Fixed-Time Stability, Uniform Strong Dissipativity, and Stability of Nonlinear Feedback Systems

In this paper, we develop new necessary and sufficient Lyapunov conditions for fixed-time stability that refine the classical fixed-time stability results presented in the literature by providing an optimized estimate of the settling time bound that is less conservative than the existing results. Then, building on our new fixed-time stability results, we introduce the notion of uniformly strongly dissipative dynamical systems and show that for a closed dynamical system (i.e., a system with the inputs and outputs set to zero) this notion implies fixed-time stability. Specifically, we construct a stronger version of the dissipation inequality that implies system dissipativity and generalizes the notions of strict dissipativity and strong dissipativity while ensuring that the closed system is fixed-time stable. The results are then used to derive new Kalman–Yakubovich–Popov conditions for characterizing necessary and sufficient conditions for uniform strong dissipativity in terms of the system drift, input, and output functions using continuously differentiable storage functions and quadratic supply rates. Furthermore, using uniform strong dissipativity concepts, we present several stability results for nonlinear feedback systems that guarantee finite-time and fixed-time stability. For specific supply rates, these results provide generalizations of the feedback passivity and nonexpansivity theorems that additionally guarantee finite-time and fixed-time stability. Finally, several illustrative numerical examples are provided to demonstrate the proposed fixed-time stability and uniform strong dissipativity frameworks.