Lyapunov Functions can Exactly Quantify Rate Performance of Nonlinear Differential Equations

Pointwise-in-time stability notions for Ordinary Differential Equations (ODEs) provide quantitative metrics for system performance by establishing bounds on the rate of decay of the system state in terms of initial condition -- allowing stability to be quantified by e.g. the maximum provable decay rate. Such bounds may be obtained by finding suitable Lyapunov functions using, e.g. Sum-of-Squares (SOS) optimization. While Lyapunov tests have been proposed for numerous pointwise-in-time stability notions, including exponential, rational, and finite-time stability, it is unclear whether these characterizations are able to provide accurate bounds on system performance. In this paper, we start by proposing a generalized notion of rate performance -- with exponential, rational, and finite-time decay rates being special cases. Then, for any such notion and rate, we associate a Lyapunov condition which is shown to be necessary and sufficient for a system to achieve that rate. Finally, we show how the proposed conditions can be enforced using SOS programming in the case of exponential, rational, and finite-time stability. Numerical examples in each case demonstrate that the corresponding SOS test can achieve tight bounds on the rate performance with accurate inner bounds on the associated regions of performance.