Том 80, № 4 (2019)

We present a survey of results on models of consensus in asynchronous multiagent systems with discrete and continuous time. We consider mathematical methods developed over recent years, which are used in the analysis of stability, stabilization, and consensus problems for linear multiagent systems with discrete time. These methods are based on the idea of using the notion of joint/generalized spectral radius of a set of matrices to analyze the rate of convergence of matrix products with factors drawn from certain sets of matrices with special properties.

This paper considers the stabilization problem of two interconnected linear subsystems of differential equations with constant delay; one of the subsystems has an exponential factor in the right-hand side. Sufficient conditions for the stability of this system are established and then used for its stabilization.

In this paper, the optimal control problem for a discrete-time stochastic system with a general-form probabilistic criterion is considered. Using dynamic programming and the properties of the Bellman function, new two-sided bounds of this function that refine the earlier results are constructed. The derived bounds are then adopted to justify the application of the modified strategy that is optimal in the two-step investment portfolio management problem under risk to the corresponding multistep problem. An example that illustrates the advantages of such a strategy over other well-known strategies is given.

Using computer simulation and a study of the asymptotic distribution, we consider the relative efficiency of M-estimates for the coefficients of the threshold autoregressive equation with respect to the least squares and least absolute deviation estimates. We assume that the updating sequence of the autoregressive equation can have Student’s, logistic, double exponential, normal, or contaminated normal distributions. We prove asymptotic normality of M-estimates with a convex loss function.

We pose and solve the problem of the synthesis of dynamic integration matrices with discrete time shift operators that simulate data processing in the integrated computational environment of the designed equipment complex. The developed approach considers a preliminary choice of the configuration of the complex by specifying a pair of configuration matrices that simulate inertialess input and output interfaces of all its heterogeneous and non-universal components. We obtain analytic expressions for complete and reduced sets of integration matrices that ensure that, with the chosen configuration matrices, the prescribed objective function for the complex remains constant. We present a methodical example that demonstrates the methodology and effectiveness of the proposed approach.

We consider smooth convex optimization problems whose full gradient is not available for their numerical solution. In 2011, Yu.E. Nesterov proposed accelerated gradient-free methods for solving such problems. Since only unconditional optimization problems were considered, Euclidean prox-structures were used. However, if one knows in advance, say, that the solution to the problem is sparse, or rather that the distance from the starting point to the solution in 1-norm and in 2-norm are close, then it is more advantageous to choose a non- Euclidean prox-structure associated with the 1-norm rather than a prox-structure associated with the 1-norm. In this work we present a complete justification of this statement. We propose an accelerated descent method along a random direction with a non-Euclidean prox-structure for solving unconditional optimization problems (in further work, we propose to extend this approach to an accelerated gradient-free method). We obtain estimates of the rate of convergence for the method and show the difficulties of transferring the above-mentioned approach to conditional optimization problems.

In this paper, a probabilistic modification of the minimal element algorithm for solving the axial three-index assignment problem is suggested. Its general idea is to extend the basic greedy-type algorithmic schemes using transition to a probabilistic setup based on variables randomization. The minimization of an objective function is replaced by the minimization of its expectation. The algorithm is implemented in three stages as follows. At the first stage, a motion in the set of random variables is defined. At the second stage, an inequality that expresses the local improvement condition is solved. At the third stage, the probabilities are recalculated, which represents an “adaptation” process. The second stage reveals a feature of the algorithm: the resulting solution depends on the “qualities” of the element itself and also on possible losses of its choice.

This paper considers two minimal-time control problems for a mechanical system that consists of a material point and an obstacle interacting with each other through a spring with elastic or viscoelastic properties. The interaction interval is determined by physical conditions of contact. It is studied how interaction time and the coefficient of restitution depend on the viscoelastic properties of the spring. A program module is developed in Python and the behavior of this material point-obstacle system is simulated.

The problem is investigated of optimal control of a system, described with the model of fractional-order double integrator, in which the initial and final conditions depend on the appropriate time-point selection. Several cases are considered, physically similar to the system transfer from a state of the rest to a state of the uniform linear motion, to a uniformly accelerated and to a periodic motion, and from a uniform to a uniformly accelerated motion. The dependencies are analyzed of the control norm from the control time and from the fractional differentiation index value.

The main stages are presented of creating an information-technological data reserve in distributed automated informational control systems. The reserve is formed on the basis of the results of the analysis of subject areas, regular requests to the system and the analysis of the execution sequence of requests processing procedures. The tasks are set of the reserve structure design and of the optimal allocation of reserve copies among computer network nodes. It is noted that this type of reserve allows reducing the processing time of typical, regular requests to the system due to the fact that the reserve contains the data sets, prepared beforehand and later used at requests processing.