Solutions to the problem of differential games with electric oscillations by finite difference methods

The article deals with the issues of electromagnetic processes from the point of view of application technology. The problem of voltage normalization in lines in the presence of opposing external forces is studied. The process of realizing the solution of this problem is based on the introduction of a mathematical model of electromagnetic processes occurring in lines as a continuous differential pursuit game described by hyperbolic equations. The finite difference method transforms the continuous pursuit problem into a discrete game. Sufficient conditions are obtained getting into neighbourhood. And with the help of a well-known a priori estimate for the difference between the solutions of a continuous differential and the corresponding discrete game, sufficient conditions are obtained for the possibility of completing the pursuit of a continuous game. A pursuer outputs terminal set, which is voltage normalization.


Introduction
When considering electromagnetic processes occurring in lines for energy transmission, as well as in telegraph and telephone lines, it must be borne in mind that the magnetic and electric fields associated with them are distributed along the entire line and that the conversion of electromagnetic energy into heat also occurs along the entire lines.Such electrical circuits, in contrast to circuits with lumped parameters, are called circuits with distributed parameters.When studying lines, it is necessary to take into account both the resistance of the wires of the line and the fact that a magnetic field is associated with them and that, therefore, the line has inductance.In addition, it must be borne in mind that electric fields exist between the line wires and that; therefore, the line wires have capacitance relative to each other.It should also be taken into account that, due to the imperfection of the insulation, the conductivity between the wires of the line, generally speaking, is not equal to zero.And therefore, if we consider the problem of voltage normalization, then when compiling the corresponding differential equations, everything that has been said will need to be taken into account.
In circuits with distributed parameters, for example, with long lines, windings of electrical machines of transformers, etc., the switching on and off of a section, as well as in circuits with lumped parameters, is accompanied by transients.With a long length of lines, a change in external electric and magnetic fields, for example, during lightning discharges, also causes transient processes.Transient processes in lines also occur during the transmission of telegraph and telephone signals, telemechanics impulses or special impulses to check the lines and identify the place of their damage.
It is clear that the passage of electric current through a wire with distributed parameters is characterized by current i and voltage z , which are functions of the position of the point and time t .
Applying the second Kirchhoff law, combining two equations into one, we obtain the so-called telegraph equation; it is an equation of the hyperbolic type.In all cases, when analyzing transient processes in ( ) continuous in the set.In addition, the terminal set It is known [3] listed above, in ), ( ), ( 1 0 x x   will be called the position or initial state of system (1), and we will denote them by , will be called a differential game.Definition 1.In game (1) maybe 0   -ending the pursuit from the starting position T   and function ( , , ) , u x t P Q   , such that for an arbitrary function
Let's split the Euclidean space ,, In this equation  are also defined in this way.
Clear 0 ) 2 Similarly anchor Original problem (1) and we have for of convergence Now, for convenience of presentation, we write problem (2) in matrix form [17] ( ) ,, Where

R
terminal set allocated M .Definition 2. We will (4) point ,, Of the second kind of degree In [1], [19] there are the following recursive relations () Next, the matrix Chebyshev polynomial from the matrixY , defined by recurrent formulas (6) we denote by Proof.Substituting (7) into equation ( 4) and taking into account equality ( 5) and ( 6), we obtain ( ) The lemma is proven.

Results
Subsection, results three they are discrete analogues of [4].

Theorem 2 .
If Assumption 3 holds, then in game (4) from the initial position Proof of Theorem 2. By the hypothesis of the theorem, there exist(12) such vectors