Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods

Explicit numerical methods have a great advantage in computational cost, but they usually fail to preserve the conserved quantity of original stochastic differential equations (SDEs). In order to overcome this problem, two improved versions of explicit stochastic Runge–Kutta methods are given such that the improved methods can preserve conserved quantity of the original SDEs in Stratonovich sense. In addition, in order to deal with SDEs with multiple conserved quantities, a strategy is represented so that the improved methods can preserve multiple conserved quantities. The mean-square convergence and ability to preserve conserved quantity of the proposed methods are proved. Numerical experiments are implemented to support the theoretical results.

Formation Control of Non-Holonomic Mobile Robots Moving on Slippery Surfaces

In this work, we analyze the decentralized formation control problem for a class of multi-robotic systems evolving on slippery surfaces. Grounded on experimental data of robots moving on a gravel surface inducing slippery, we show that a deterministic model cannot capture the uncertainties resulting from the kinematics of the robots while, instead, a model incorporating stochastic noise is capable of emulating such perturbations on wheel driving speed and turn rate. To account for these uncertainties, we consider a second order non-holonomic unicycle model to capture the full dynamics of individual vehicles where both actuation force and torque are subject to stochastic disturbances. Upon reducing the input-output dynamics of individual robot to a stochastic double integrator, we investigate the effects of these perturbations on the control input using concepts from stochastic stability theory and through numerical simulations. We demonstrated the applicability of the proposed scheme for formation control notably by providing sufficient conditions for exponential mean square convergence and we numerically determined the range of noise intensities for which team of robots can achieve formation stabilization. The promising findings from this work are expected to aid the design of robust control schemes for formation control of non-holonomic robots on off-road or un-paved surfaces.

Spherically Restricted Random Hyperbolic Diffusion

This paper investigates solutions of hyperbolic diffusion equations in R 3 with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere S 2 are studied. All assumptions are formulated in terms of the angular power spectrum or the spectral measure of the random initial conditions. Approximations to the exact solutions are given. Upper bounds for the mean-square convergence rates of the approximation fields are obtained. The smoothness properties of the exact solution and its approximation are also investigated. It is demonstrated that the Hölder-type continuity of the solution depends on the decay of the angular power spectrum. Conditions on the spectral measure of initial conditions that guarantee short- or long-range dependence of the solutions are given. Numerical studies are presented to verify the theoretical findings.