Finite-time stability for Caputo–Hadamard type fractional differential systems without and with proportional delays

As a regularization of the Hadamard type fractional derivative and a natural extension of the Caputo-Hadamard fractional derivative, the Caputo-Hadamard type fractional derivative exhibits exceptional compatibility, serving as a tractable tool for precise characterization of ultra-slow varying dynamical processes. Compared with Lyapunov stability within the framework of an infinite-time horizon, achieving prescribed performance in finite-time is imperative for practical applications. Herein, this paper concentrates on the finite-time stability of Caputo-Hadamard type fractional differential systems [C-HTFDSs] under two scenarios: systems without delays and systems with proportional delays. To achieve this, for both linear (homogeneous/nonhomogeneous) and nonlinear cases without time delays, the finite-time stability criteria are established leveraging a modified Laplace transform technique and an adaptive fractional Gronwall type inequality, respectively. Then, with regard to the homogeneous and nonhomogeneous linear C-HTFDSs with proportional delays, two novel proportional delayed Mittag-Leffler matrix functions are designed separately, leading to the sound formulations of their fundamental solutions. Finally, as to the nonlinear C-HTFDS with proportional delay, a compatible proportional retarded fractional Gronwall type inequality with two integral terms is constructed and demonstrated in detail. Not only that, several indispensable numerical simulations are implemented to validate the effectiveness and practicality of the theoretical findings.