First we consider the scalar case of problems when extraneous fields (currents) in the radiating system is linearly polarized [7,13], and created by their DP (1) has only one component. Let the operator A acts from some Hilbert functional space into the complex space of continuous and square integrable functions in domain (or).

In space we introduce the scalar product and norm

,

where is a point of integration.

Along with the Chebyshev norm

in the space we introduce scalar product and the generated by it norm and metric as follows:

, ,

Note, space is a Banach space relatively uniform norm (4) and it is incomplete space concerning norm defined according to (5) [14].

We consider also that the operator has the izometric property or it is completely continuous. Let the given amplitude DP is real positive (nonnegative) continuous function which different from nonzero in some limited closed domain and identically equal zero on complement. Let is isometric operator, that is for any and equality is satisfied

In [15-17] the synthesis problem of given amplitude DP is formulated (is investigated) as a minimization problem in the Hilbert space of the functional

characterizing the value of mean-square deviation of modules of given and synthesized DP in domain. For the formulated problem occurs [18,19]

Theorem 2.1. Let the linear operator

is isometric relatively mean-square metric and it is continuous operator from into

concerning uniform metric of the space, and the given amplitude DP is a real positive (nonnegative) continuous function in the domain.

Then at least one point of absolute minimum of the functional (7) exists in the space and a subsequence which converges weakly to one of points of absolute minimum can be selected from any minimizing sequence.

On the base of the necessary condition for an extremum of the functional (zero equality of its Hato differential [20]), we obtain the equation with respect to the optimal distribution of excitation sources

where is the conjugate with operator.

Let the set of zeros of the operator A consists of only one zero element. Then acting on both parts of (8) by operator, we obtain the equivalent to (8) equation in space:

By solutions of this equation the optimal distribution of excitation sources in radiating system are defined by the formula

From Theorem 2.1 and the properties of functional follows Corollary 2.1. Since functional is differentiable in by Hato, it is growing [21] and according to Theorem 2.1 has at least one point of absolute minimum, then (8) in the space and (9) in the space have at least one solution.

Lemma 2.1. Between solutions of (8) and (9) there exists bijection, that is if is the solution of (8), then is the solution of (9). On the contrary, if is the solution of (9), then the corresponding solution of (8) is defined by (10).

The possibility of investigation of solutions of synthesis problems, using (8) or (9) follows from Lemma. Note, Equation (9) is simpler as (8), since the latter contains the operator exponent.

Note, solutions of synthesis problems according to the prescribed amplitude DP are determined with precision to value (is arbitrary constant), since

. So if there exists the solution of Equations (8) (or (9)), then there is also generated by its family of solutions in which one solution different from another by phase constant. For the uniqueness of desired solutions additional conditions impose on the functions or.

In the case completely continuous operator describing DP of radiating system the smoothing Tikhonov type functional [11]

which includes requirements as to the mean-square deviation of DP, so to norm of current, is used [18,22] for the synthesis of various types of antennas. The parameter α can be viewed as regularization parameter [11,23] or as a weighing coefficient, by means of which can control ratio between the first and second summand of functional.

Theorem 2.2. Let the linear operator acts from the complex Hilbert space into the complex space of continuous functions and it is completely continuous, and given DP is real positive (nonnegative) and continuous function in (or in).

Then at least one point of absolute minimum of functional exists in and a subsequence which converges weakly to one of the points of absolute minimum can be selected from any minimizing sequence.

Differentiating functional by Hato and performing appropriate transformations, we obtain the equation [22,24]

in the space.

Equation with respect to synthesized DP based on equality and (12) has the form

Lemma similar to Lemma 2.1 is valid for (12) and (13).

From Theorem 2.2 and properties of functional follows [7,18]

Corollary 2.2. Since differentiable in by Hato functional is growing and according to Theorem 2.2 has at least one point of absolute minimum, then (12) in space and (13) in the space have at least one by one solution.

Consider the more general case when the excitation fields (or currents) in the radiating system and generated by it DP have vector character [5,6]. In this case, we set that the operator is completely continuous and it acts from complex space of square integrable in the domain vector-valued functions, into the complex space of continuous functions on the compact vector-valued functions

equipped by scalar product. We introduce the scalar product and generated by it norm in:

(14)

We define module of vector as following:

.

In the space along with the Chebyshev norm

where

we introduce the scalar product and generated by it the mean-square norm and metric

(17)

If DP of radiating system has two components, , i.e. it is described by (1), then as the optimization criterions are used the following functionals [7]:

.

(20)

In the functional (20), , are the given amplitude of components of required DP. At that

, and functions,

can be given with account of existing requirements to polarization characteristics of emitted field.

If in the synthesis problem functional (19) is used as the optimization criterion, the problem on finding the minimum points is reduced to finding the solutions of the equation

in the space. Equivalent to (21) equation with respect to vector DP in space has the form

In this case the following theorem is valid [7].

Theorem 2.3. Let linear completely continuous operator acts from the complex Hilbert space

into the complex space of continuous functions equipped by the scalar product, and given DP is a real positive continuous function on the compact.

Then in there exists at least one point of absolute minimum of functional and a subsequence which converges weakly to one of the points of absolute minimum can select from any minimizing sequence.

For the functional (20) is true Theorem 2.4. At conditions of Theorem 2.3 functional in the space has at least one point of absolute minimum and subsequence which converges weakly to one of points of absolute minimum can be selected from any minimizing sequence.

For minimizing of the functional in the space we obtain equation [7]

.

(23)

Equivalent to (23) equation with respect to synthesized DP in the space has the form

The existence of solutions of (23) in space and (24) in the space, respectively, follows from necessary condition of functional minimum.

If necessary the weight function [15,17] can introduce in the functionals and by means entering appropriate scalar products and affect on quality of the approximation of synthesized and given DP in a certain range of angles.

The synthesis problems with optimization of geometry of radiating system are more complicated class of problems. These problems need to find a configuration of the radiating system and amplitude-phase distribution of excitation fields (currents) in it [25,26]. Moreover, the operator depends on two functions: function describing the geometry of the system, and amplitude-phase distribution function of excitation sources i.e.

In addition, the function has, as rule, vector character, and the operator is a nonlinear concerning the function. Later on we shall consider the synthesis problem of a flat aperture, in which in addition to amplitude-phase distribution (APD) desired is too the function that describes the boundary of aperture. The basis of the formulation of such problems can be put the functionals (7), (11) and (20), expanding their by corresponding requirements to geometry of radiating system.

In many radio engineering systems antenna arrays (AR) have gained widespread use. Antenna array is [4,6,27] antenna, which consists of identical (or differenttype) radiators placing corresponding way in space and they collate by common system of power and control. In [28-37] investigations of nonlinear synthesis problems and planar antenna arrays according to the prescribed amplitude DP are presented.

To describe the electromagnetic characteristics of antenna arrays are used different by precision mathematical models [38-42]. In the base of construction of mathematical models is imposed [40,42] that the excitation of each radiator is characterized by a unique complex number -complex amplitude of excitation. It’s the physical meaning depends on the type of radiating system. On the base on the linearity of the Maxwell’s equations the complex excitation amplitudes enter linearly in the expression for DP of array, that is

where is vector DP -th radiator. Vector is called vector excitation of array or vector of amplitude-phase distribution of currents in the array.

In general, the construction of high-accuracy mathematical models of array is reduced to solving the corresponding exterior boundary problem of high-frequency electrodynamics for system of the Maxwell’s equations in multiply-connected domain [1,39-41]. In the particular case, where the elements of the array are ideally leading talamy accounting of the mutual influence is based on the method of induced electromotive forces (IEF) and it is reduced to solution the corresponding system of linear integral equations [42]

where is matrix-integral linear operator; I is complexvalued vector distribution function of surface currents on radiators; is vector-valued function describing the outside fields (voltage) which is necessary to create in the system of power of array. Allocating in the space compact class of solutions (where (27) is correct), solution of (27) is written as

Here it is assumed that the corresponding regularized system¹ exists for the system of (27). Then on the basis of (28) formula for DP of array takes the form

This relation allows to formulate the synthesis problem of antenna array according to the prescribed amplitude DP with account of the mutual influence of elements as the problem on finding the vector minimizing the functional

in space.

At need to take into account the component-wise approximation of modules of given and synthesized DP’s, the functional [43-45]

analogously to (20) can be used as optimizing criterion.

By the desired solutions of this problem the optimal vector of extraneous voltages on inputs of radiators is determined on the basis of (27).

Thus, the basic requirements for synthesis problems of different types of radiating systems according to the prescribed amplitude DP are formulated. Note that recorded functionals is not convex [21], and therefore may have nonunique extreme point. In further the above statements of problems allow to obtain relatively simple nonlinear integral or matrix equations for the study and solution of which can be applied numerical methods of nonlinear analysis methods.

Integral equations method [40,42] is used widely in such classes of problems. The method of synthesis of antenna arrays from cylindrical dipoles with account of mutual influence is proposed in [43-45]. Analysis of problem of nonuniqueness solutions is studied there by means computational experiments.

In spite of the fact that from the amplitude DP is easy to obtain the DP by power and vice versa, in the mathematical aspect the synthesis problems of given amplitude DP and given energy DP are different tasks. For example, if is the optimal solution of some variational synthesis problem of amplitude DP, then will not be the optimal solution of a similar problem for the given DP. On this basis, in [46-49] on the operator level are considered statements of synthesis problems with use of two types of stabilizing functionals, in which the vector character of the electromagnetic fields takes into account.

Consider the synthesis problem of given energy DP. Taking into account the expression for DP

in the simplest form this problem can be formulated as a problem on finding the solutions of nonlinear operator equation of the first kind

where is a real positive continuous function in

(at that) which can not belong to the domain of values of the operator. It is known [11] that (32) is severely ill-posed problem. In this connection, we consider the problem on best meansquare approximation of the real positive continuous (in the domain) function by function

(,).

Formulate it as minimizing problem of functional [49]

in the space. In this functional the first summand characterizes mean-square deviation of given and synthesized DP by power. Second summand imposes restrictions on norm of currents in the radiating system. Real parameter we shall consider as a weighing multiplier. The existence of at least one point of minimum functional in the space states [7,49]

Theorem 2.5. Let the linear operator acts from the space into and it is completely continuous, is given nonnegative continuous the function in, at that.

Then in the space H_U there exists at least one point of absolute minimum of the functional and subsequence that converges weakly to one of the points of absolute minimum can be selected from any minimizing sequence.

On the base of necessary condition of minimum functional is obtained the equation [7]

with respect to optimal currents in the space. This equation is a nonlinear operator equation having in the right part linear operator along with the Hammerstein type operator. If the set of zeros consists of only the zero element, then acting on both parts of (34) by operator, we obtain equation an equivalent to (34) with respect to synthesized DP in the space

In [49] is shown that the functional has -property [50], that is the minimum point of the functional is interior point of some convex weakly closed set of the space. On this basis from Theorem 2.5 follows [7]

Corollary 2.3. Since the functional is differentiable in by Hato, has at least one minimum point and property, then (34) in the space and

(35) in the space have at least by one solution.

Lemma 2.2. At conditions of Theorem 2.5 and limited values of parameter operator

is compact in the space.

Since for elements relatively compact set of normalized space the strong and weak convergence coincide [51] then with Theorem 2.5 and Lemma 2.2 follows Corollary 2.4. If is minimizing sequence of the functional converging weakly to the minimum point, then the sequence converges uniformly in to.

We put that the linear antenna is linear electric conductor of length, sizes of cross-section of which are much less than the wavelength. Due to these limitations the excitation currents in the antenna shall have only directaxis current [5]. Introduce the Cartesian and connected with spherical coordinate systems such that the origin of coordinates coincides with the middle of the antenna. We direct the axis along the antenna. Then the current vector in a Cartesian coordinate system will have only z-component. We shall introduce the dimensionless coordinates, , and parameter

connecting the electric size of antenna with angle, outside of which given amplitude DP identically zero. Then the formula for DP of linear antenna takes the form [7]

For DP of antenna the Parseval equality [8]

is valid, that is the operator is isometric.

Taking into account that is finite function with compact carrier and expressions for the operators and, we obtain the expanded form of (8):

On the basis of (9), (38) we obtain the Hammerstein type equation concerning optimal DP

in the space, where

The existence of at least one solution of (40) in the space and (41) in the space follows from Corollary 2.1.

We shall present three important properties of (41).

1) If is the solution of (41), then complexconjugate function is the solution of (41) too.

2) If is the solution of (41), then is the solution of (41) too, where β is an arbitrary real constant.

3) For even functions nonlinear operator, which is in the right part of (41), transfers even phase DP in even, and odd—in an odd. That is the operator is invariant with respect to the type of parity of function. Due to this property the existence of fixed points of the operator -solutions of (41) is possible in the classes of even and odd phase DP’s.

In [16,52] is shown that (41) has two solutions in the class of real functions:

which is called the primary solution of the first type and

is the primary solution of the second type. Point is determined from the condition. For the even , that is the solution (44) is a real odd function (the corresponding to it amplitude DP is even function).

These solutions are effective only at small values of parameter. With the growth of this parameter there exist branching points at which more effective (in the sense value of functional) complex solutions branch-off from real solutions.

Consider according to [7,16] results of investigation of branchings of primary solution of the first type of solutions of (41). Using the procedure of decomplexification of the space [14] from (41) we move to the equivalent system

On the base of the branching theory of solutions [53] the problem on finding such values of parameter (branching points) and all different from solutions of system of (45) satisfying the conditions

are considered. Condition (46) means that it is necessary to find small continuous solutions

converging uniformly to zero at. Putting in (45)

and expanding the integrand, in the vicinity of the point in the power series by, and, and taking into account that the function is its solution, we obtain system of integral equations of Lyapunov-Schmidt type [53] with respect to small solutions,:

where. On the base of the left part of (49) we obtain linear homogeneous integral equation

for finding the points of possible branching of solutions.

Equation (50) is a nonlinear one-parameter spectral problem concerning parameter. It is shown in [7,16] that for a given even amplitude DP there exist branching points of two types: eigenvalues of multiplicity two correspond to the first type, eigenvalues the multiplicity of three—to the second type. It is found in [54] analytical expressions for eigenfunctions of (50) and are obtained systems of transcendental equations for finding the possible branching points.

Using for finding the solutions of branching equation the Newton diagram method, it is shown [53] that two complex-conjugate between themselves solutions, which in the first approximation, have the form

branch-off from the real solution in the branching points of the first type. Here,

are even by s functions which are obtained by means corresponding transformations,

is the second eigenfunction of

(50) at the points.

In [7] it is shown also that the branching-off solutions branch-off too. Analogous investigations are performed in branching points of the second type.

To estimate the effectiveness of different types of solutions we consider value of the functional depending on the parameter, which it takes on these solutions. For example, in Figure 1 are shown the values of the functional for. The most effective solution images envelope which: on the segments I corresponds to the primary solution, on II-branching-off solutions at the point with odd phase DP, on III-solutions and branching-off solutions from theseon IV-solutions and branching-off solutions at the point with even phase, на V-branching-off at the point solutions of the type, on VI-solutions.

Thus, the analytical investigations [7,16,52,54] and the results of numerical experiments enable to describe the general structure of the solutions of the problem dependeing on change of the value of the parameter.

Because the values of the functional on some types of solutions in a given interval of change of the parameter may be equal, the curves shown in Figure 1, does not map the full structure of the existing solutions. For greater clarity this structure can be represented schematically as a “tree” of solutions. In Figure 2 it is shown for the case of even DP. The primary solution is central. The solution with odd phase DP

branches first. At the point solution

branches from branching solution, and solution enter in a real solution in a neighborhood of the point at, tending to. At the same point solution with even phase DP

branches from primary solution. The solution of the type with an odd phase DP branches at the point, which is located directly behind. The solutions of the type and the type form basic branches of “tree”. The possible branching points of branching-off solutions are shown on these branches.

In [56,58,73,74] using the found branching lines and eigenfunctions, the analytical investigations of branching of the primary solution of the first type of (65) for the case when the the kernel has the form (61), and the multiplicity of eigenvalues of the linear Equation (73) at the branching points is two, are presented.

The study of solutions of (65) is realized on the beam belonging to the domain. Let

be eigenvalue of (73). We assign to parameter the small disturbance

, and consider the problem on finding all different from solutions of (65), which at satisfy the conditions

,

.

The system of (65) by means of expanding the integrand functions is reduced to the corresponding system of Lyapunov-Schmidt equations, similar to (71) and (72). Desired solutions are found in the form

,

.

As a result we obtain [74] that at the points

from the primary solution

branch-off two complex-conjugate solutions having in the first approximation the form

The imaginary part being determined by the properties of eigenfunctions. Functions

, obtained on the base of (93), determine the properties of the phase DP and APD of the field in aperture. The properties obtained in the first approximation of solutions agree with numerical results.

For example, in Figure 7 are shown the values of the functional at, which it takes on the primary (curve 1) and branching-off (curves 2, 3, 4) solutions on the beam. Note, that on the segment the branching-off solutions with an odd phase DP, to which the nonsymmetric amplitude-phase distribution of the field in aperture corresponds, are the most effective. On the segment the most effective is the solution of 4 with properties,

. The symmetric but complex APD of the field in aperture corresponds to it. From the analysis of Figure 7 follows that the same efficiency of the synthesis can be achieved on the branching-off solutions at smaller sizes of aperture and smaller values of parameters, , than on the primary solution. The linear size of aperture can be decreased by the amount or at realization of branching-off solution.

Numerical examples of synthesis of given funnelshaped amplitude DP defined in the domain by (90), are given in Figures 8 and 9. The branching lines of solutions of (63) for this DP are shown in Figure 4. The given DP and optimum synthesized DP are presented in Figures 8(a) and (b), respectively, at, . The optimum amplitude distribution of the field in an aperture, which creates given in Figure 8(b) the synthesized DP, is shown in Figure 9. From the analysis of these figures we see that the symmetric amplitude DP (Figure 8(b)) can be created by different distributions of the field in aperture of radiating system, including real and nonsymmetric distribution (Figure 9).

As an example of the scalar problem we consider iterative process of solving the equation of type (9), in the base of which we put the successive approximations method [7,75]

Obviously, the successive approximations method (103) is equivalent to the following iterative process

In [75] it is shown that the sequences and generated by iterative process (104), are relaxational for functional. Relaxation properties of (104) states Theorem 5.1. The sequence is generated by the iterative process (104), it is relaxation for functional, and the values which it takes on form a convergent numerical sequence.

Formulate also the properties of the operator entering in (41) that complement the properties of 1˚ - 3˚ solutions of (41), presented in Section 3.1.

Theorem 5.2. Nonlinear operator, defined by (41), acts in the space of continuous complex-  valued functions, it is a compact and maps set it into itself, where

that is.

From the proved theorem follows, in particular, the following fact. Since the solutions of (41) are fixed points of the operator B, from the relation follows that all solutions of this equation belong to the set. In addition, is valid [75]

Corollary 5.1. If the sequence which is generated by the iterative process (104), is minimizing for the functional, then from can be selected a subsequence converging uniformly to the minimum point of the functional.

Note that Theorem 5.2 and Corollary 5.1 are extended to the case of synthesis problem of a flat aperture with use of equation of the type (63).

In the base of construction of iterative processes of solving the nonlinear operator equations of the type (12) and (13) we put implicit scheme of the successive approximations method [7,76]. In a general case, the iterative process of solution of (12) has the form

where is an identity operator acting in the space.

The implicit scheme of iteration process for (13) with respect to synthesized DP ³ has the form similar to (106)

Note that the implicit schemes (106) and (107) are characterized by the fact that linear operator equation is solved on every iteration step. In addition the question of solvability of (106) and (107) appears, to which a positive answer gives a theorem about the solvability of the functional equation of the second kind of the type

in the Banach space, where is a linear compact operator [69].

Theorem 5.3 [69]. In order that (108) have the solution at an arbitrary, it is necessary and sufficiently that homogeneous equation have a unique solution (obviously, that).

For a sequence obtained by (106), is valid Theorem 5.4. Let be a completely continuous operator, be a continuous real nonnegative function in and at there exists the inverse operator, in additionthe dimension of the space of zeros.

Then the sequence generated by the iterative process (106), is a minimizing for the functional

in the space.

We denote the operator in right part of (106) as:

For the operator is valid Lemma 5.1. Let be a completely continuous operator. Then the operator defined by (110), is compact and it transfers any bounded set into its relatively compact part at

.

Thus, it is shown that there is true Corollary 5.2. If is operator continuous in some neighborhood of the point, then from Theorem 5.2 and Lemma 5.1 follows that the subsequence converges to some solution of (12) by the norm of the space if.

Dependent on the choice of initial approach the successive approximations (106) can converge to the solutions of various types [56-58].

First we shall consider the iterative process of solution the equation of type (34) in the Hilbertian space under certain restrictions on the parameter. This equation is written as

where is an identity operator,

is completely continuous operator. We denote. Note that the scalar product and the corresponding norm in the space are defined by (14) and (15), and the Chebyshev and mean-square norms in the space are introduced by Formulas (16)-(18).

Henceforth we shall consider completion of the space

relatively to the norm [14], which is the Banach space and coincides with the Hilbertian space, the norm in which we shall denote by symbol. We assume that

is a completely continuous operator and in the space the domain of its values is a set of continuous functions.

Taking into account the equality we shall consider the expression in (111) as an operator of multiplication by the function:

acting in the space where is real nonnegative continuous function on the compact, in addition. Obviously, that (112) is a linear bounded operator, and.

If, then there exists the inverse operator

, the norm of which satisfies the inequality [14].

In this case, Equation (111) we shall write as

Here we shall show that the solution of (114) can be obtained as a limit of successive approximations of the iterative process [61]:

where is some fixed number with the interval. In addition successive approximations can converges to different solutions of (114) depending on the choice of the initial approximation.

To determine the conditions and to justify convergence of (115), we shall use the Theorem 4.1 with [61, p. 68], according to which: if nonexpanding operator converts a closed convex set of strictly convex Banach space into its compact part, then successive approximations

where is any fixed number from the interval, converges to some solution of the equation at some.

Since the Hilbertian space is strictly convex Banach space (see [61, p. 67]), then to satisfy of the conditions of this theorem concerning (114), it is sufficiently to show that a closed convex set exists in the space, where the operator is nonexpanding and completely continuous. In addition there is such relation

.

Satisfication of these conditions results from lemmas, proved in [7,49].

Lemma 5.2. Let be a linear completely continuous operator and the domain of its values is a set of continuous functions,. Then is a nonexpanding operator on

, where

that is, for any the inequality

is satisfied.

Lemma 5.3. Let be a linear completely continuous operator and the domain of its values is a set of continuous functions,. Then, defined by (114), is a completely continuous operator for which the relation

(is a closed convex set, defined by

(116)), is satisfied.

In this section we shall consider the synthesis problem of a flat aperture according to the prescribed amplitude DP for the case when the form of aperture and amplitudephase distribution of the field (currents) in it is optimized simultaneously, limiting by the case of linear polarization [25,26,77]. We shall consider a special case when the field in the aperture is linearly polarized along one of the coordinate axes, and DP has only one component. We introduce inside of aperture the polar coordinate system:,. Let be a function of the boundary of aperture. Then DP which is formed by amplitude-phase distribution of the field in the aperture, is given by the formula [7,16]

Later on we omit the index in definition of. Let the given amplitude DP be different from identical zero in some limited closed domain and it is identically equal to zero at. The problem of simultaneous synthesis of the aperture shape and amplitude-phase distribution of the field in it is considered as the problem on finding the functions and minimizing the functional

in which the first two summands describe the meansquare deviation of modules of given and synthesized DP’s in space, and the third one—imposes restrictions on the square of aperture. We shall consider the parameter as a weight coefficient.

We introduce into consideration the following functional spaces: is a space of square integrable complex functions in the domain,

is a space of square integrable real functions on the segment, is a space of square integrable complex functions in the domain. Scalar products and generated by it norms we shall introduce as follows:

,

,

,

,

,

Taking into account the introduced norms, the last summand in (119) and Parseval’s equality have the form

,

.

On this base the functional is presented as:

We shall consider the iterative process of numerical minimization of (121). In it base we shall put the ideas similar, as at minimization of functions of two variables by a coordinate descent method. Let be a minimum point of the functional and

be an initial approximation chosen from some neighborhood of the point. We shall denote by the initial shape of aperture, that is described by the function. Substitute in (121) and consider its restriction in the space:

From the necessary condition of the functional minimum we obtain equation of type (9). Numerically we solve it by successive approximations method, given in pt. 5.1:

As a result, we find the function, and obtain the first approximation of the solution by the formula of type (10).

We shall pass to finding the function that describes the boundary of aperture. We fix the function extending its analytically according to (10) to the plane in (121), and consider the functional

which depends only on the function. With the necessary minimum condition:

, where

is an arbitrary element of the space, we obtain the equation

which is a nonlinear functional equation with respect to the function.

We shall find numerically solutions of (124), using the Newton-Kantorovich method [69]:

where is the partial Frechet derivates of operator by the function. We assume that. Equation (125) is a linear integral equation of the form

which can be reduced to the Fredholm equation of the second kind at. Solving (127) we find the first approximation for the function that describes a boundary of aperture of the radiating system.

Continuing finding in turn the approximations of functions and, we obtain the sequence

that is relaxational for (119). In more detail the problem of choice of initial approximations and justification of relaxation for functional is given in [7,56,57].

First we shall consider the numerical results of synthesis of flat aperture with optimization of its geometry. In Figures 15 and 16 the examples of synthesis of amplitude DPs, which in cross section have quasi-square and

quasi-triangular shapes, are given. The optimal shapes of apertures are given there too.

Note that the problems of such class arise, in particular, at the synthesis of contour DPs of fixed and variable forms for satellite antenna systems needed for uniform irradiation of a given territorial zone from the board of artificial satellite, where multi-beam antennas are used often [77- 82].

If multibeam antenna (MBA) has a radiating aperture of circular shape, and partial beams in the cross section have the shape of a circle and nonuniform distribution of radiated energy inside of the section, then on the junction of three neighboring rays with a circular cross section the so-called critical zones (Figure 17) with low level of radiated energy occur. One of the possible ways of solution of this problem is passage to alternative forms of apertures that on the base of the optimal APD will form rays that have rectangular, triangular or hexagonal shapes and close to constant (inside of contour) coefficient of directed action in rectangular cross section.

Obviously that on the base of such partial beams it is easy to synthesize given summary DP without critical

zones. Below the results of synthesis of triangle-beam contour DP with partial beams with circular (Figure 18(a)) and quasi-rectangular (Figure 18(b)) contours are presented.

From the analysis of the figures we see that in the summary DP which is obtained on the base of quasisquare of contours, critical zones are absent, and variation of radiated energy inside of the contour does not exceed 2 dB.