Temperamental fractional calculus extends the framework of traditional fractional calculus by introducing a tempered parameter to control the decay rate of the memory kernel, that is, the tempered parameter \( e^{-\rho (t-s)}\) can coordinate the decay rate of the memory kernel, the model established by this parameter is an effective model for describing remote interactions, abnormal diffusion and non-local effects. In this paper, based on Krasnosel’skii fixed point theorem, Hölder inequality and so on, the controllability of the tempered type fractional differential system subjected to nonlocal delay and impulsive effect is studied. Due to the presence of the attenuation kernel, we encountered some difficulties when proving the equicontinuity. We reorganized the formulas by adding and removing terms, thereby completing the proof of the equicontinuity of the operator with the attenuating kernel. During the process of proving total continuity, we will encounter another difficulty, which is the difficulty brought about by the order \(\alpha \) where \(0< \alpha < 1\) , and we have overcome this hardship by using the H \(\ddot{o}\) lder inequality, etc. Krasnosel’skii fixed point theorem cleverly combines the strengths of both the Banach contraction mapping principle (which requires a strict contraction) and Schauder’s fixed point theorem (which requires compactness and continuity). It handles operators that are neither strictly contractive nor compact/continuous on their own, but can can be decomposed into the sum of two operators, one being a contraction operator and the other being a compact operator, therefore, we adopt this method in this article. Finally, one example is given to illustrate the validity of the theory.
