The role of symmetry in the regularity properties of optimal controls
The role of symmetry is well studied in physics and economics, where many great contributions have been made. With the help of Emmy Noether's celebrated theorems, a unified description of the subject can be given within the mathematical framework of the calculus of variations. It turns out that Noether's principle can be understood as a special application of the Euler-Lagrange differential equations. We claim that this modification of Noether's approach, has the advantage to put the role of symmetry on the basis of the calculus of variations, and in a key position to give answers to some fundamental questions. We will illustrate our point with the interplay between the concept of invariance, the theory of optimality, Tonelli existence conditions and the Lipschitzian regularity of minimizers for the autonomous basic problem of the calculus of variations. We then proceed to the general nonlinear situation, by introducing a concept of symmetry for the problems of optimal control, and extending the results of Emmy Noether to the more general framework of Pontryagin's maximum principle. With such tools, new results regarding Lipschitzian regularity of the minimizing trajectories for optimal control problems with nonlinear dynamics are obtained.