Noether symmetry theorem of Herglotz type for time-delayed fractional Birkhoffian system are studied. Firstly, based on the fractional derivative of Riemann-Liouville, the Herglotz variational principle of time-delayed fractional Birkhoffian system is established, and the time-delayed Birkhoff′s equation of Herglotz type is derived. Secondly, the definition and criterion of Herglotz type Noether symmetric transformation of time-delayed fractional Birkhoffian system are established. Thirdly, the Noether′s theorem of the system is proposed and proved, in addition, the inner relationship between Noether symmetries and conservation is accurately explored. Next, the special case of the theorem is discussed, in other words, when the Herglotz generalized variational principle is reduced to the classical variational principle, the result of this paper is degraded into the Noether symmetry theorem of the time-delayed fractional Birkhoffian system. Finally, an example is given.

研究了时滞分数Birkhoffian系统的Herglotz型Noether对称定理。首先，基于Riemann-Liouville的分数导数，建立了时滞分数Birkhoffian系统的Herglotz变分原理，并推导了Herglotz型时滞Birkhoff方程。其次，建立了时滞分数Birkhoffian系统的Herglotz型Noether对称变换的定义和判据。第三，提出并证明了系统的Noether定理，此外，还精确地探索了Noether对称性与守恒性之间的内在联系。接下来，讨论定理的特殊情况，换句话说，当将Herglotz广义变分原理简化为经典变分原理时，本文的结果被降级为时滞分数Birkhoffian系统的Noether对称定理。最后给出一个例子。