Uniform stabilization of a nonlinear beam by nonlinear boundary feedback

We consider the planar motion of a uniform prismatic beam of length L. We want to derive a model that reflects the effect of stretching on bending, which necessarily leads to nonlinear partial differential equations for the motion of the beam. We will, however, assume that the constitutive equations for bending are linear. This is in agreement with existing engineering literature (see, for example [S] and the bibliography therein). It should be remarked that the effect of stretching on bending becomes significant if, in particular, a rigid rotation is superimposed on the motion. We do not consider such a rotation here, even though it could be handled within the present framework. We assume that the beam, in its reference state, occupies the region described in rectangular coordinates by