Incremental nonlinear stability analysis of stochastic systems perturbed by Lévy noise

Abstract

We present a theoretical framework for characterizing incremental stability of nonlinear stochastic systems perturbed by either compound Poisson shot noise or finite-measure Lévy 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 toward 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 Lévy 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 Lévy-Khintchine theorem.We demonstrate both empirical and analytical computation of the error ball using several numerical examples, and illustrate how varying the parameters of the system affect the tightness of the bound.

Reduced Graphene Oxides Modified Bi2Te3 Nanosheets for Rapid Photo‐Thermoelectric Catalytic Therapy of Bacteria‐Infected Wounds

“…As shown in Figure a, all the X‐ray powder diffraction (XRD) peaks of Bi 2 Te 3 and rGO‐Bi 2 Te 3 were well matched with the characteristic peaks of Bi 2 Te 3 crystal, including (015), (110), (1010), (205), (125), (1115), (0210), (0015), and (2110). [ 16 ] Besides, in the XRD pattern of rGO, the (002) diffraction peak reappeared, while the (001) diffraction peak in the original GO disappeared, proving the successful reduction of GO. [ 17 ] In Figure 2b, the Raman spectra of rGO and rGO‐Bi 2 Te 3 indicated two main characteristic peaks of rGO, which was ascribed to the D ‐Band at 1353 cm −1 and G ‐Band at 1592 cm −1 .…”

“…The O 1s spectra (Figure 2f) displayed only carbonoxygen peaks (CO) in GO, but tellurium oxygen peak (TeO) and tellurium hydroxyl peak (TeOH) at 533.6 and 535.3 eV in rGO‐Bi 2 Te 3 , which might be related to the oxidation of tellurium atoms on the surface of rGO‐Bi 2 Te 3 . [ 16c,23 ]…”

“…Finally, the Bi 2 Te 3 nanosheets were dispersed in an isopropyl alcohol solution and used for further experiments. [ 16a ]…”

An Overview of Soft Robotics

“…This method involves linearizing the nonlinear system dynamics of soft robots in a high-dimensional latent state space so that the linear dynamics can be used for control strategies. This concept has been extended to model and control stochastic system dynamics using a deep stochastic Koopman operator (DeSKO) (101) with a stability-assured control method for soft robots.…”

“…In soft robotics, proprioception provides information about the current state of a soft robot, which is crucial for proper feedback control. Motion capture systems are often used to measure soft robots' posture, especially in research works that deal with modeling and control (70,90,95,96,100,101). In this method, reflective markers are placed along soft robots' bodies and detected by infrared cameras, enabling real-time ground-truth measurements of robots' posture.…”