Stability of dynamic systems with aftereffect with due regard for Markov perturbations

Lyapunov-Krasovskii functionals and infinitesimal operators are employed to analyze a system for asymptotic stochastic stability on the whole and asymptotic stability on the whole. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 134–146, November–December 2007.     

Lyapunov function method for investigation of stability of stochastic Ito random-structure systems with impulse Markov switchings. I. General theorems on the stability of stochastic impulse systems

The asymptotic stochastic stability on the whole and asymptotic p-stability on the whole are investigated with the help of Lyapunov functions and the concept of an infinitesimal operator over solutions to a system (to calculate this operator, it suffices to know only the coefficients of the system). The stability under constant perturbations is considered. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 135–145, March–April 2009.     

Robust dissipativity of discrete systems and its investigation with the help of a sequence of sets of Lyapunov functions

A generalized concept of robust dissipativity is introduced and theorems on the analysis of this property are formulated and proved with the help of the Lyapunov functions method. It is proposed to use a specially constructed sequence of sets of Lyapunov functions that can allow one to improve the initial estimate of the boundary set of a dissipative system up to the establishment of the asymptotic robust stability property in the limiting case. An example of investigating the dissipativity of a linear discrete system with unknown parameters under additive perturbation is given. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 13–23, March–April 2008.     

Lyapunov function method for investigation of stability of stochastic ito random-structure systems with impulse markov switchings. II. First-approximation stability of stochastic impulse systems with markov parameters

The asymptotic stochastic stability in the large and asymptotic p-stability in the large are investigated with the help of the Lyapunov functions and an infinitesimal operator over solutions to a system (to calculate this operator, it suffices to know only the coefficients of the system). The stability under persistent disturbances is considered. Continued from Cybernetics and Systems Analysis, Vol. 45, No. 2, 2009. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 146–158, May-June 2009.     

Convergence of a jump procedure in a semi-Markov environment in diffusion-approximation scheme

The sufficient convergence conditions are obtained for a jump stochastic approximation procedure in a semi-Markov environment in a diffusion approximation scheme with balance conditions for a singular perturbation of the regression function. To this end, a singular perturbation problem is solved for the asymptotic representation of the compensating operator of an augmented Markov renewal process. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 124–133, November–December 2007.     

Continuous stochastic approximation with semi-Markov switchings in the diffusion approximation scheme

The sufficient conditions of convergence are obtained for a continuous stochastic approximation procedure in the diffusion approximation scheme under balance conditions imposed on singular perturbations of the regression function in a semi-Markov environment. To this end, a singular perturbation problem is solved for the asymptotic representation of a compensating operator of a Markov renewal process. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 170–178, July–August 2007.     

Mathematical modeling of the dynamics of consolidation processes with relaxation effects

The consolidation of salt-saturated porous media is modeled with space-time nonlocality effects and with/without the relaxation of the filtration rate taken into account. A numerical algorithm for modeling the dynamics of the process is proposed and an asymptotic analysis of the problem for excess head is performed for weak spatial nonlocality. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 59–66, November–December 2008.     

Analysis of stability of stochastic dynamic systems under Poisson perturbations. II. Stability of solutions in quadratic mean of systems of linear stochastic differential equations under Poisson perturbations

Necessary and sufficient conditions of asymptotic stability in quadratic mean are obtained for trivial solutions of systems of linear stochastic differential equations under Poisson perturbations. Model problems are analyzed. Part I of this article is published in No. 4 (2005). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 50–66, November–December 2005.     

Stability of an equilibrium state in a multiinfectious-type SIS model on a truncated network

In a homogeneous constant population, the basic SIS model potentially has an epidemic equilibrium state with global asymptotic stability since it can be reduced to the logistic equation. On the basic SIS model with a nonhomogeneous constant population, viewed as a multitype SIS model, the global or local asymptotic stability of an epidemic equilibrium state has also been studied.^1–4 However, this kind of analysis in other models with nonhomogeneous populations has rarely been developed, even though the corresponding models with homogeneous populations are well known. In addition, recent studies of complex networks have revealed that heterogeneity of the link number of vertices drastically changes the epidemic thresholds.^5–9 For these reasons, figuring out the roles of heterogeneity is a major topic in epidemic modeling. Here, we consider a multiinfectious-type SIS model on a network, and show the (local or global) asymptotic stability of an epidemic equilibrium state whenever it exists. This work was presented in part at the 11th International Symposium of Artificial Life and Robotics, Oita, Japan, January 23–25, 2006     

Stability of solutions of dynamic systems with randomly structured aftereffect

The relation is established between the asymptotic stochastic stability of a linear functional differential equation and exponential stability of the trivial solution to this equation. The direct and inverse Lyapunov theorems on the stability of linear differential equations are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 31–38, March–April 2006.     

Asymptotic efficiency of kernel support vector machines (SVM)

The paper analyzes the asymptotic properties of Vapnik’s SVM-estimates of a regression function as the size of the training sample tends to infinity. The estimation problem is considered as infinite-dimensional minimization of a regularized empirical risk functional in a reproducing kernel Hilbert space. The rate of convergence of the risk functional on SVM-estimates to its minimum value is established. The sufficient conditions for the uniform convergence of SVM-estimates to a true regression function with unit probability are given. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 81–97, July–August 2009     

Linear regression with nonstationary variables and constraints on its parameters

The problem of estimating parameters of a linear regression with allowance for inequality constraints on the parameters is considered in the special case when its variables have a trend. A parameter estimation algorithm is described. The consistency of parameter estimates is proved and their asymptotic distribution is found. Consistent estimates are proposed for the mean-square error matrix of estimates of regression parameters and noise dispersion under rather general assumptions on the law of noise distribution. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 50–64, May–June 2009.     

Queuing system evolution in a diffusion approximation scheme

An asymptotic diffusion approximation scheme is investigated as applied to the requirement evolution in semi-Markov queuing systems. In proving the diffusion approximation theorem, the compensating operator of the corresponding extended Markov process is used. This problem is solved with the help of a phase merging procedure. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 136–145, May–June 2009. Original article submitted August 19, 2008.     

On delay-dependent stability for a class of nonlinear stochastic delay-differential equations

Global asymptotic stability conditions for nonlinear stochastic systems with state delay are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov–Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of the system coefficients. Nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given.     

Models with conjugation conditions and high-accuracy methods of their discretization

New classes of problems with discontinuous solutions are considered. Corresponding generalized problems are obtained. Increased-accuracy computational schemes for their discretization are proposed. Numerical schemes are proposed with asymptotic accuracy not worse than that of similar schemes for problems with smooth solutions. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 110–131, January–February, 2000.     

Nonparametric estimate of almost periodic signals

The asymptotic properties of least-squares estimates of almost periodic signals under random noise are investigated. The consistency and strong consistency of the estimates are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 57–63, May–June 2007.     

Profit maximization and risk minimization in semi-Markovian networks

The problem of profit maximization and risk minimization under stationary conditions is considered for multichannel semi-Markovian networks. Systems of integral equations that describe data handling and profit accumulation in the network are subjected to asymptotic analysis. The objective functions of the optimization problems are expressed explicitly in terms of network parameters. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 65–79, March–April 2007.     

Generic Complexity of Presburger Arithmetic

Fischer and Rabin proved in (Proceedings of the SIAM-AMS Symposium in Applied Mathematics, vol. 7, pp. 27–41, 1974) that the decision problem for Presburger Arithmetic has at least double exponential worst-case complexity (for deterministic and for nondeterministic Turing machines). In Kapovich et al. (J. Algebra 264(2):665–694, 2003) a theory of generic-case complexity was developed, where algorithmic problems are studied on “most” inputs instead of set of all inputs. A question rises about existing of more efficient (say, polynomial) generic algorithm deciding Presburger Arithmetic on a set of closed formulas of asymptotic density 1. We prove in this paper that there is not an exponential generic decision algorithm working correctly on an input set of asymptotic density exponentially converging to 1 (so-called strongly generic sets).     

The asymptotic properties of estimates of the parameters of nonlinear time series

Asymptotic properties of nonlinear time series parameter estimators constructed on trajectories of stochastic systems under stationary and transient conditions are studied with the use of the least-squares method. The investigation method is based on the study of asymptotic properties of extremal sets of random functions. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 62–72, March–April, 2000