Stochastic Stabilization of Dynamical Systems Using Lévy Noise

Abstract

We investigate the perturbation of the non-linear differential equation) by random noise terms consisting of Brownian motion and an independent Poisson random measure. We find conditions under which the perturbed system is almost surely exponentially stable and estimate the corresponding Lyapunov exponents.

Classical robots perturbed by Lévy processes: analysis and Lévy disturbance rejection methods

“…In [6], stochastic exponential stability of differential equations perturbed by Lévy noise that can be represented as the sum of Brownian motion and a superposition of Poisson processes is investigated. Under local Lipschitz conditions on the coefficients of the SDE, they prove zero crossing theorems as well as Martingale convergence theorems which are interpreted as strong laws of large numbers.…”

“…They then perturb the system by large jump Poisson processes and prove that the system is exponentially stable, i.e., almost sure asymptotic boundedness of log |x(t)| t . We can apply the results of [6] to our problem, since [6] deals with vector-valued processes driven by vector-valued Poisson processes. In order to do so, we must verify appropriate Lipschitz conditions.…”

“…Applebaum and Siakalli [6] presents a beautiful computation of the Lyapunov exponent lim sup t→∞ log |x(t)| t , and we need to apply it to the twolink system. Moreover in [6,7], a special form of the coefficient matrices of the Lévy noise has been chosen to prove exponential stability. We need to generalize this for general coefficients appearing in robotics.…”

Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise

“…A number of meritorious results concerning SDEs with non-Gaussian Lévy noise has been presented in existing literatures [7,8,[15][16][17]. Among them, conditions which can guarantee the existence and uniqueness of the solutions to the SDEs with non-Gaussian Lévy noise are to be assumed as the one of the most basic and important Lipschitz condition.…”

The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps

“…SDEs driven by Levy jumps have become extremely popular for modeling financial, physical and biological phenomena and many results about such equations have been studied intensively for many authors. For example, Ikeda [37], Applebaum [38] and Rong [39] are devoted to the study of such equations both in theory and their applications; Bass [40] and Albeverio [41] focused on the study about existence and uniqueness of SDEs with Poisson random measure; Oksendal [42] have studied the optimal control, optimal stopping and impulse control for jump diffusion processes; Li [43] discussed the almost sure stability of linear stochastic differential equations with jumps; Applebaum [44] investigated almost sure exponential stability and moment exponential stability of SDEs with Lévy noise; Applebaum [45] showed that perturbed system with Brownian motion and Poisson random measure is almost surely exponentially stable and estimated the corresponding Lyapunov exponents; Zhu [46] studied the asymptotic stability in the p-th moment and almost sure stability for SDEs with Lévy jump.…”