Almost periodic parameters for the second moment stability of linear stochastic systems

The technical note presents conditions to assure the uniform second moment stability for a class of linear time-varying stochastic systems. The system parameters are assumed to be almost periodic, a concept that is weaker than the periodic one. Under the existence of the long-run average cost associated with the stochastic system, we apply the almost periodicity to prove the desired stability result. An application illustrates the usefulness of the approach by implementing an almost periodic state-feedback strategy to control the velocity of a DC motor device

Exponential Stabilisation of Continuous-time Periodic Stochastic Systems by Feedback Control Based on Periodic Discrete-time Observations

Since Mao in 2013 discretised the system observations for stabilisation problem of hybrid SDEs (stochastic differential equations with Markovian switching) by feedback control, the study of this topic using a constant observation frequency has been further developed. However, time-varying observation frequencies have not been considered. Particularly, an observational more efficient way is to consider the time-varying property of the system and observe a periodic SDE system at the periodic time-varying frequencies. This study investigates how to stabilise a periodic hybrid SDE by a periodic feedback control, based on periodic discrete-time observations. This study provides sufficient conditions under which the controlled system can achieve pth moment exponential stability for p > 1 and almost sure exponential stability. Lyapunov's method and inequalities are main tools for derivation and analysis. The existence of observation interval sequences is verified and one way of its calculation is provided. Finally, an example is given for illustration. Their new techniques not only reduce observational cost by reducing observation frequency dramatically but also offer flexibility on system observation settings. This study allows readers to set observation frequencies according to their needs to some extent

Finite-size and correlation-induced effects in Mean-field Dynamics

The brain's activity is characterized by the interaction of a very large number of neurons that are strongly affected by noise. However, signals often arise at macroscopic scales integrating the effect of many neurons into a reliable pattern of activity. In order to study such large neuronal assemblies, one is often led to derive mean-field limits summarizing the effect of the interaction of a large number of neurons into an effective signal. Classical mean-field approaches consider the evolution of a deterministic variable, the mean activity, thus neglecting the stochastic nature of neural behavior. In this article, we build upon two recent approaches that include correlations and higher order moments in mean-field equations, and study how these stochastic effects influence the solutions of the mean-field equations, both in the limit of an infinite number of neurons and for large yet finite networks. We introduce a new model, the infinite model, which arises from both equations by a rescaling of the variables and, which is invertible for finite-size networks, and hence, provides equivalent equations to those previously derived models. The study of this model allows us to understand qualitative behavior of such large-scale networks. We show that, though the solutions of the deterministic mean-field equation constitute uncorrelated solutions of the new mean-field equations, the stability properties of limit cycles are modified by the presence of correlations, and additional non-trivial behaviors including periodic orbits appear when there were none in the mean field. The origin of all these behaviors is then explored in finite-size networks where interesting mesoscopic scale effects appear. This study leads us to show that the infinite-size system appears as a singular limit of the network equations, and for any finite network, the system will differ from the infinite system

Dynamical Models of Extreme Rolling of Vessels in Head Waves

Rolling of a ship is a swinging motion around its length axis. In particular vessels transporting containers may show large amplitude roll when sailing in seas with large head waves. The dynamics of the ship is such that rolling interacts with heave being the motion of the mass point of the ship in vertical direction. Due to the shape of the hull of the vessel its heave is influenced considerably by the phase of the wave as it passes the ship. The interaction of heave and roll can be modeled by a mass-spring-pendulum system. The effect of waves is then included in the system by a periodic forcing term. In first instance the damping of the spring can be taken infinitely large making the system a pendulum with an in vertical direction periodically moving suspension. For a small angular deflection the roll motion is then described by the Mathieu equation containing a periodic forcing. If the period of the solution of the equation without forcing is about twice the period of the forcing then the oscillation gets unstable and the amplitude starts to grow. After describing this model we turn to situation that the ship is not anymore statically fixed at the fluctuating water level. It may move up and down showing a motion modeled by a damped spring. One step further we also allow for pitch, a swinging motion around a horizontal axis perpendicular to the ship. It is recommended to investigate the way waves may directly drive this mode and to determine the amount of energy that flows along this path towards the roll mode. Since at sea waves are a superposition of waves with different wavelengths, we also pay attention to the properties of such a type of forcing containing stochastic elements. It is recommended that as a measure for the occurrence of large deflections of the roll angle one should take the expected time for which a given large deflection may occur instead of the mean amplitude of the deflection

Concepts for a theoretical and experimental study of lifting rotor random loads and vibrations. Phase 6-A: Effects of blade torsion, of blade flap bending flexibility and of rotor support flexibility on rotor stability and random response

The effects of lifting rotor blade torsion, blade flap bending flexibility and rotor support flexibility on rotor stability and random response are described. The subjects discussed are: (1) blade representation and method of analysis, (2) random gust response statistics for coupled torsion-flapping rotor blade vibrations, (3) flap bending corrections to the rigid blade analysis of lifting rotors, and (4) effects of rotor support flexibility. The response of linear periodically time varying systems to random excitation is examined

Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions

In this manuscript we analyze the collective behavior of mean-field limits of large-scale, spatially extended stochastic neuronal networks with delays. Rigorously, the asymptotic regime of such systems is characterized by a very intricate stochastic delayed integro-differential McKean-Vlasov equation that remain impenetrable, leaving the stochastic collective dynamics of such networks poorly understood. In order to study these macroscopic dynamics, we analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics and sigmoidal interactions. In that case, we prove that the solution of the mean-field equation is Gaussian, hence characterized by its two first moments, and that these two quantities satisfy a set of coupled delayed integro-differential equations. These equations are similar to usual neural field equations, and incorporate noise levels as a parameter, allowing analysis of noise-induced transitions. We identify through bifurcation analysis several qualitative transitions due to noise in the mean-field limit. In particular, stabilization of spatially homogeneous solutions, synchronized oscillations, bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow further exploring the role of noise in the nervous system.Comment: Updated to the latest version published, and clarified the dependence in space of Brownian motion