CONSERVED QUANTITIES ALONG THE PONTRYAGIN EXTREMALS OF QUASI-INVARIANT OPTIMAL CONTROL PROBLEMS D. F. M. Torres Departamento de Matem´ atica Universidade de Aveiro 3810-193 Aveiro, Portugal fax: +351 234382014 e-mail: delfim@mat.ua.pt Keywords: optimal control, Pontryagin maximum principle, Noether theorem, conservation laws, in- variance up to first-order terms in the parameters. 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 incor- porates the possibility to consider a family of trans- formations depending on several parameters and, what is more important, to deal with quasi-invariant and not necessarily invariant optimal control prob- lems. We trust that this latter extension provides new possibilities and we illustrate it with several ex- amples, not covered by the previous known optimal control versions of Noether’s theorem. 1 Introduction The work on invariant problems of the calculus of variations was initiated in 1918 by Emmy Noether, in the gorgeous paper [13, 14]. The universal prin- ciple described by Noether’s theorem (see e.g. [5, pp. 262–266] or [18, §4.3.]), asserts that invariance of the integral functionals with respect to a family of transformations result in existence of a certain conservation law or equivalently the first integral of the corresponding Euler-Lagrange differential equa- tions. This first integral is computed in terms of the Lagrangian and the family of transformations. This result is of great importance in physics, engi- neering, systems and control and their applications (see [17, 9, 12, 1]). One important application of the Noether theorem is, for example, to the n-body problem. For a discussion of this problem, and inter- pretation of the respective first integrals from invari- ance under Galilean transformations and application of Noether’s theorem, we refer the reader to [11] and [7, pp. 190–192] or [9, Ch. 2]. In the optimal control setting, the relation between invariance of a problem and the existence of ex- pressions which are constant along any of its ex- tremals, has been obtained following the classical Noether’s approach based on the transversality con- ditions 1 (see [3, 4] and references therein). Using the original paper of Emmy Noether [13, 14] and the more simpler and direct approach of Andrzej Traut- man [23], Hanno Rund [16] (see also [10]) and John David Logan [9] for insight and motivation, exten- sions to the previous known optimal control versions of Noether’s theorem were obtained by the present author in [19, 21, 22]. Here we extend the very con- cept of invariance (Definition 3.1), allowing several parameters and equalities up to first-order terms in the parameters. This extension allows one to for- 1 In the calculus of variations, transversality conditions are expressed by the so called general variation of the functional (see e.g. [6, Sec. 13] or [7, p. 185]). Proceedings of the 10th Mediterranean Conference on Control and Automation - MED2002 Lisbon, Portugal, July 9-12, 2002.