Noether's Theorem for Control Problems on Time Scales
Agnieszka B. Malinowska, Moulay Rchid Sidi Ammi
备注：
This is a preprint of a paper whose final and definite form is published in International Journal of Difference Equations ISSN 0973-6069, Vol. 9 (2014), no. 1, 87--100
We prove a generalization of Noether's theorem for optimal control problems defined on time scales. Particularly, our results can be used for discrete-time, quantum, and continuous-time optimal control problems. The generalization involves a one-parameter family of maps which depend also on the control and a Lagrangian which is invariant up to an addition of an exact delta differential. We apply our results to some concrete optimal control problems on an arbitrary time scale.
We establish a generalization of Noether theorem for stochastic optimal control problems. Exploiting the tools of jet bundles and contact geometry, we prove that from any (contact) symmetry of the Hamilton-Jacobi-Bellman equation associated to an optimal control problem it is possible to build a related local martingale. Moreover, we provide an application of the theoretical results to Merton's optimal portfolio problem, showing that this model admits infinitely many conserved quantities in the form of local martingales.
Fractional Noether's Theorem with Classical and Caputo Derivatives: constants of motion for non-conservative systems
由于开创性的工作，埃米诺特是众所周知的，所有守恒定律在物理学，\textrm{e.g.}，守恒的能量或守恒的动量，是直接关系到不变性的行动下的家庭的转变。然而，经典的Noether定理不能给出关于非保守系统的运动常数的信息，因为经典的变分法不可能为这类系统制定物理上有意义的拉格朗日方程。另一方面，近年来，拉格朗日函数中的分数阶变分法（fractional calculus of variation）依赖于分数阶导数，已经成为研究非保守系统的一种优雅的选择。在目前的工作中，我们得到了一个推广的Noether定理的拉格朗日依赖于混合经典和Caputo衍生物，可用于获得耗散系统的运动常数。此外，我们还得到了分数阶最优控制问题的Noether条件.
In the present work, we formulate a generalization of the Noether Theorem for action-dependent Lagrangian functions. The Noether's theorem is one of the most important theorems for physics. It is well known that all conservation laws, \textrm{e.g.}, conservation of energy and momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether theorem cannot be applied to study non-conservative systems because it is not possible to formulate physically meaningful Lagrangian functions for this kind of systems in the classical calculus …
