IEEE/CAA Journal of Automatica Sinica
Citation:  Z. Song and P. Li, “General Lyapunov stability and its application to timevarying convex optimization,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 11, pp. 2316–2326, Nov. 2024. doi: 10.1109/JAS.2024.124374 
In this article, a general Lyapunov stability theory of nonlinear systems is put forward and it contains asymptotic/finitetime/fast finitetime/fixedtime stability. Especially, a more accurate estimate of the settlingtime function is exhibited for fixedtime stability, and it is still extraneous to the initial conditions. This can be applied to obtain less conservative convergence time of the practical systems without the information of the initial conditions. As an application, the given fixedtime stability theorem is used to resolve timevarying (TV) convex optimization problem. By the Newton’s method, two classes of new dynamical systems are constructed to guarantee that the solution of the dynamic system can track to the optimal trajectory of the unconstrained and equality constrained TV convex optimization problems in fixed time, respectively. Without the exact knowledge of the time derivative of the cost function gradient, a fixedtime dynamical nonsmooth system is established to overcome the issue of robust TV convex optimization. Two examples are provided to illustrate the effectiveness of the proposed TV convex optimization algorithms. Subsequently, the fixedtime stability theory is extended to the theories of predefinedtime/practical predefinedtime stability whose bound of convergence time can be arbitrarily given in advance, without tuning the system parameters. Under which, TV convex optimization problem is solved. The previous two examples are used to demonstrate the validity of the predefinedtime TV convex optimization algorithms.
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