Optimal control of pulse compensators of oscillatory phenomena in a network oil and gas pipeline system

Within the framework of oil and gas engineering, the problem of optimal control of pulse compensators that counteract harmful oscillatory phenomena in a continuous medium during transportation via network gas-hydraulic carriers is considered. Powerful compressor units that create high pressure in the carrier of a continuous medium, to a large extent contribute to the formation of undesirable oscillatory phenomena (pulsations) that occur at the output of these compressors. These ripples are transmitted to the network carrier environment, which significantly reduces the efficiency of compressor units and even causes accidents in the networks of gas and hydraulic carriers. The latter means that the software engineering of the oil and gas industry should include research in the direction of improving the reliability of operation of compressor units and gas-hydraulic carriers. In the presented study, the mathematical description of the oscillatory process of a continuous medium is carried out by formalisms of a differential-difference system of hyperbolic equations with distributed parameters on a graph. At the same time, the mathematical model contains a fairly accurate mathematical description of controlled pulse compensators. The problem of controlling pulse compensators of an oscillatory process is considered as the problem of a point control action on a controlled differential-difference system at the places where continuous medium vibration dampers are connected to a network carrier. This is a characteristic feature of the presented study, which is quite often used in practice when engineering the processes of transporting various kinds of continuous media through network oil and gas carriers. The study essentially uses the conjugate state and the conjugate system for a differential-difference system - the relations determining the optimal point control are obtained. The results of the work are applicable in the framework of oil and gas engineering to the study of issues of stabilization and parametric optimization.

Convergence of approximate solutions for singular difference systems with maxima

In this paper, by introducing a new singular fractional difference comparison theorem, the existence of maximal and minimal quasi-solutions are proved for the singular fractional difference system with maxima combined with the method of upper and lower solutions and the monotone iterative technique. Finally, we give an example to show the validity of the established results.

Minimal wave speed in an HIV-1 virus integrodifference system

This paper is concerned with the minimal wave speed of nonconstant traveling wave solutions in an HIV-1 virus integrodifference system. Here, the traveling wave solution models the spatial spreading process of infected cells and virus. When the basic reproduction ratio of the corresponding ordinary differential system or difference system is larger than one, we establish the existence of nonconstant traveling wave solutions if the wave speed is not less than a threshold, and if the speed is smaller than the threshold, we prove the nonexistence of nonconstant traveling wave solutions. Moreover, when the basic reproduction ratio of the corresponding ordinary differential system or difference system is not larger than one, we also confirm the nonexistence of nonconstant traveling wave solutions.

Dynamics of a second-order nonlinear difference system with exponents

In this paper, we study the persistence, boundedness, convergence, invariance and global asymptotic behavior of the positive solutions of the second-order difference system $$\begin{aligned} x_{n+1}&= \alpha _1 + a e ^{-x_{n-1}} + b y_{n} e ^{-y_{n-1}},\\ y_{n+1}&= \alpha _2 +c e ^{-y_{n-1}}+ d x_{n} e ^{-x_{n-1}} \quad n=0,1,2,\ldots \end{aligned}$$ x n + 1 = α 1 + a e - x n - 1 + b y n e - y n - 1 , y n + 1 = α 2 + c e - y n - 1 + d x n e - x n - 1 n = 0 , 1 , 2 , … where $$\alpha _1, \alpha _2, a, b , c,d$$ α 1 , α 2 , a , b , c , d are positive real numbers and the initial conditions $$x_{-1},x_0, y_{-1}, y_0$$ x - 1 , x 0 , y - 1 , y 0 are arbitrary nonnegative numbers.

Point control of a differential-difference system with distributed parameters on the graph

The article considers the problem of point control of the differential-difference equation with distributed parameters on the graph in the class of summable functions. The differential- difference system is closely related to the evolutionary differential system and moreover the properties of the differential system are preserved. This connection is established by the universal method of semi-discretization in a time variable for a differential system, which provides an effective tool in order to find conditions for unique solvability and continuity on the initial data for the differential-difference system. For this differential-difference system, a special case of the optimal control problem is studied: the problem of point control action on the controlled differential-difference system is considered by the control, concentrated at all internal nodes of the graph. At the same time, the restrictive set of permissible controls is set by the means of conditions depending on the nature of the applied tasks. In this case, the controls are concentrated at the end points of the edges adjacent to each inner node of the graph. This is a characteristic feature of the study presented, quite often used in practice when building a mechanism for managing the processes of transportation of different kinds of masses over network media. The study essentially uses the conjugate state of the system and the conjugate system for a differential-difference system — obtained ratios that determine optimal point control. The obtained results underlie the analysis of optimal control problems for differential systems with distributed parameters on the graph, which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.

q-Difference Systems for the Jackson Integral of Symmetric Selberg Type

We provide an explicit expression for the first order q-difference system for the Jackson integral of symmetric Selberg type. The q-difference system gives a generalization of q-analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system we use a set of the symmetric polynomials introduced by Matsuo in his study of the q-KZ equation. Our main result is an explicit expression for the coefficient matrix of the q-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials we compute the coefficient matrix.