We give the sufficient conditions to fixed-time stability for nonautonomous ODEs for which the right-hand side is only Lebesgue measurable with respect to time t. Using dual tools, we present completely new approach to fixed-time stability by defining new class of functions $\mathcal{M}\mathcal{B}$ and consideration of dual Hamilton-Jacobi inequality with the use of Lyapunov function. Duality means that we move all main notions as trajectories, Lyapunov function, Hamiltion-Jacobi inequality to dual space which is determined by us. Then we study stability of the dual trajectories and next we come back to our primal space and infer suitable stability for the original problem. As an application of the main theorems we give numerical examples of fixed-time stable neural network.

我们给出了非自治常微分方程的固定时间稳定性的充分条件，其中右手边对于时间t仅是勒贝格可测的。使用双重工具，我们通过定义新的函数类来提出全新的固定时间稳定性方法$\mathcal{米}\mathcal{乙}$并考虑使用李雅普诺夫函数的对偶 Hamilton-Jacobi 不等式。二元性意味着我们将所有主要概念作为轨迹、Lyapunov 函数、Hamiltion-Jacobi 不等式移动到由我们决定的对偶空间。然后我们研究双轨迹的稳定性，接下来我们回到我们的原始空间并推断原始问题的合适稳定性。作为主要定理的应用，我们给出了固定时间稳定神经网络的数值例子。