**Delfim F. M. Torres**

## A Noether Theorem on Unimprovable
Conservation Laws for Vector-Valued

Optimization Problems in Control Theory

**Abstract:**

We obtain a version of Noether's invariance theorem for optimal control problems
with a finite number of cost functionals. The result is obtained by formulating
E. Noether's result for optimal control problems subject to isoperimetric
constraints, and then using the unimprovable (Pareto) notion of optimality. It
was A. Gugushvili who drew the author's attention to a result of this kind that
was posed as an open mathematical question of a great interest in applications
of control engineering.

**Keywords:**

Multicriteria optimal control systems, Noether symmetry principle, optimization
with vector-valued cost, necessary conditions for unimprovable-Pareto-optimality,
isoperimetric constraints.

**MSC 2000:** 49K15, 49N99, 93C10