Conservation laws in optimal control theory

Abstract

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's theorem are given, in the sense which enlarges the scope of its application. The dissertation looks at extending the second Noether's theorem to optimal control problems which are invariant under symmetry depending upon k arbitrary functions of the independent variable and their derivatives up to some order m. Furthermore, we look at the Conservation Laws, i.e. conserved quantities along Euler-Lagrange extremals, which are obtained on the basis of Noether's theorem. And finally we obtain a generalization of Noether's theorem for optimal control problems. The generalization involves a one-parameter family of smooth maps which may depend also on the control and a Lagrangian which is invariant up to an addition of an exact differential.