Vol 78, No 10 (2017)

Consideration was given to the problem of operation speed for the class of linear autonomous systems with infinite-dimensional state vector. The statements about the properties of the Minkowsky sum for the convex sets were formulated and proved. For the boundary points of the 0-controllability set, the necessary and sufficient conditions for solvability of the problem of operation speed were established. The optimality conditions were set down for the boundary points in terms of the discreet principle of maximum. Nonuniqueness of the optimal control and degenerate nature of the principle of maximum were proved for the internal points. An algorithm to solve the problem of operation speed for the internal points was developed by reducing it to the allowed case of the boundary points. Examples were given.

Stability conditions for fixed points of the dynamic systems in the space of nonempty convex compacts in finite-dimensional space were obtained. For the sets of dynamic systems, a methods was proposed to construct the Lyapunov vector functions. The study was carried out using the Matrosov method of comparison in combination with the methods of the convex geometry. In some special cases, the results obtained are applicable to the study of the sets of trajectories of dynamic systems.

We consider dynamical systems defined by autonomous and periodic differential equations that depend on two scalar parameters. We study the problems of constructing boundaries of stability regions for equilibrium points in the plane of parameters. We identify conditions under which a point on the boundary of a stability region has one or more smooth boundary curves coming through it. We show schemes to find the basic scenarios of bifurcations when parameters transition over the boundaries of stability regions. We distinguish types of boundaries (dangerous or safe). The main formulas have been obtained in the terms of original equations and do not require to pass to normal forms and using theorems on a central manifold.

We consider the problem of non-asymptotical confidence estimation of linear parameters in multidimensional dynamical systems defined by general regression models with discrete time and conditionally Gaussian noises under the assumption that the number of unknown parameters does not exceed the dimension of the observed process. We develop a non-asymptotical sequential procedure for constructing a confidence region for the vector of unknown parameters with a given diameter and given confidence coefficient that uses a special rule for stopping the observations. A key role in the procedure is played by a novel property established for sequential least squares point estimates earlier proposed by the authors. With a numerical modeling example of a two-dimensional first order autoregression process with random parameters, we illustrate the possibilities for applying confidence estimates to construct adaptive predictions.

Consideration was given to construction of a unilateral asymptotic confidence interval for an unknown conditional probability of the event A under the condition B. Analytical expressions for the boundaries of such interval were obtained for various a priori restrictions on the probabilities of the events A and B such that the probability of the event B is separable from zero by a certain value, as well that there exists an upper a priori restriction on the probability of the event A. The interval estimates obtained were compared in precision.

We consider an exponential queueing network that differs from a Gelenbe network (with the usual positive and so-called negative customers), first, in that the sojourn time of customers at the network nodes is bounded by a random value whose conditional distribution for a fixed number of customers in a node is exponential. Second, we significantly relax the conditions on possible values of parameters for incoming Poisson flows of positive and negative customers in Gelenbe’s theorem. Claims serviced at the nodes and customers leaving the nodes at the end of their sojourn time can stay positive, become negative, or leave the network according to different routing matrices. We prove a theorem that generalizes Gelenbe’s theorem.

Miniaturization and improved performance of computers, sensors, and executive devices open up new possibilities for intelligent control over complex mechatronic systems on transition processes and under turbulence. Previously, adaptive control problems in a changing environment and with state space structure changing in time have virtually not been studied due to limited capabilities for a practical implementation. The change of the state space structure (dimension) is possible under active interaction with the environment. In this work, with the example of an airplane with a large number of “feathers” distributed on the surface, i.e., elements with pressure sensors and executive turning devices, we show that in complex systems adaptation to changes in the structure of external disturbances can be done with internal selforganization, similar to multiagent systems. Under turbulence, to solve the problem of equating the influence of disturbing forces on different elements of the wing and transforming airplane flow to a mode close to laminar, we have considered the possibility to apply a multiagent protocol. The operability of the new approach is illustrated with a specially made experimental testbed that allows for real physical experiments.

This paper considers a simple model of a cooperative patrolling game with coalition structure. It is shown that the Shapley value coincides with the Owen and Aumann–Dreze values under an odd number of defenders.

This paper introduces an extension of the vehicle routing problem by involving several decision makers in competition. Each customer is characterized by demand and distance to the warehouse. The problem is described in form of a cooperative transportation game (CTG). We consider customers as players in the game. Their strategies are the routes for a vehicle they should rent in a coalition to deliver goods subject to their demand with minimal transportation costs, under the assumption that transportation costs are allocated between the players according to the Nash arbitration scheme. For each profile in coalitional strategies, we define a coalitional structure of players and the costs of each player. A strong equilibrium is found for the cooperative transportation game. In addition, we develop a procedure to calculate the strong equilibrium. This procedure is illustrated by a numerical example.