Quasi-invariant optimal control problems

Delfim F.M. Torres

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 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.

Keywords: optimal control; Pontryagin maximum principle; Noether theorem; conservation laws; invariance up to first-order terms in the parameters.

Classification (MSC2000): 49K15.

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