Active Control of Structures

In recent years a large large amount of work on the active control of the flexible structures has been done. The concrete engineering problems that motivate this study come from very different fields: aerospace, structures, meteorology, nanotechnology, etc. An overview can be found in the book of Fuller, Elliot and Nelson [5]. The problem of active vibration control (AVC) can be set in the framework of mathematical theory of optimal control as stated in the book of Lions [9]. It corresponds to a minimization problem governed by a partial differential equation. The numerical analysis of this kind of problems is an area of active research (see [6], [7] and references therein). In this project we study the numerical approximation of two problems of AVC applied to a wave equation following the structure presented in [6]. The goal of these problems is the reduction of the vibration in two types of sensors: pointwise and distributed. The control variable is given by the real amplitudes of the secondary forces, which are modeled by Dirac’s deltas. In this way, the problems consist of choosing the optimal real amplitudes of these secondary forces. Since our interest is the computational implementation we study an approximation of the wave equation in the frequency domain, then we study the approximation of the optimal control for two problems of AVC.