Feedback Linearization and Semilinearization for Smart Elastic Structures

Abstract

This paper shows how the geometrically exact quasilinear equations of motion of non-linearly elastic and viscoelastic rods and shells whose response is sensitive to ambient magnetic, electric, or thermal fields can be converted to semilinear or linear equations by suitable feedback controls of the ambient fields. Indeed, in certain cases, the feedback can make the response of a nonhomogeneous structure be like that of a homogenous structure, enlarge or diminish the isotropy group of the structure, increase or decrease the internal dissipation in the structure, and cause naturally different wave speeds to be the same. The availability of such controls indicates that the shocks to which quasilinear hyperbolic partial differential equations for nonlinear elastic structures are susceptible need cause no difficulty in control problems. In particular, if the structure is subject to additional controls that cause it to perform specific tasks, then these additional controls are treated by the theory for (semi)linear partial differential equations, for which there is an extensive development.