Incremental Nonlinear Stability Analysis of Stochastic Systems Perturbed by L\'{e}vy Noise

TL;DR

This paper develops a theoretical framework to analyze the incremental stability of nonlinear stochastic systems affected by Le9vy noise, providing bounds on trajectory deviations and demonstrating results through numerical case studies.

Contribution

It introduces a novel stability analysis framework for systems with compound Poisson and Le9vy noise, linking stability properties to noise parameters and system dynamics.

Findings

Mean-squared error between trajectories converges exponentially to a bounded error ball.

Convergence rate matches that of the nominal system with exponential stability.

Error bounds for Le9vy noise are additive based on shot and white noise components.

Abstract

We present a theoretical framework for characterizing incremental stability of nonlinear stochastic systems perturbed by compound Poisson shot noise and finite-measure L\'{e}vy noise. For each noise type, we compare trajectories of the perturbed system with distinct noise sample paths against trajectories of the nominal, unperturbed system. We show that for a finite number of jumps arising from the noise process, the mean-squared error between the trajectories exponentially converge towards a bounded error ball across a finite interval of time under practical boundedness assumptions. The convergence rate for shot noise systems is the same as the exponentially-stable nominal system, but with a tradeoff between the parameters of the shot noise process and the size of the error ball. The convergence rate and the error ball for the L\'{e}vy noise system are shown to be nearly direct sums of the respective quantities for the shot and white noise systems separately, a result which is analogous to the L\'{e}vy-Khintchine theorem. We demonstrate our results using several numerical case studies.