Vladimir Kolmanovskii

Stability of epidemic model with time delays influenced by stochastic perturbations

Many processes in automatic regulation, physics, mechanics, biology, economy, ecology etc. can be modelled by hereditary equations (see, e.g. [1–6]). One of the main problems for the theory of stochastic hereditary equations and their applications is connected with stability. Many stability results were obtained by the construction of appropriate Lyapunov functionals. In [7–11], the procedure is proposed, allowing, in some sense, to formalize the algorithm of the corresponding Lyapunov functionals construction for stochastic functional differential equations, for stochastic difference equations. In this paper, stability conditions are obtained by using this procedure for the mathematical model of the spread of infections diseases with delays influenced by stochastic perturbations.

Some peculiarities of the general method of Lyapunov functionals construction

The general method of Lyapunov functionals construction for stability investigation of stochastic hereditary systems which was proposed and developed before is considered. Some features of this method for difference systems which allow one to use the method more effectively are discussed.

States estimate of hereditary stochastic systems

Numerous phenomena in the theory of automatic regulation, mechanics, radiophysics, biology, immunology, economics, robototechnics, etc. need for theiccorrect qualitative and quantitative description to use equations with time lag. One problem which here arises is connected with states estimate of stochastic hereditary systems [1]. This paper is a survey of some results and problems of filtering, interpolation and smoothing for arbitrary Gaussian stochastic processes for the case of linear observations with delays. In particular, the estimate problems of solutions of linear stochastic differential equations with delays and integral equations of Volterra are considered. Besides new method for solution of optimal control problem with incomplete state information is adduced. The survey is based on the papers [2-14]

Asymptotic properties of the solutions for discrete Volterra equations

Conditions of the boundedness of the solutions, stability in the first approximation, and asymptotic equivalence for discrete Volterra-type equations are proven. All are formulated in terms of the characteristics of the equations, using an operator approach.

Numerical solution of some optimal control problems

Numerical solutions of the stochastic time optimal control problem and the problems with probability criterium for mathematical pendulum and rigid body are obtained

Filtering of Gaussian Stochastic Processes by Observations with Delays

The problem which arises in stability and optimal control theory for stochastic hereditary systems [1–4] is connected with states estimate [5–8]. In this paper the equation for optimal in mean square estimate of Gaussian stochastic process by observations with delay is obtained and dependence of the estimate error on the delay in observations is investigated.

AN ESTIMATION PROBLEM FOR LINEAR STOCHASTIC EQUATIONS WITH MEMORY

We consider optimal estimation of solutions of linear equations with memory. The observation process contains a delay. Expressions are derived for the optimal estimate and the estimation error. Their dependence on the problem parameters is analyzed.

Estimation of the solutions of linear stochastic integral equations

Abstract The problem of the optimal estimate (filtering) (optimal in the mean-square sense) of a partially observed process specified by a linear stochastic Volterra equation is solved. Numerical examples are given.