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## Optimal Control Problem in a Stochastic Model with Periodic Hits on the Boundary of a Given Subset of the State Set (Tuning Problem)

In this paper, a general stochastic model with controls applied at the moments

when the random process hits the boundary of a given subset of the state set is

proposed and studied. The general concept of the model is formulated and its

possible applications in technical and economic systems are described. Two versions

of the general stochastic model, the version based on the use of a continuous-time

semi-Markov process with embedded absorbing Markov chain and the version based

on the use of a discrete-time Markov process with absorbing states, are analyzed.

New representations of the stationary cost index of the control quality are obtained

for both versions. It is shown that this index can be represented as a linear-fractional

integral functional of two discrete probability distributions determining the control

strategy. The results obtained by the author of this paper about an extremum

of such functionals were used to prove that, in both versions of the model, the

control is deterministic and is determined by the extremum points of functions of

two discrete arguments for which the explicit analytic representations are obtained.

The perspectives of the further development of this stochastic model are outlined.

We study the structure of the functional of accumulation defined on the trajectories of semi-Markov process with a finite set of states. As t -> ∞ this functional increases linearly and the coefficient is linear-fractional functional relative to the probability measure, defining homogeneous Markov randomized control strategy.

The functionals constructed on trajectories of the controlled semi- Markov process with a finite set of states are investigated. Theorems of functionals’ structure (dependences on the probability measures defining the Markov homogeneous randomized strategy of control) and of structure of probability measures on which the extremum of these functionals is reached, are formulated. Examples are given.

We construct the optimal strategy for management system with variable structure, which is described by a controlled semi-Markov process with a finite set of states. An algorithm for solving the problem and the results of numerical experiments

Consideration was given to optimization of the queue control strategy in the MlGl1l queuing system where decision about continuing or stopping admission of customers is made at the service completion instants of each customer in compliance with the distribution on the set of decisions depending on the number of customers remaining in the system. The mean specific income in the stationary mode was used as the efficiency criterion and the set of permissible strategies coincided with the set of homogeneous randomised Markov strategies. It was proved that if there exists an optimal strategy, then it is degenerate and threshold with one point on control switching, that is, if the number of customers in the system exceeds a certain level, then admission of customers must be stooped or, otherwise, it must be continued.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

The optimal strategy of the structure control in queue and reliability model is studied by using controlled semi-Markov processes. The optimal strategy has been proved to be looked for in the class of threshold strategies.

The functioning of different systems can be described using queueing models. Optimization is used to increase the efficiency of the system functioning. The research is devoted to CBSMAP-flow. Note that it is very reasonable to change the characteristics of arrival flows and others characteristics in various queueing models for optimization of its functioning. In the research a process of system functioning is investigated at an controlled arrival process (flow).

Algorithmization of the problem is necessary because choice of optimum strategy at a large number of the controlled objects is very labor-intensive process. The research is very interesting and actual. The researched model and algorithm describe a lot of mathematical problems such as control of telecommunication networks, transport management and other subjects...

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.