Fixed-time stability of ODE and fixed-time stability of neural network

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 and consideration of dual...

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Bibliographic Details
Published in:International journal of control 2021-12, Vol.94 (12), p.3332-3338
Main Authors: Michalak, Anna, Nowakowski, Andrzej
Format: Article
Language:English
Subjects:
dual fixed time stability
dual Lyapunov stability
fixed-time stability
Liapunov functions
neural network
Neural networks
Stability
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Summary: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 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.
ISSN:0020-7179
1366-5820
DOI:10.1080/00207179.2020.1763469