  We study in optimal control the important relation between invariance of the
problem under a family of transformations, and the existence of preserved
quantities along the Pontryagin extremals. Several extensions of Noether
theorem are provided, in the direction which enlarges the scope of its
application. We formulate a more general version of Noether's theorem for
optimal control problems, which incorporates the possibility to consider a
family of transformations depending on several parameters and, what is more
important, to deal with quasi-invariant and not necessarily invariant optimal
control problems. We trust that this latter extension provides new
possibilities and we illustrate it with several examples, not covered by the
previous known optimal control versions of Noether's theorem.
