Almost sure asymptotic stabilizability for deterministic systems with wiener processes

Let dx(t) = y(t) dt, where y(t) is a one-dimensional Wiener process. In this note, we obtain a formula for the moment-generating function of y(T), where T is the 1/2-winding time about the origin of the integrated Wiener process x(t).

This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x > 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.

In this paper, a SIR model with two delays and general nonlinear incidence rate is considered. The local and global asymptotical stabilities of the disease‐free equilibrium are given. The local asymptotical stability and the existence of Hopf bifurcations at the endemic equilibrium are also established by analyzing the distribution of the characteristic values. Furthermore, the sufficient conditions for the permanence of the system are given. Some numerical simulations to support the analytical conclusions are carried out. At last, some conclusions are given. Copyright © 2014 John Wiley & Sons, Ltd.

The purpose of this paper is to give a survey of the results proved in Florchinger (1993) concerning the stabilizability problem for control stochastic nonlinear systems driven by a Wiener process. Sufficient conditions for the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium point of the system, are provided by means of stochastic Lyapunov-like techniques. The notion of dynamic asymptotic stability in probability of control stochastic differential systems is introduced, and the stabilization by means of dynamic controllers is studied. >

A one-dimensional Wiener plus independent Poisson control process has an integrated, discounted non-quadratic cost function with asymmetric bounds on the non-anticipative control, assumed to be a function of the current state. A Bellman equation and maximum principle for partial differential-difference equations may be used to obtain the optimal closed-loop control if some assumptions on the asymptotic behaviour of certain partial differential-difference equations are met. The finite and infinite integral cases are treated separately.

<p>This paper investigates rumor propagation in a multilingual environment, taking into account language usage variations. Firstly, a 2I2S2R model is proposed within a heterogeneous network framework that incorporates both immunologic and cross-transmitted mechanisms. Secondly, the paper calculates the basic reproduction number $ R_0 $ by the next-generation matrix method. Thirdly, the local asymptotic stability and the global asymptotic stability are further explored, which indicate that whether the rumor continuously spreads or becomes extinct is determined by the threshold. Finally, the numerical simulation and sensitivity analysis are given to illustrate the effectiveness of theoretical results and the influence of model parameters on rumor spreading.</p>

In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction–diffusion predator–prey system with mutual interference described by the Crowley–Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction–diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction–diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.

Asymptotic stabilization of nonlinear systems with long input delay via memoryless feedback: A linearization method

We study stability for dynamical systems specified by autonomous stochastic differential equations of the form dX(t) = f(X(t))dt + g(X(t))dW(t), with (X(t)) t≥0 an ‐valued Itô process and (W(t)) t≥0 an ‐valued Wiener process, and the functions and are Lipschitz and vanish at the origin, making it an equilibrium for the system. The concept of asymptotic stability in probability of the null solution is well known and implies that solutions started arbitrarily close to the origin remain close and converge to it. The concept therefore pertains exclusively to system properties local to the origin. We wish to address the matter in a more practical manner: Allowing for a (small) probability that solutions escape from the origin, how far away can they then be started? To this end we define a probabilistic version of the basin of attraction, the γ‐BOA, with the property that any solution started within it stays close and converges to the origin with probability at least γ. We then develop a method using a local Lyapunov function and a nonlocal one to obtain rigid lower bounds on γ‐BOA.

In this paper, we study the synchronization of identical nonlinear systems over large-scale network with uncertainty in the interconnections. We consider a special class of nonlinear systems which have a dissipative nonlinearity and the stability of such systems can be analyzed using absolute stability theory tools like the Positive Real Lemma, Bounded Real Lemma and dissipativity theory. We extend this analysis to the stochastic setting over a network where the interconnection weights are drive by Wiener process with given mean and variance. To capture the stability of the synchronized state, we study the notion of mean square stability from stochastic stability theory and formulate a network size-independent sufficient condition based on the theory of stochastic dissipative systems. We also compute a heuristic margin of synchronization for the networked systems to indicate the tolerance to stochastic uncertainty in interconnection links.

In the present paper we first obtain the comparison principle for the nonlinear stochastic differential delay equations with Markovian switching. Later, using this comparison principle, we obtain some stability criteria, including stability in probability, asymptotic stability in probability, stability in the pth mean, asymptotic stability in the pth mean and the pth moment exponential stability of such equations. Finally, an example is given to illustrate the effectiveness of our results.

In our life there are various infectious diseases which lead to negative effects on the economy. Therefore, it is highly important to develop mathematical models to describe the spread of the pandemic. In this study, we first develop a deterministic SEI1I2R epidemic model incorporating double epidemic hypothesis. We first verify that there exists a unique global positive solution of the system with any positive initial value, then we investigate the local and global asymptotical stability of the equilibria. Following that, we incorporate the inherent stochasticity of the virus spread by reinterpreting the previous SEIR-type epidemic model of infectious diseases as stochastic systems modelled via the log-normal standard Ornstein–Uhlenbeck process. After that we study the stochastic dynamics of the model in detail. More precisely, we first analyze the existence and uniqueness of the global solution of the system. Then by using the Markov semigroup theory, we establish sufficient criteria for the existence and uniqueness of an invariant probability measure of the system, which implies the persistence of the disease in a community. Especially, under the same conditions as the local asymptotical stability of the endemic equilibrium, we get the specific form of the local probability density function around the quasi-endemic equilibrium of the stochastic system. Meanwhile, we also verify that the invariant global probability density function can be approximated by the local probability density function as the noise intensity tends to zero. In addition, we give some sufficient conditions for elimination of the disease. Finally, numerical simulations are carried out to demonstrate the validity of the derived results and the availability of the proposed models by using MATLAB. This work can improve our insight into epidemic dynamics and help us implement interventions to regulate the disease dynamics.

A class of chemical reaction networks is described with the property that each positive equilibrium is locally asymptotically stable relative to its stoichiometry class, an invariant subspace on which it lies. The reaction systems treated are characterized primarily by the existence of a certain factorization of their stoichiometric matrix and strong connectedness of an associated graph. Only very mild assumptions are made about the rates of reactions, and, in particular, mass action kinetics are not assumed. In many cases, local asymptotic stability can be extended to global asymptotic stability of each positive equilibrium relative to its stoichiometry class. The results are proved via the construction of Lyapunov functions whose existence follows from the fact that the reaction networks define monotone dynamical systems with increasing integrals.

In this paper, a delayed SEIR dynamic model with relapse and the nonlinear incidence rate is considered. The basic reproductive rate, R0, is derived and the existence of equilibria is established. It has been shown that if R0 < 1, the disease-free equilibrium is locally asymptotically stable and the disease dies out. The effect of the time delay on stabilities of the endemic equilibrium has been studied. If R0 > 1, the unique endemic equilibrium is local asymptotical stability when time delays are small enough, and when increasing of time delay, it has been shown that the existence of Hopf bifurcations by analyzing the root distribution of the transcendental characteristic equation. Moreover, a sufficient condition has been obtained to establish the local stability properties of the endemic equilibrium. At the end of the paper, some numerical simulations have been presented to confirm the correctness of theoretical analyses.

New results on stability of random coupled systems on networks with Markovian switching