Nonlinear Analysis in Synthesis Theory of Radiating Systems

Volume 6, Issue 4, August 2021     |     PP. 76-131      |     PDF (2441 K)    |     Pub. Date: August 29, 2021
DOI: 10.54647/physics14359    64 Downloads     5096 Views  

Author(s)

Petro Sаvenko, Department of numerical methods of the Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU,3 «B» Naukova Str., 79060, Lviv, Ukraine

Abstract
A brief overview of the works of the author and his co-authors on the application of nonlinear analysis methods for solving the nonlinear inverse problems (synthesis problems) for optimizing the different types of radiating systems are presented in the paper. The synthesis problem 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.

Keywords
nonlinear inverse problems, radiating systems, energy characteristics of directivity, nonlinear integral equations of the Hammerstein type, nonuniqueness and branching (or bifurcation) of solutions, numerical finding of optimal solutions, contour diagrams of directivity.

Cite this paper
Petro Sаvenko, Nonlinear Analysis in Synthesis Theory of Radiating Systems , SCIREA Journal of Physics. Volume 6, Issue 4, August 2021 | PP. 76-131. 10.54647/physics14359

References

[ 1 ] G. T. Markov, B. M. Petrov and G. P. Grudinskaja, Electrodynamics and Propagation of Radio Waves, Sov. Radio, Moscow, 1979.
[ 2 ] G. T. Markov and A. F. Chaplin, Excitation of Electromagnetic Waves, Radio and communications, Moscow, 1983.
[ 3 ] V. B. Zhukov, Calculation of the Hydroacoustic Antennas according to the Directivity Pattern, Sudostroienije, Leningrad, 1977.
[ 4 ] A. F. Chaplin, Analysis and Synthesis of Antenna Array, Vyshchaya shkola, Lvov, 1987.
[ 5 ] G. T. Markov and D. N. Sazonov, Antennas, Energiya, Moscow, 1975.
[ 6 ] E. G. Zelkin and V. G. Sokolov, Synthesis Methods of Antenna Arrays and Antenna with continuous aperture, Sov. Radio, Moscow, 1980.
[ 7 ] P. O. Savenko, Nonlinear Problems of Radiating Systems Synthesis (Theory and Methods of Solution), IAPMM of NASU, Lviv, 2002.
[ 8 ] M. Minkovich and V. P. Yakovlev, Theory of Synthesis of Antennas, Sov. Radio, Moscow, 1969.
[ 9 ] B. Z. Katsenelenbaum, “Approximability Problem of the Electromagnetic Field”, Uspekhi phys. nauk, vol. 164, no. 9, pp. 983-993, 1994.
[ 10 ] G. V. Aleksejev, “About Incorrect of the First Kind Nonlinear Operator Equation of Antennas Synthesis Theory”, J. of Comp. Math. and Math. Phys., vol. 19, no. 6, pp. 1590-1595.
[ 11 ] Т N. Tikhonov and V. Y. Arsenin, The Methods of Solution of Incorrect Problems, Nauka, Moscow, 1979.
[ 12 ] V. K. Ivanov, V. V. Vasin and V. P. Tanana Иванов, The Theory of Linear Incorret Problems and its Applications, Nauka, Moscow, 1978..
[ 13 ] M. I.  Andriychuk, N. N. Voitovich, P. A. Savenko and V. P. Tkachuk, Antenna Synthesis According to Amplitude Directivity Pattern: Numerical Methods and Algorithms, Naukova Dumka, Kiev, 1993.
[ 14 ] V. A. Trenogin, Functional Analysis, Nauka, Moscow, 1980.
[ 15 ] N. N. Voitovich and P. A. Savenko, “Generalized Criterion of Proximity Diagram in the Antennas Synthesis Problem by Method V. V. Semjonova”, Radiotekhn. and electron., vol. 17, no. 9, pp. 1794-1798, 1973.
[ 16 ] N. N. Voitovich and P. A. Savenko, “Branching of Solutions of the Antenna Synthesis Problem According to the Prescribed Amplitude Directivity Pattern”, Radiotekhn. and electron., vol. 21, no. 4, pp. 723-729, 1976.
[ 17 ] N. N. Voitovich and P. A. Savenko, “Synthesis of Antennas According to the Prescribed Amplitude Directivity Pattern and Associated Problems of Quasi-optics (review)”, Radiotekhn. and electron, vol. 24, no. 8, pp. 1465-1500, 1979.
[ 18 ] P. A. Savenko, “Numerical Solution of One Class of Nonlinear Problems of Synthesis Theory of Radiating Systems”, J. of Comp. Math. and Math. Phys., vol. 40, no. 6, pp. 929-939, 2000.
[ 19 ] P. O. Savenko, “About the Existence of Solutions of One Class of Nonlinear Inverse Problems of Mathematical Physics Concerning Synthesis of Radiating Systems”, Doklady Akad. Nauk of Ukraine, no. 8, pp. 48-52, 1994.
[ 20 ] N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, 1968.
[ 21 ] М. М. Vainberg, “Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations”, Nauka, Moscow, 1972.
[ 22 ] P. A. Savenko, “Branching of Solutions of the Antenna Synthesis Problem According to the Prescribed Amplitude Directivity Pattern with Use of Regularizing Functionals”, Izv. Vysch. uch. zaved. Radioelectronica, vol. 39, no. 2, pp. 35-50, 1996.
[ 23 ] F. Verlan and V. S. Sizikov, Integral Equations: methods, algorithms, programs. Reference textbook, Naukova Dumka, Kyiv, 1986.
[ 24 ] P. O. Savenko, “Synthesis of Linear Antenna and Antenna Arrays with Use of Regularizing Functionals”, Proc. of Int. Seminar/Workshop "Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory", ІAPММ of NASU, Lviv, pp. 62-63, 1995.
[ 25 ] P. O. Savenko. and V. J. Anokhin, “Numerical Optimization of Shape of Flat Aperture and Field According to the Prescribed Amplitude Directivity Pattern”, Teoret. Electrotekh., Lviv, LDU, vyp. 53, pp. 118-126, 1996.
[ 26 ] P. O. Savenko and V. J. Anokhin, “Synthesis of the Surface Antenna for a Given Power Pattern”, VI-th Int. Conf. "Mathematical Methods in Electromagn. Theory", Lviv, Ukraine, Sept. 10-13, 1996, Lviv, pp. 525-528, 1996.
[ 27 ] V. I Dmitriev and N. I. Berezina, Numerical Methods for Solving the synthesis of Radiating Systems, Izd. MGU, Moscow, 1986.
[ 28 ] P. A. Savenko, “Synthesis of Linear Antenna Arrays According to the Prescribed Amplitude Directivity Pattern”, Izv. Vysch. Uch. Zaved., Radiophys., vol. 22, no. 12, pp. 1498-1504,1979.
[ 29 ] P. A. Savenko and L. M. Pasnak, “Numerically-analytical Method of Synthesis of Linear Antenna Arrays of Vibrators According to the Prescribed Amplitude Directivity Pattern”, Izv. Vysch. Uch. Zaved., Radiophys., vol. 40, no. 12, pp. 11-25, 1997.
[ 30 ] P. A. Savenko and L. M. Pasnak, “The Numerical Solution of Nonlinear Synthesis Problem of Antenna Array of Curved Radiator with Account of their Mutual Influence”, Izv. Vysch. Uch. Zaved., Radioelectr., vol. 43, no. 8, pp. 1-15, 2000.
[ 31 ] P. A. Savenko and M. D. Tkach, “Synthesis of Flat Microstrip Antenna Array According to the Prescribed Energy Directivity Pattern, Izv. Vysch. Uch. Zaved., Radioelectr., vol. 44, no. 6, pp. 19-32, 2001.
[ 32 ] L. M. Pasnak  and P. O. Savenko , “Nonlinear Synthesis Problems of Antenna Arrays Considering Mutual Influence of Scatterers”, Millennium Conference on Antennas & Propagation, Vol. 1. Davos, Switzerland, 9-14 April 2000, Davos, p. 201, 2000.
[ 33 ] L. M. Pasnak  and P. O. Savenko , “Synthesis of Antenna Arrays with Regard to the Radiators Mutual Coupling and Vector Characteristics of Electromagnetic Fields”, III Int. Conf. on Antenna Theory and Techniques (ICATT’99). Sevastopol, September 8-11, 1999, Sevastopol, pp. 135-137, 1999.
[ 34 ] L. M. Pasnak  and P. O. Savenko , “Numerical Solution of Phase Synthesis Problem of Linear Array of Vibrators with Account their Mutual Influence”, Izv. Vysch. Uch. Zaved., Radioelectr.,vol. 40, no. 9, pp. 50-62, 1997.
[ 35 ] V.F.Kravchenko, L. P. Protsakh, P. A. Savenko and M. D. Tkach, “The Mathematical Features of the Synthesis of Plane Equidistant Antenna Arrays According to the Prescribed Amplitude Directivity Pattern”, Antennas, no. 3 (154), pp. 34- 48, 2010.
[ 36 ] P. A. Savenko and M. D. Tkach, “The Numerical Solution of the Nonlinear Synthesis Problem of Microstrip Antenna Array with Account the Mutual Influence of Radiators”, Radiotekhn. And Electron., vol. 46, no. 1, pp. 58-65, 2001.
[ 37 ] P. O. Savenko, “Structure of Solutions of Synthesis Problem of Microstrip Array According to the Prescribed Energy Directivity Pattern”, Matematychni Metody ta Fizyko-Mekhanichni Polya, vol. 46, no. 3, pp. 151-163, 2003.
[ 38 ] L. D. Bakhakh and S. D. Kremenetskii, Synthesis of Radiating Systems (theory and methods of calculations). Sov. Radio, Moscow, 1974.
[ 39 ] V. I. Dmitriev, A. S. Il’inskii and A. G. Sveshnikov, “Development of Mathematical Methods for the Study of Direct and Inverse Problems of Electrodynamics”, Uspekhi math. Nauk, vol. 31, no. 6, pp. 123-141, 1976.
[ 40 ] V. I. Dmitriev and P. P. Sereda, “Mathematical Models and Method of Integral Equations in the Theory of Spiral Wire Antennas”, Chislen. metody electodinam., Moscow, MGU, vyp. 4, pp. 55-65, 1980.
[ 41 ] A. S. Il’inskii, “Electrodynamic Theory of Antenna Arrays”, Izv. Vysch. Uch. Zaved., Radioelectr., vol. 21, no. 2, pp. 9-21, 1978.
[ 42 ] A. S. Il’inskii and I. V. Berezhnaja, “Investigation of the Current Distribution in the System Arbitrarily Located Vibrators”, Vychislit. Metody and Program., Moscow, MGU, vyp. 20, pp. 263-268, 1973.
[ 43 ] L. M. Pasnak  and P. O. Savenko , “Synthesis of Linear Equidistant Antenna Array of Interacting Vibrators According to the Prescribed Amplitude Directivity Pattern”, Radiotekhn. fnd Electron., vol. 42, no. 5, pp. 567-572, 1997.
[ 44 ] P. O. Savenko  and L. M. Pasnak , “Synthesis of Linear Antenna Array for According to the Prescribed Amplitude DP with Account of their Mutual Influence”, Visnyk Lvivs. Univ., vyp. 42, pp. 69-74, 1995.
[ 45 ] P. A. Savenko and L. M. Pasnak , “Numerical Solution of the Nonlinear Synthesis Problem of Antenna Array of Curved Radiators with Account their Mutual influence”, Izv. Vysch. Uch. Zaved., Radioelectr., vol. 43, no. 8, pp. 1-15, 2000.
[ 46 ] P. O. Savenko “About one Method of the Synthesis Problems of Radiating Systems under the Given Power Directivity Pattern”, Int. Conf. "Mathemat. Methods in Electromagn. Theory", Kharkiv, Ukraine, September 11-15, 2000, Kharkiv, pp. 618-620, 2000.
[ 47 ] P. A. Savenko, “About Structure of the Solutions of the Synthesis of Linear Antenna under the Given Energy Directivity Pattern”, Radiophys. and Radioastronomiya, vol. 5, no. 4,pp. 405-415, 2000.
[ 48 ] P. O. Savenko, “Numerical Solution of Synthesis Problems of Radiating Systems According to the Prescribed Power Directivity Pattern”, Proc. of 6th Int. Sem. "Direct and Inverse Problem of Electromagnetic and Acoustic Wave Theory", IAPMM of NASU, Lviv, pp. 141-145, 2001.
[ 49 ] P. A. Savenko, The Numerical Solution of the Inverse Synthesis Problems of Radiating Systems According to the Prescribed Power Directivity Pattern”, J. of Comp. Math. and Math. Phys., vol. 42, no. 10, pp. 1495-1509, 2002.
[ 50 ] M. M. Vainberg, Variational Methods of Investigation of Nonlinear Operators, Gos. Izd. Tekhn.-teoret. Liter., Moscow, 1956.
[ 51 ] M. M. Vainberg, Functional Analysis, Prosveschenije, Moscow, 1979.
[ 52 ] P. O. Savenko, “Branching of solutions of the synthesis problem of linear antenna according to the prescribed amplitude directivity pattern in the space of Helder”, Matematychni Metody ta Fizyko-Mekhanichni Polya, vol. 41, no. 2, pp. 35-44, 1998.
[ 53 ] М. М. Vainberg and V. А. Trenogin, Theory of branching of solutions to nonlinear equations, Nauka, Moscow, 1969.
[ 54 ] N. N. Voitovich and P. A. Savenko, “About one integral equation of antenna synthesis theory”, Matematychni Metody ta Fizyko-Mekhanichni Polya, vyp. 2, pp. 161-163, 1975.
[ 55 ] P. A. Savenko, “About the branching of the solutions of the synthesis problems of radiating systems with flat aperture according to the prescribed amplitude directivity pattern”, Izv. Vysch. Uch. Zaved., Radiophys, vol. 29, no. 3, pp. 339-345, 1986.
[ 56 ] P. O. Savenko, L. P. Protsakh  and M. D. Tkach, “About the mean-square approximation of real finite non-negative continuous function with respect to two variables by module of double Fourier integral”. I, Matematychni Metody ta Fizyko-Mekhanichni Polya, vol. 51, no. 1, pp. 53-64, 2008.
[ 57 ] P. O. Savenko, L. P. Protsakh  and M. D. Tkach, “About the mean-square approximation of real finite non-negative continuous function with respect to two variables by module of double Fourier integral”. ІI // Matematychni Metody ta Fizyko-Mekhanichni Polya, vol. 51, no. 4, pp. 73-78, 2008.
[ 58 ] L. P. Protsakh , P. O. Savenko and M. D. Tkach, “Branching of solutions of the problem of mean-square approximation of real finite function with respect to two variables by modulus of double Fourier transformation”, Matematychni Metody ta Fizyko-Mekhanichni Polya, vol. 53, no. 3, pp. 3-7, 2010.
[ 59 ] I. I. Liashko, V. F. Yemelianov and A. K. Boyarchuk, Bases of classical and modern mathematical analysis, Vyshcha shkola, Kyiv, 1988.
[ 60 ] I. I. Privalov, Introduction to the theory of functions of complex variables, Nauka, Moscow, 1984.
[ 61 ] М. А. Krasnosel’skyy, G. М. Vainikko and P. P. Zabreiko, Approximate solution of operational equations, Nauka, Moscow, 1969.
[ 62 ] A. A. Abramov, L. F. Yukhno,” About a general nonlinear self-adjoint spectral problem for ordinary differential equations systems", J. of Comput. Math. And Math. Phys., vol. 49, no. 4, pp. 624-627, 2009.
[ 63 ] S. I. Solov’yev, “The Finite Element Method for Symmetric Eigenvalue Problems with Nonlinear Occurrence of Spectral Parameter”, J. of Comput. Math. And Math. Phys., vol. 37, no. 11, pp. 1311-1318, 1997.
[ 64 ] L. P. Protsakh , P. O. Savenko , M. D. Tkach , “Numerical solving of two-dimensional nonlinear spectral problem arising in antenna synthesis with flat radiating aperture”, Proc. of Хth Int. Seminar/Workshop "Direct and inverse problems of electromagnetic and acoustic wave theory". Lviv, September 23-25, 2005. Lviv, pp. 199-203, 2005.
[ 65 ] P. A. Savenko, L. P. Protsakh , “Implicit function method in solving the two-dimensional nonlinear spectral problem”, Izv. Vysch. Uch. Zaved., Mathematics, no. 3 (546), pp. 41-44, 2007.
[ 66 ] L. P. Protsakh , P. O. Savenko, “Implicit functions methods in solving the two-parameter linear spectral problems”, Matematychni Metody ta Fizyko-Mekhanichni Polya, vol. 52, no. 2, pp. 42-49, 2009.
[ 67 ] P. O. Savenko, L. P. Protsakh , “Numerical solution of two-point boundary problem with two-dimensional nonlinear spectral parameter”, Matematychni Metody ta Fizyko-Mekhanichni Polya, vol.1, n0. 1, pp. 48-56, 2011.
[ 68 ] G. M. Vainikko, Analysis of discretized methods, Таrtus. Gos. Universitet, Tartu, 1976.
[ 69 ] L. V. Kantorovich and G. P. Akilov, Functional analysis, Nauka, Moscow, 1977.
[ 70 ] V. I. Smirnov, Course of high mathematics, Vol. 2, Nauka, Moscow, 1974.
[ 71 ] P. O. Savenko, “The numerical algorithm of solving the generalized eigenvalues problem for completely continuous self-adjoint operators with nonlinear spectral parameter”, Matematychni Metody ta Fizyko-Mekhanichni Polya, vol. 40, no. 1, pp. 146-150, 1997.
[ 72 ] P. O. Savenko, L. P. Protsakh , “Variational approach to solving the nonlinear vector spectral problem for the case of self-adjoint positive semi-definite operators”, Dopovidi of NASU, no.6, pp. 26-31, 2004.
[ 73 ] P. A. Savenko, “The branching of the solutions of the synthesis problems of radiating systems with flat aperture according to the prescribed amplitude directivity pattern”, Izv. Vysch. Uch. Zaved., Radiophys, vol. 29, no. 3, pp. 339-345, 1986.
[ 74 ] L. P.Protsakh , P. O. Savenko, Tkach M. D. “Branching of solutions for the problem of mean square approximation of a real finite function of two variables by the modulus of double Fourier transformation”, Journal of Mathematical Sciences, Vol. 180, No. 1, pp. 51-67, 2012.
[ 75 ] P. A. Savenko, “The numerical solution of one class of nonlinear synthesis problems of radiating systems”, // J. of Comput. Math. And Math. Phys., vol. 40, no. 6, pp. 929-939, 2000.
[ 76 ] P. O. Savenko, “About Convergence of the Iterative Process in a Nonlinear Synthesis Problems of Radiating Systems under the Use Smoothing Functionals”, Matematychni Metody ta Fizyko-Mekhanichni Polya, vol. 43, no. 2, pp. 105-111, 2000.
[ 77 ] P. O. Savenko, V. J. Anokhin , “Synthesis of Amplitude-Phase Distribution and Shape of a Plane Antenna Aperture for a Given Power Pattern”, IEEE Trans. on Antennas and Propag, vol. 45, no. 4, pp. 744-747, 1997.
[ 78 ] V. J. Anokhin , P. O. Savenko, “Synthesis of the Contour Directivity Pattern Without Critical Zones”, URSI Int. Symp. Electromagn. Theory, St. Peterburg, Russia, May 23-26, 1995, St. Peterburg., pp. 197-199, 1995.
[ 79 ] P. O. Savenko, V. J. Anokhin , “Synthesis of the Surface Antenna for a Given Power Pattern”, 6th Int. Conf. "Mathematical Methods in Electromagn. Theory", Lviv, Ukraine, Sept. 10-13, 1996, Lviv, pp. 525-528, 1996.
[ 80 ] B. Podlevskyi, P. Savenko, “Synthesis of the Multibeams Reflector Antenna Directivity Patterns on the Given Magnitude”, 6th Int. Conf. "Mathematical Methods in Electromagn. Theory", Lviv, Ukraine, Sept. 10-13, 1996, Lviv, pp. 521-524, 1996.
[ 81 ] B. Podlevskyi, P. Savenko, “Synthesis of the Variable Shape Contoured Directivity Patterns of the Hybrid Reflector Antennas”, Trans Black Sea Region Symposium on Applied Electromagnetism (April 17-19, 1996, Metsovo, Greece)., N.T.U.A. PRES, Athens., p. 20, 1996.
[ 82 ] B. Podlevskyi, P. Savenko, “Synthesis of the Variable Shape Contoured Directivity Patterns of the Hybrid Reflector Antennas”, Electromagnetics, vol.18, no. 5, pp. 507-527,1998.