DEVELOPMENT OF METHODS FOR SOLVING THE TASKS OF THE CONTINIUM LINEAR PROGRAMMING USING LEGENDRE POLYNOMIALS

Abstract The article demonstrates the theoretical foundations of the mathematical apparatus − ЭСО continuum of linear programming. It demonstrates a technique for solving problems with the use of orthogonal systems of functions. The article was an exact solution of the problem of variational calculus to linear constraints. The purpose of the work is to develop accurate methods of solving the problem in the class of Legendre polynomials. The study demonstrates an ability to build the exact solution of the problem and the conditions under which the decision is allowed. Based on the properties of Legendre polynomials, an exact solution of the problem of continual linear programming is provided, in which the integrands and functional limitations are presented in rows of finite degree. Analytically, it is proven that the solution obtained is a limiting case of the linear combination of delta functions. It is shown that the parameters of the optimization problem of finding the unknown functions plan contains half of the variables than in the canonical method. Recommendations are given for the construction of the optimization algorithm. There is a possibility of extending the proposed technology solution in the direction of using other systems of orthogonal polynomials.


Introduction
The simplest of these tasks can be solved by conventional means, if it is the class of functions in which a decision is based [1, p.25].It was found that the quality of the solution is higher the more "a delta" has a character of its analytical solution.It is shown that the exact solution of the simplest problem of continual linear programming must be sought in the class of delta functions [1, p.48].In general, the synthesis problem can be formulated as follows: Let the input of a linear system with a known frequency response signal is, the spectral power is described by the function.Find a function that maximizes functional: and satisfying the limits for the spectral power of the where -are integrated in the R function.The research in this article is dedicated to the development of exact methods for solving the above problem of continual linear programming (1)-( 4).

The formulation of the problem in the class of Legendre polynomias
LОЭ's introduce the variable t [1, p.24]: with provision of which, we will get the next presentation of our problem: Using the labeling for c(t) where c(t),a(t) because of integrability of on the interval are the integrable functions on the interval   , when the problem is reduced to the form (1)-(3).

Solution of the problem in the class of Legendre polynomials
The solution in the form of an expansion in Legendre polynomials   t P n [4, p.64-74]: Representing the specified functions   and аО'ХХ РОЭ ЭСО sЭКЭОЦОЧЭ ШП ЭСО ЩrШЛХОЦ, аСТМС consists in maximizing function when   n N  is even integer, which contains the finite number N LОРОЧНrО's ЩШХвЧШЦТКХs   t P n .

The usage of
In the view of orthogonal polynomials   t P k expression (26) can be presented as: By substituting (28) into ( 27) we obtain The solving is next: where n  the constants of integration.The right side of formula (30) can be written as follows: that allows to present the solution (30) this way: Indeed, integrating with the left side of (29), taking into account (15) we obtain: On the other hand, using (32), a similar result can be written: An important fact is that if the coefficients n  of one and quite simply determined by ( 11), (25), 18) with the unambiguous representation (28), (31).It should be noted that by virtue of (28) to determine n . This condition limits the range of acceptable solutions (10)-(12).

An algorithm for solving the simplest problem
Consider the exact solution of the problem (1)- (36) Using (8) and (9) represent the task (36) in the form: Since (42)   t a 0 is represented by one member of the decomposition   Figure 2 shows the behavior of the functional value of M (47) depending on the number of polynomials in the solution   Pihnastyi O.M.Development of methods for solving the tasks of the continium linear programming using Legendre polynomials // р Pihnastyi O.M.Development of methods for solving the tasks of the continium linear programming using Legendre polynomials // р

Fig. 1 .
Fig. 1.LОРОЧНrО's ЩШХвЧШЦТКХs   ) LОЭ's НОКХ аТЭС ЭСО solution of the problem (10)-(12) for the case when the functions   R C ,   R A i are represented by polynomials of degree N1, N2: These include the problem of the LPC, which limits (2) defines the moment of order .M.Development of methods for solving the tasks of the continium linear programming using Legendre polynomials // р Pihnastyi O.M.Development of methods for solving the tasks of the continium linear programming using Legendre polynomials // р to write the problem (37)-(39) as (18)-(20) as follows: function has a maximum value of the functional (37) is reached at   min 2 functional change its value remains constant.This result is due to the finite number of polynomials   t P n in the expansion of   t c (40).

Fig. 2 .
Fig. 2. The dependence of the value of the functional M(n) (46) on the number n of the Legendre polynomials   t P n M.Development of methods for solving the tasks of the continium linear programming using Legendre polynomials // р

Fig. 3 . 2 .
Fig. 3. Accuracy representation   t x (47) for the case 1 0  S according to the n in the expansion of Legendre polynomials   t P n