Attraction and Stochastic Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Respect to Semimartingales

Abstract

In this paper, we will establish new results on the attraction for solutions to stochastic functional differential equations with respect to semimartingale. Most of the existing results stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in the study of attraction. Moreover, from our results on the attraction follow several new criteria on almost surely asymptotic stability and boundedness of the solutions.     

Stability in Probability of Nonlinear Stochastic Volterra Difference Equations with Continuous Variable

Abstract

The general method of Lyapunov functionals construction, that was proposed by Kolmanovskii and Shaikhet and successfully used already for functional-differential equations, difference equations with discrete time, difference equations with continuous time, and is used here to investigate the stability in probability of nonlinear stochastic Volterra difference equations with continuous time. It is shown that the investigation of the stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to investigation of the asymptotic mean square stability of the linear part of this equation.     

Robust Control of Markovian Switching Stochastic Systems with Noise Dependent States and Inputs

A problem of state feedback stabilization of discrete-time stochastic processes under Markovian switching and random diffusion (noise) is considered. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the rate vector and the diffusion term. Sufficient conditions based on linear matrix inequalities (LMI's) for stochastic stability is obtained. The robustness results of such stability concept against all admissible uncertainties are also investigated. An example is given to demonstrate the obtained results.     

An H2-Type Norm of a Discrete-Time Linear Stochastic System with Periodic Coefficients Simultaneously Affected by an Infinite Markov Chain and Multiplicative White Noise Perturbations

This article investigates the problem of the definition and computation of an H₂-type norm for discrete-time time-varying periodic stochastic linear systems simultaneously affected by multiplicative white noise perturbations and random jumping according to a Markov chain with an infinite countable number of states. Also, we solve an optimization problem that contains, as a special case, the H₂ optimal control problem for the considered class of stochastic systems under the assumption of perfect state measurements.     

Modeling treatment of cancer using virotherapy with generalized logistic growth of tumor cells

Abstract

Virotherapy is an effective strategy in cancer treatment. It eliminates tumor cells without harming the healthy cells. In this article, a deterministic mathematical model to understand the dynamics of tumor cells in response to virotherapy is formulated and analyzed by incorporating cytotoxic T lymphocytes (CTLs). The basic reproduction number and the immune response reproduction number are computed and different equilibria of the proposed model are found. The local stability of different equilibria is discussed in detail. Further, the proposed model is extended to stochastic model. Numerical simulation is performed for both deterministic and stochastic models. It is observed that when both the reproduction numbers are greater than one, which corresponds to existence of unique nontrivial equilibrium point, dynamics of deterministic and stochastic models are almost same. The deterministic model shows a very complex dynamics when one or both the reproduction numbers are below one. The system exhibits both backward bifurcation and Hopf-bifurcation for suitable sets of parameters and in this situation it is not easy to predict the dynamics of cancer cells and virus particles. The existence of backward bifurcation demonstrates the fact that partial success of virotherapy can be achieved even if the immune response reproduction number is less than one.     

Stability of random impulsive coupled systems on networks with Markovian switching

Abstract

This article is intended to study global asymptotical stability in probability for random impulsive coupled systems on networks with Markovian switching. Two cases are considered. (1) Continuous dynamics are stable while impulses are unstable; (2) impulses are stable while continuous dynamics are unstable. To begin with, based on Lyapunov method as well as graph-theoretic technique, several new stability criteria in two cases are derived, that are, the Lyapunov-type criteria and the coefficients-type criteria. Then main results are used for a class of random impulsive coupled oscillators. Finally, the effectiveness of the obtained results is verified by numerical simulations.     

On the Recursive Sequence x_{n + 1} = {{ \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {Cx_{n-1} + Dx_{n-2}}}

We investigate the global character of solutions of the equation in the title with positive parameters and positive initial conditions. We obtain results about the global attractivity of the equilibrium, the existence and attractivity of the period-two solution and the semicycles.     

The Asymptotic Stability of x(n + k) + ax(n) + bx(n-l) = 0

We established necessary and sufficient conditions for the asymptotic stability of the difference equation where the coefficients a and b are real numbers and k and l are nonnegative integers.