the well-posedness and regularity of some systems od multidimensional partial differential equa- tions can be found in [17,24,25]. The regularity of a multidimensional wave 2000 Mathematics Subject Classification. 93C20, 34G10, 93D25, 35L20. Key words and phrases. Euler–Bernoulli beam, boundary control, well-posedness, regularity. This work was supported by the National Natural Science Foundation of China and the National Research Foundation. 405 1079-2724/06/0700-0405/0 c  2006 Plenum Publishing Corporation