Translational symmetry of certain integrals with application to the calculus of variations and optimal control

Following an earlier paper [1], translational symmetry and transformations for certain integrals are further investigated to obtain a necessary condition for translational symmetry and a partial differential equation for an augmented value function. An inclusion lemma and global optimal control theorems are derived in order to extend the method to scalar optimal control problems. In some cases, this leads to the construction of a globally contracting stationary Lyapunov function. Examples are presented to illustrate the results.