Mean square stability with general decay rate of nonlinear neutral stochastic function differential equations in the $ G $-framework

<abstract><p>Few results seem to be known about the stability with general decay rate of nonlinear neutral stochastic function differential equations driven by $ G $-Brownain motion ($ G $-NSFDEs in short). This paper focuses on the $ G $-NSFDEs, and the coefficients of these considered $ G $-NSFDEs can be allowed to be nonlinear. It is first proved the existence and uniqueness of the global solution of a $ G $-NSFDE. It is then obtained the trivial solution of the $ G $-NSFDE is mean square stable with general decay rate (including the trivial solution of the $ G $-NSFDE is mean square exponentially stable and the trivial solution of the $ G $-NSFDE is mean square polynomially stable) by $ G $-Lyapunov functions technique. In this paper, auxiliary functions are used to dominate the Lyapunov function and the diffusion operator. Finally, an example is presented to illustrate the obtained theory.</p></abstract>

Numerical Spline Method for Simulation of Stochastic Differential Equations systems: طريقة شرائحية عددية لمحاكاة حل نظم من المعادلات التفاضلية العشوائية

In this paper, numerical spline method is presented with collocation two parameters for solving systems of multi-dimensional stochastic differential equations (SDEs). Multi-Wiener's time-continuous process is simulated as a discrete process, and then the mean-square stability of proposed method when applied to a system of two-dimensional linear SDEs is studied. The study shows that the method is mean-square stability and third-order convergent when applied to a system of linear and nonlinear SDEs. Moreover, the effectiveness of our method was tested by solving two test linear and non-linear problems. The numerical results show that the accuracy and applicability of the proposed method are worthy of attention.

Sign Stability of Dual Switching Linear Continuous-Time Positive Systems

This paper investigates 1-moment exponential stability and exponential mean-square stability (EMS stability) under average dwell time (ADT) and the preset deterministic switching mechanism of dual switching linear continuous-time positive systems when a numerical realization does not exist. The signs of subsystem matrices, but not their structures of magnitude, are key information that causes a qualitative concept of stability called sign stability. Both 1-moment exponential stability and EMS stability, which are the traditional stability concepts, are generalized intrinsically. Hence, both 1-moment exponential sign stability and EMS sign stability are introduced and are proven based on sign equivalency. It is shown that they are symmetrically and qualitatively stable. Notably, the notion of stability can be checked quantitatively using some examples.

Statistical-mechanical analysis of adaptive filter with clipping saturation-type nonlinearity

Adaptive signal processing is used in broad areas. In most practical adaptive systems, there exists substantial nonlinearity that cannot be neglected. In this paper, we analyze the behaviors of an adaptive system in which the output of the adaptive filter has the clipping saturation-type nonlinearity by a statistical-mechanical method. To represent the macroscopic state of the system, we introduce two macroscopic variables. By considering the limit in which the number of taps of the unknown system and adaptive filter is large, we derive the simultaneous differential equations that describe the system behaviors in the deterministic and closed form. Although the derived simultaneous differential equations cannot be analytically solved, we discuss the dynamical behaviors and steady state of the adaptive system by asymptotic analysis, steady-state analysis, and numerical calculation. As a result, it becomes clear that the saturation value S has the critical value SC at which the mean-square stability of the adaptive system is lost. That is, when S > SC, both the mean-square error (MSE) and mean-square deviation (MSD) converge, i.e., the adaptive system is mean-square stable. On the other hand, when S < SC, the MSD diverges although the MSE converges, i.e., the adaptive system is not mean-square stable. In the latter case, the converged value of the MSE is a quadratic function of S and does not depend on the step size. Finally, SC is exactly derived by asymptotic analysis.<br>

Statistical-mechanical analysis of adaptive filter with clipping saturation-type nonlinearity

Adaptive signal processing is used in broad areas. In most practical adaptive systems, there exists substantial nonlinearity that cannot be neglected. In this paper, we analyze the behaviors of an adaptive system in which the output of the adaptive filter has the clipping saturation-type nonlinearity by a statistical-mechanical method. To represent the macroscopic state of the system, we introduce two macroscopic variables. By considering the limit in which the number of taps of the unknown system and adaptive filter is large, we derive the simultaneous differential equations that describe the system behaviors in the deterministic and closed form. Although the derived simultaneous differential equations cannot be analytically solved, we discuss the dynamical behaviors and steady state of the adaptive system by asymptotic analysis, steady-state analysis, and numerical calculation. As a result, it becomes clear that the saturation value S has the critical value SC at which the mean-square stability of the adaptive system is lost. That is, when S > SC, both the mean-square error (MSE) and mean-square deviation (MSD) converge, i.e., the adaptive system is mean-square stable. On the other hand, when S < SC, the MSD diverges although the MSE converges, i.e., the adaptive system is not mean-square stable. In the latter case, the converged value of the MSE is a quadratic function of S and does not depend on the step size. Finally, SC is exactly derived by asymptotic analysis.<br>

The exponential behavior and stabilizability of quasilinear parabolic stochastic partial differential equation

In this paper, we study the stability of quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone. The exponential mean square stability and pathwise exponential stability of the solutions are established. Moreover, under certain hypothesis on the stochastic perturbations, pathwise exponential stability can be derived, without utilizing the mean square stability.

Exponential mean-square stability properties of stochastic linear multistep methods

The aim of this paper is the analysis of exponential mean-square stability properties of nonlinear stochastic linear multistep methods. In particular it is known that, under certain hypothesis on the drift and diffusion terms of the equation, exponential mean-square contractivity is visible: the qualitative feature of the exact problem is here analysed under the numerical perspective, to understand whether a stochastic linear multistep method can provide an analogous behaviour and which restrictions on the employed stepsize should be imposed in order to reproduce the contractive behaviour. Numerical experiments confirming the theoretical analysis are also given.