Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time. In this paper, we address finite time stability of discrete-time dynamical systems. Specifically, we show that finite time stability leads to uniqueness of solutions in forward time. Furthermore, we provide Lyapunov and converse Lyapunov theorems for finite-time stability of discrete autonomous systems involving scalar difference fractional inequalities and minimum operators. In addition, lower semicontinuity of the settling-time function capturing the finite settling time behavior of the dynamical system is studied and illustrated through several examples. In particular, it is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov theorems for finite time stability of discrete-time systems can only assure the existence of lower semicontinuous Lyapunov functions.

有限时间稳定性涉及动力学系统，其轨迹在有限时间内收敛到Lyapunov稳定平衡状态。在本文中，我们讨论了离散时间动力系统的有限时间稳定性。具体来说，我们证明了有限的时间稳定性会导致正向时间解的唯一性。此外，我们提供了Lyapunov和逆Lyapunov定理，用于标量差分分数不等式和最小算子的离散自治系统的有限时间稳定性。此外，还研究了捕获时间函数的较低半连续性，该函数捕获了动态系统的有限建立时间行为，并通过几个示例进行了说明。特别地，表明了李雅普诺夫函数的规律性和稳定时间函数的规律性是相关的。所以，