The Palais-Smale condition for variationally defined curves in complete Riemannian manifolds

We present a general framework based on the methods of El{\'{\i}}asson for verifying the Palais-Smale condition for a large class of one dimensional variational problems in complete Riemannian manifolds. Motivated by engineering applications, we include two examples of interpolation problems in manifolds and in each case demonstrate how our results can be used to prove the existence of solutions.