Department of Mathematics

University of Aveiro

3810-193 Aveiro, PORTUGAL

E-mail: delfim@mat.ua.pt

**The role of symmetry in the regularity properties of
optimal controls**

**Abstract:**

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.