Representation of Feedback Operators for Hyperbolic Systems

Abstract

The purpose of this article is to extend the representation theorem in [1] and [7] to certain classes of damped hyperbolic systems. The original motivation for our study of hyperbolic systems comes from the work by Lupi, Chun, and Turner [8]. The approach in [8] is interesting because they make no prior assumptions regarding the form of the controls and actuators so that the gains operators produced by an optimal design could be used to make decisions about where actuators and sensors are best placed. In particular, in [8] it was assumed that the input operator was the identity and the elastic system was not damped. By solving the LQR problem with the input operator equal to the identity, one can gain insight into the type and location of practical distributed controllers for structural control. This insight comes from explicit knowledge of the kernels (so called functional gains) that describe the integral representations of feedback gain operators. Even with no damping the LQR problem has a solution since the input operator is the identity (the system is exactly controllable). However, as we see below the problem with no damping is extremely complex. Basic questions concerning the existence and smoothness of functional gains remain open and yet these issues are important in the applications proposed in [8]. Consequently, as a first step we take the middle ground and consider damped systems with distributed control.