_{1}

^{*}

A brief review of the works of the author and his co-authors on the application of nonlinear analysis, numerical and analytical methods for solving the nonlinear inverse problems (synthesis problems) for optimizing the different types of radiating systems, is presented in the paper. The synthesis problems are formulated in variational statements and further they are reduced to research and numerical solution of nonlinear integral equations of Hammerstein type. The existence theorems are proof, the investigation methods of nonuniqueness problem of solutions and numerical algorithms of finding the optimal solutions are proved.

In many practical applications at the optimal design of various types of radio and acoustic radiating systems the requirements are only to the energy characteristics of the directivity of the radiated field (amplitude directivity pattern (DP) or DP by the power). Therefore there is a need to approximate real finite functions by modules of onedimensional or two-dimensional and discrete Fourier transform dependent on the real physical parameters. At the same time the absence of requirements to phase characteristics of field is used to improve the quality of approximation of synthesized DP to given.

Later on the variational formulations of different types of inverse problems in mean-square approach, which in further are reduced to investigation and numerical solution of one-dimensional or two-dimensional nonlinear integral equations of the Hammerstein type with separate module and argument of desired complex-valued function, are considered. Nonuniqueness and branching (or bifurcation) of solutions dependent on the change of the physical parameters characterizing the radiating system are characteristic features of such equations. Problems on finding the set of branching points (bifurcation) are not investigated nonlinear one-parameter or two-parameter spectral problems.

The existence of connected components of the spectrum, which in the case of real parameters are of spectral lines, is essential difference between the two-dimensional and one-dimensional spectral problems. The problem on finding the spectral lines is reduced to numerical solution of the Cauchy problem for an ordinary differential equation of the first order.

The degenerate of kernels in linear operators of the Hammerstein type equations is feature of the synthesis problems of antenna arrays. It allows to reduce nonlinear two-parameter spectral problems on finding the set of branching points of solutions to the corresponding systems of linear algebraic equations with nonlinear occurrence of the spectral parameters in the coefficients of system.

In the basis of construction of numerical algorithms for finding the optimal solutions are taken such princeples: localization of existing solutions dependent on the value of the physical parameters of the problem by means the use of numerical methods of solving the non-linear one-parameter and two-parametric spectral problems, and methods of the branching theory of solutions—construction and justification of the convergence of iterative processes for numerical finding the various types of existing solutions of basic equations (equations of Hammerstein type)—analysis of the effectiveness of found solutions.

In general case, the analysis problems (direct problems) of radio (or acoustic) radiating systems are reduced to solution the corresponding boundary problems of electrodynamics (acoustic) at a given excitation sources of fields [1-4] on the basis of Maxwell’s equations (wave equation). The directivity pattern is one of the basic characteristics of the emitted field on large distances from the radiating system. It describes the properties of the field in space dependent on the angular coordinates of a spherical coordinate system. In general case, the DP is a vector of complex-valued function which has the form [5,6]

Here is linear operator acting from some functional Hilbert space (the space of square integrable functions in the domain, describing the distribution of extraneous fields (currents) in volume) into the space of complex-valued continuous functions defined in some domain (or) [

In the simplest form the inverse problem (the synthesis problem) according to the prescribed amplitude DP can be formulated as the problem on finding the solutions of nonlinear operator equation of the first kind

where is a given amplitude DP. In staged thus the synthesis problem all three correct conditions of problems by Hadamard [10-12]: existence of the solution, uniqueness of the solution, continuous dependence of the solution of the input data, can be violated simultaneously. Violation of condition (1) in the first place is connected with the fact that the given DP can not belong to the class realized, that is to the domain of values of the operator. In other words, such DP can not be obtained at any distribution of field in the aperture of the radiating system belonging to the space. Trying to recreate the DP just leads to effect of superdirectivity [

Condition (2) is violated due to the nonlinearity of the problem.

Therefore, the variational formulations of problems, which in addition to the requirements of the basic characteristics of DP also contain requirements to the distribution function of the currents (fields) in the aperture of the radiating system, are considered. At that is required not complete coincidence obtained DP with given, but only the best (in the sense of the selected criterion) approximation to it.

An important feature of the variational formulation of synthesis problems is the fact that in the optimization criterion can introduce functionals describing certain other requirements to amplitude-phase distribution (APD) of outside excitation sources. Their mean-square deviation, as a rule, will be used as the criterion of proximity of amplitudes of the given and synthesized DP.

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) [

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 [

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 [

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 [

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 [

.

(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 [

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 [

.

(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 [

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 system^{1} 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 [

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 [

(,).

Formulate it as minimizing problem of functional [

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 [

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 [

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 [

Here on the example of scalar synthesis problems of linear radiator and radiating system with a flat aperture are presented the results of investigation of nonuniqueness problem of solutions corresponding to these tasks nonlinear integral equations of Hammerstein type depending on the change of the physical parameters.

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 [

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 [

For DP of antenna the Parseval equality [

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 [

On the base of the branching theory of solutions [

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 [

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 [

Using for finding the solutions of branching equation the Newton diagram method, it is shown [

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 [

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

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

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.

Consider according with [55-58] the synthesis problems of a flat aperture assuming that form of aperture is known and a field has elliptical polarization. Let the plane in which aperture is located, coincides with the plane of the Cartesian coordinate system. Then the radiated field in the far zone can be represented by the formula [

where

,

is radial ort of spherical coordinate system, , is a vector DP of flat aperture. Since, function describes the tangential component of the electric vector or vector of current flowing through the aperture:

Introducing in a far zone special coordinate system [

, ,

enables the vector synthesis problem to reduce to two independent scalar synthesis problems.

Obviously, the system, , is orthonormal, and transformation (54) is rotation of spherical coordinate system on an angle around the vector. At that vector in the coordinate system (54) has the form [

where

For mappings (56) the Parseval equality [

are valid, that is operators, are isometric.

Consider the synthesis problem of a flat aperture, in which component-wise deviation of modules given and synthesized diagrams is taken into account. As optimization criterion we choose the functional type

where, are modulus of components of the given amplitude DP in closed domain

. This criterion provides not only proximity of modules of given and synthesized DP’s, but it allows certain to influence on the polarization characteristics of the radiated field [

where

(60)

is a kernel. In the case of rectangular aperture the kernel takes the form

where

are real numeric parameters characterizing the sizes of aperture, in wavelengths, is wave number, , are angles that characterize the domain (solid angle), in which different from the identity components of amplitude DP are given.

Later on we omit index in (59) and shall investigate the solutions of one equation

where for reduction of records we introduce the following notations

, ,.

Thus, the synthesis problem of flat radiating system with arbitrary polarization of irradiation according to the prescribed amplitude DP is reduced to two independent and simpler synthesis problems with linearly polarized fields in the aperture.

Equation (63) is a nonlinear two-dimensional integral equation of the Hammerstein type and it has nonunique solutions. Their quality and properties depend on the form of aperture, the values of parameters and properties of given amplitude DP.

On the base of decomplexification [

in the domain. The elements of this space are written as:,

,. Norms in these spaces have the form:

Equation (63) in the decomplexified space is reduced to equivalent to it system of equations

,

.

(65)

Denote the closed convex set of continuous functions as setting that

,

,

.

Consider one of the properties of the function that included in (62) at. Obviously,

is a continuous function if and

are continuous functions, at that

for any. If

and simultaneously, then is a complex zero with undefined argument by definition [60, p. 20]. On this basis at and we redefine as function, module of which is equal to one and argument is undefined.

Theorem 3.1. The operator determined by the formulas (65) maps a closed convex set of the Banach space in itself and it is completely continuous.

As the corollary from the Theorem 3.1 follows satisfaction of conditions of the Schauder principle [14, p.

411] according to which the operator has a fixed point belonging to the set. This point is a solution of a system of (65) and (63), respectively.

Easily to be convinced that function

is one of solutions of (63) in the case of symmetric domain.

In [56,58] it is shown that the operator

is positive on the cone of nonnegative functions of the space [

To find the branching lines and complex solutions of (63) that branch-off from the real (primary) solution, we shall consider the problem on finding such a set of parameter values and all different from solutions of (65) that at

satisfy the conditions

These conditions indicate the need to find such small continuous in solutions,

which converge uniformly to zero as. At that it should take into account also the direction of movement of vector to vector.

Set, , and desired solutions we find in the form

We write the system of nonlinear integral equations of Lyapunov-Schmidt with respect to small solutions, as

(71)

(72)

Here, are coefficients of expansion of integrand functions of (65) in uniform convergent power series.

The problem on finding the set of possible branching points of solutions of (71) and (72) is reduced [56,58] to find the eigenvalues of two-dimensional linear homogeneous integral equation

at condition. Eigenfunctions of (73) are used [

Note that (73) in the general case is a nonlinear two-parameter spectral problem. For the numerical finding the approximate solutions it is necessary to construct its digitization and consider the corresponding problem in finite-dimensional spaces. It should be noted that in the literature, in particular in [62,63], more attention is given to the construction of numerical methods for solving the nonlinear one-parameter problems.

In [64-67] a general method for finding the approximate solutions of (73), which may be applicable to a wide range of nonlinear two-parameter spectral problems is proposed.

Denote the spectral parameters as. Let E and V are complex Banach spaces, and the vector parameter belongs to domain (open connected set) of the complex space, where, , is some real constant. Consider the operator-function, where to every is put in correspondence operator.

Here the space of linear bounded operators [

We shall consider the nonlinear two-parameter spectral problem of the form

where necessary to find the eigenvalues

and corresponding eigenvectors

such that.

Let the Banach spaces and, be given and also a system of linear bounded operators such that

Operators are called connecting [14,68]. Note, by the principle of uniform boundedness [

Let element is selected in each space. Writing these elements in ascending numerical order, we form sequence.

Definition 3.1 [

Definition of different types of convergence of operators to is given in [^{2} to is presented.

Discretization of initial problem (74), the choice of the space and definition of operators can be differentially. In particular, one of the approaches to the digitization of (74) if the operator-function

is described by formulawhere is a linear continous operator and is unique operator in the separable (infinite-dimensional) Hilbert space, consists in following. Take an arbitrary complete orthonormal system of functions in. Each element is represented as a series

, where is Fourier coefficient of element. Since is linear continuous operator acting in separable Hilbert space, it admits the matrix representation [

where. At that sequence of the Fourier coefficients of element is obtained from the sequence of Fourier coefficients of element x by means transformation matrix.

Using the matrix representation of the operator in particular case (concerning Equation (73)) the spectral problem (74) is formulated as

where is identity matrix in the space of sequences. Thus, the operators and are equivalent in the sense that they put in correspondence one and the same element, but we obtain the Fourier coefficients of element as a result of operation of operator on element. Obviously, that the spectrums of these operators coincide, that is the spectral problem (77) and the problem

are equivalent.

According to [14,68] applying to the problem (74) other discretization methods, including the following: quadrature (cubature) processes in the case of homogeneous integral equations and changing the derivatives by difference analogues in differential equations, we obtain the approximation problems for approximate finding the eigenvalues and eigenfunctions in finite-dimensional spaces

At that the problem on finding the eigenvalues is reduced to finding the roots of the -th order determinant, i.e. the roots of the equation

Consider the necessary in further auxiliary one-parameter spectral problem as a particular case of (74). Set that variable in the operator-function is expressed by some unique differentiable function

mapping domain in some subdomain. In the simplest case we put , where is a real parameter. Introduce into consideration at operator function

(narrowing of operator-function). One-parameter nonlinear spectral problem

is connected with it. Here to each value

operator is put in correspondence.

Analogously to (78) we consider approximating sequence of discretizing problems (80) at

The spectrum of operator-function is denoted as. Suppose that. For spectral of (74) holds [56,67].

Theorem 3.2. Let the following conditions be satisfied:

1) operator-function is holomorphic, and;

2) operator-functions are holomorphic and for any closed bounded set the following inequality

is valid;

3) operators,

are the Fredholm operators with zero index for any;

4) spectrum and a sequence of functions are differentiable in the domain;

5) is stable for any

.

Then the following statements are true:

1) every point of spectrum is isolated, it is eigenvalue of the operator, the finite-dimensional eigensubspace and the finite-dimensional root subspace correspond to it;

2) for each there exists a sequence

from , such that

;

3) each point is a spectrum point of the operator-function;

4) if in some small -neighborhood of the point

at all n larger any number

(corresponding, according to definition of limit of sequence p. 2)) the sequence of partial derivates

is nonzero, then in an arbitrarily small -neighborhood of point there exists a continuous differentiable function

, which is solution of (89), at that

and at the point

however little differs from point of a spectrum of auxiliary one-parameter problem (91); that is in some bicylindrical domain

there exists a connected component of spectrum of the operator-function (,are small real constants).

Proof. The proof of the theorem is given in [

If the points are the eigenvalues of

(78) and derivatives, in these points are nonzero, to find connected components of the spectrum of this problem on the base of (79) Cauchy problem [56,57,65] we solve the in a neighborhood of each point

Return to finding the solutions of (73), in which, are spectral parameters. Let,

, where. By direct check we set that for arbitrary values of the parameters the function

is one of the eigenfunctions, that is there exists a connected set of the spectrum, coinciding with the domain. As a result of this, the condition is not satisfied. To find another connected components of spectrum we exclude eigenfunction (73) from the kernel of integral equation, namely, consider the equation

where

Here is adjoint with (73) eigenfunction of equation of From Lemma Schmidt [53, p. 132] follows that from spectrum of operator is excluded coherent component coinciding with the domain and the corresponding to the function.

Using to (73) certain convergent cubature process with coefficients and nodes and rejecting in it remainder, we obtain homogeneous system of linear algebraic equations (SLAE)

where.

The presence of such values of parameters, , which are the solutions of the equation

is necessary condition of the existence different from zero solutions of (87). We consider (88) as the problem on finding the implicitly given function, reducing it to the Cauchy problem (82) and (83). Putting in (85), we shall consider the auxiliary oneparameter spectral problem

solutions of which we use as initial conditions in the Cauchy problem (83). Corresponding this equation SLAE has the form

and the problem on finding the eigenvalues of this system is reduced to finding the roots of the equation

. For the numerical solution of the Cauchy problem (82) and (83) are used the Runge-Kutta and Adams methods.

We shall present numerical examples of finding the solutions of (73) for two given amplitude DP’s. In Figures 3 and 4 are shown spectral lines of (73), corresponding the given DP and given DP which is defined by the formula:

(90)

Note that to each point of the spectral lines given in these figures correspond the eigenfunctions of (73) with the characteristic properties for each line. For example, below are shown the eigenfunctions that correspond to points of intersection of the spectral lines 1 and 2 (Figures 3 and 4) with the beam.

In [71,72] along with the implicit functions method a variational approach to solution of the nonlinear oneparameter and two-parameter spectral problems on finding the eigenvalues and eigenelements of equation

in the real Hilbert space for the case when is a linear positive definite

c_{2} c_{2} = c_{1}

self-adjoint operator nonlinearly depending on the parameters, is proposed. Variational problem is formulated as the problem on finding such values of parameter and such functions on which functional

becomes minimum. The equivalence of the spectral problem (91) and put it in correspodence of variational problem (92) is proved. Based on the method of generalized coordinate descent iterative process for the numerical finding one of the eigenvalues and the corresponding eigenfunction of (91) is suggested. Local convergence is proved.

Example of use of a variational approach to finding the eigenvalues and eigenfunctions of (73) is shown in

Found by numerical method form and properties of eigenfunctions in the possible branching points are used to determine of the properties of branching-off in these points of solutions of nonlinear systems of (65).

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 [

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

. The symmetric but complex APD of the field in aperture corresponds to it. From the analysis of

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

The investigations of nonlinear synthesis problems of linear and planar antenna arrays (AA) according to the prescribed amplitude DP are presented partially in [7,24, 28-37]. In the basis of construction of mathematical models it is assumed [6,27] that the excitation of each radiator is characterized by a single complex number - complex amplitude of excitation, the physical meaning of which depends on the type of radiating system. Taking into account the linearity of Maxwell’s equations, the complex amplitudes of excitation enter linearly in the expression for DP of array, that is

Here is a vector DP of -th radiator. Vector is called the vector of excitation of array or vector of amplitude-phase distribution of currents in the array. Such formulation of DP of array is used in the synthesis problems with regard for mutual influence of radiators [27, 29,30]. Thus the problem on finding the functions is reduced to solution of the corresponding boundary problem of electrodynamics in multiply connected domains [2,4,39,40]. The method of integral equations [40,42] is used widely in such classes of problems. The synthesis method of antenna arrays with cylindrical dipoles with account of mutual influence is proposed in [29,34]. Analysis of nonuniqueness problem of solutions is studied there by means of computational experiments.

In the problems of analysis and synthesis of antenna arrays with many elements is used simplified mathematical model of AA [5,6]. It is assumed [

Here are the generalized angular coordinates,

,

are dimensionless numerical parameters characterizing the distance between the radiators and the domain (solid angle) G, in which the required amplitude DP is given. Since in (95) only the second multiplier depends on the vector APD of excitation currents in the array:

only the synthesis problem of factor of AA is considerd. Function is -periodic by argument and -periodic by. We consider also (96) as the action of the operator from a finite-dimensional space (is number of radiators) into the finite-dimensional subspace of the space where is the domain corresponding to the period of array. Let the amplitude DP be given in the domain and on the set is identically equal to zero. The synthesis problem is to minimize the functional [

The basic of synthesis equations of multiplier of AA have the form

the equation concerning APD of currents in AA, where is conjugate with operator, and

is equation concerning of synthesized DP. Here

, ,; is the kernel the form of which depends on the distribution of elements in AA. In particular, in the case of a rectangular array with number of elements

the kernel is written as

To find the possible branching lines of solutions of (99) a linear homogeneous integral equation

(101)

is obtained where is a primary solution of (99).

Note that the kernel is degenerate. Consequently, Equation (101) is reduced to the corresponding homogeneous SLAE what in a special case of rectangular array has the form

Coefficients of this system depend nonlinearly on the spectral parameters and on the given amplitude DP. In [35,65] the conditions are determined and the existence theorem of connected components of the spectrum of (101) is proved. To find the spectral lines the implicit function method (82) and (83), is used.

Consider the numerical results of finding the solutions of the branching lines in the synthesis problems of a plane equidistant antenna array with radiators for two given in the domain

amplitude DPs (

The prescribed and synthesized amplitude DPs (with phase DP odd by argument) at, , are shown in

The above results show that the nonlinear synthesis problems according to the prescribed amplitude DP and given energy DP have nonunique solutions. Application of the methods of branching theory of solutions to nonlinear integral equations allows to determine the quantity of existing solutions, to find solutions in the first approximation and to determine their quality characteristics. To find the complete solutions of these equations numerical methods [7,29,36,49,75] are applied. The defined properties of solutions obtained by analytical investigations make it possible to choose the initial approximation having the basic properties of the desired solutions and they are placed in certain neighborhoods of complete solu-

tions.

Conditionally the process of numerical solution of synthesis problem can be divided into two stages. The first of them is described partially above and it consists in finding the points (lines) of branching and determination of types of existings solutions depending on the value of physical parameters. The second stage consists in solving the basic synthesis equations by iterative methods.

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 [

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 [

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 [

Theorem 5.3 [

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 [

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 [

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 [

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 [

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 (

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 (

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.

Note the main features and problems arising at investigations of othe class of problems reviewed in the article:

• The investigation of nonuniqueness and branching of existing solutions, which depend on the physical parameters of the problem is the main difficulty at solving this class of problems. As follows from the researches presented, in particular, for a special case in [7,13,16], when, the quality of existing solutions increases significantly with growth of parameters. However, to obtain the best approximation to the given amplitude DP at relatively small values of the parameters describing the sizes of aperture, allowing to confine by investigations of the first few branching points (lines), is essential in the synthesis problems of radiating systems.

• At finding the solutions of (45) by the successive approximations method in the case of the even by both arguments (or one argument) functions to obtain solution of certain type it is necessary to choose an initial approximation, which belongs to the corresponding invariant set of nonlinear operators and.

• On the base of computational experiments it is revealed that the branching-off complex solutions, which exist at small sizes of aperture, increase the efficiency of synthesis within 20% - 40% compared with the real (primary) solutions. The presence of different by structure but identical by efficiency solutions (in the sense of value of the corresponding functional), provides for practice the possibility of choosing one of them that has a simpler physical realization. In addition, the branching-off solutions at conservation of the same efficiency which corresponds to real solutions allow to reduce the linear size of the aperture in the limits of 10 to 20 percent.

• The proposed numerical method of solution of nonlinear two-parameter spectral problems arising at investigation of nonlinear integral equations can be successfully applied, in particular, to solving the linear and nonlinear two-parameter spectral problems concerning matrix equations and ordinary differential equations of the second -th order with nonlinear occurrence of the spectral parameters into coefficients of equations and boundary conditions.

• A mathematical analogy between the synthesis problems of acoustic and electromagnetic antennas and synthesis problems of radio allows to use developed methods and numerical algorithms in the above sections of acoustics, radio physics and radio engineering.

• In mathematical aspect the synthesis problems of radiating systems formulated in paragraph 2, belong to problems of non-linear approximation of real finite functions by modules of one-dimensional or two-dimensional, or else discrete Fourier transforms [56,57]. In this connection, the above results can be directly applicable to solving the problems of mean-square approximation of nonnegative finite functions.