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首页> 外文会议>2nd IFAC (International Federation of Automatic Control) Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control 2003 Apr 3-5, 2003 Seville, Spain >A PROPER EXTENSION OF NOETHER'S SYMMETRY THEOREM FOR NONSMOOTH EXTREMALS OF THE CALCULUS OF VARIATIONS
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A PROPER EXTENSION OF NOETHER'S SYMMETRY THEOREM FOR NONSMOOTH EXTREMALS OF THE CALCULUS OF VARIATIONS

机译：变量演算的非光滑极值的Noet对称性定理的正确推广

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摘要

For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations.
机译：对于不光滑的Euler-Lagrange极值，Noether的守恒定律不再有效。我们证明，只要将Euler-Lagrange极值限制为满足DuBois-Reymond必要条件的极值，在更广泛的Lipschitz函数类中，艾美奖Noether的微积分定理仍然有效。在光滑情况下，所有Euler-Lagrange极值都是DuBois-Reymond极值，其结果适当地扩展了经典的Noether定理。这与Noether对称定理在最佳控制设置方面的最新发展形成了鲜明对比，当针对变化微积分的问题指定时，对称定理导致了不正确的扩展。

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