Existence of variationally defined curves with higher order elliptic Lagrangians

We present a method for proving the existence of solutions to a class of one dimensional variational problems, namely those where the Lagrangian is strongly elliptic. The method is demonstrated by some examples of optimal interpolation problems which are motivated by applications to the mechanics and control of rigid bodies. In each case the key step is to show that the variational problem satisfies the Palais–Smale condition. We do so in a general setting, showing that the number of initial conditions required depends on the higher order energy bounding properties of the Lagrangian.