Invex Energies on Riemannian Manifolds

The problems tackled in this chapter are directly or indirectly concerned with new trends in multitime optimal control. More precisley, we analyze the Riemannian convexity of energy functionals connected to the volumetric energy and the kinetic energy. The tools we use are those, not only of Riemannian geometry, but also of variational calculus and geometric dynamics. Some of the Lagrangians we discuss about are obtained by considering Euler–Lagrange prolongations of a PDE system of order one, such that, the solutions of the prolongation are ultra-potential maps. The problems solved here include: Riemannian convexity of kinetic energy functional, convex functions generated by convex functionals, invexity of kinetic and volumetric energy functionals, invexity for least square Lagrangians etc. A major result of this theory consists in relating the convexity of the energy functionals with the geometry of the underlying manifolds. The main key for our original approach is the new notion of sub-Killing vector fields on Riemannian manifolds.