Vol 80, No 5 (2019)

We describe mathematical methods for analyzing the stability, stabilizability and consensus of linear multiagent systems with discrete time. These methods are based on the idea of using the notion of joint/generalized spectral radius of matrix sets to analyze the rate of convergence of matrix products with factors from the sets of matrices with special properties. This is a continuation of the survey by the same authors named “Consensus in Asynchronous Multiagent Systems”; the first part was published in [1].

The AC optimal power flow (AC OPF) problem is considered and five convex relaxations for solving this problem—the semidefinite, chordal, conic, and moment-based ones as well as the QC relaxation—are overviewed. The specifics of the AC formulation and also the nonconvexity of the problem are described in detail. Each of the relaxations for OPF is written in explicit form. The semidefinite, chordal and conic relaxations are of major interest. They are implemented on a test example of four nodes.

Nonlinear affine systems with constrained vector control that are represented in a canonical (normal) form and are closed by feedbacks linearizing the system in a neighborhood of the origin, are considered. For the nonlinear closed-loop system, the problem is set to construct an estimate of the attraction domain of an equilibrium position. A method for constructing an estimate of the attraction domain, which is based on results of absolute stability theory, is suggested. The estimate is sought as a Cartesian product of positive invariant sets of the subsystems composing the system. In the case of ellipsoidal invariant sets, construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.

A general class of the nonlinear time-varying systems of Itô stochastic differential equations is considered. Two problems on the partial stability in probability are studied as follows: 1) the stability with respect to a given part of the variables of the trivial equilibrium; 2) the stability with respect to a given part of the variables of the partial equilibrium. The stochastic Lyapunov functions-based conditions of the partial stability in probability are established. In addition to the main Lyapunov function, an auxiliary (generally speaking, vector-valued) function is introduced for correcting the domain in which the main Lyapunov function is constructed. A comparison with the well-known results on the partial stability of the systems of stochastic differential equations is given. An example that well illustrates the peculiarities of the suggested approach is described. Also a possible unified approach to analyze the partial stability of the time-invariant and time-varying systems of stochastic differential equations is discussed.

For the time series that describe the amounts of crude and load oil, the Fisher and Shannon entropies as well as other information measures are calculated. Their dynamics are used for performing an early diagnosis of the temporal limits of the oil field evolution, including its separate zones, and also for evaluating the state of oil production in response to planned bed stimulation. The suggested method is tested on the Forties Oil Field data. In accordance with the experimental results, oil field exploitation without considering the dynamics of these information measures reduces oil production, increasing the output of water-cut oil and causing an inefficient mode of bed stimulation.

We consider the problem of formally finding the complexity parameters of the environment for a moving object operating on a plane in the presence of obstacles. We give a mathematical justification for the method of calculating the complexity, introduce the concepts of local and integral complexities of the environment, give analytic formulas for calculating the complexity, and show simulation results.

We considered linear dynamical systems subjected to the uncertainty in its matrix. Using the linear matrix inequality technique, upper bounds on the deviations in linear dynamical systems were obtained; also, the problem of minimization of deviations in linear control systems by means of a linear static state feedback was analyzed. The results of numerical simulations testified to a low conservatism of the obtained estimates.

The paper is devoted to the formal statement and solution of a problem arising when assigning the locomotives for freight transportation realization in accordance with preset schedule. The goal is to determine whether the number of locomotives is sufficient at a specified initial allocation of them to perform all transport operations. The solution is presented in the form of an algorithm that builds the coverage of the schedule: the complete one, if it exists, or else the partial one being the maximal independent. The theorem is proved on one-to-one correspondence between the existence of the complete coverage and the sufficiency of the number of locomotives.

This paper considers finite-dimensional concave games, i.e., noncooperative n-player games in which the objective functionals are concave in their “own” variables. For such games, we investigate the design problem of numerical search algorithms for Nash equilibrium that have guaranteed convergence without additional requirements on the objective functionals such as convexity in the “other” variables or similar hypotheses (weak convexity, quasiconvexity, etc.). Two approaches are described as follows. The first approach, being obvious enough, relies on the Hooke-Jeeves method for residual function minimization and acts as a “standard” for comparing the efficiency of alternative numerical solution methods. To some extent, the second approach can be regarded as “an intermediate” between the relaxation algorithm and the Hooke-Jeeves method of configurations (with proper consideration of all specifics of the objective functions). A rigorous proof of its convergence is the main result of this paper, for the time being in the case of one-dimensional sets of players strategies yet under rather general requirements to objective functionals. The results of some numerical experiments are presented and discussed. Finally, a comparison with other well-known algorithms is given.