ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Spectral Method of Nonlinear Stochastic Control System Analysis

Author(s):

A. V. Panteleev

Moscow Aviation Institute
(National Research University)
125993, Moscow, Russia,
GSP-3, A-80,
Volokolamskoe shosse, 4

Konstantin A. Rybakov

Moscow aviation institute (national research university),
Mathematical cybernetics department, associate professor
associate professor, candidate of physico-mathematical sciences

rkoffice@mail.ru

I. L. Sotskova

Moscow Aviation Institute
(National Research University)
125993, Moscow, Russia,
GSP-3, A-80,
Volokolamskoe shosse, 4

Abstract:

This book presents methods of analysis for nonlinear continuous stochastic control systems with fixed and random structure, based on the spectral form of mathematical description and focused on the use of modern high-performance computers. It is intended for technicians and engineers interested in modern problems of control theory, as well as undergraduate and graduate students of technical colleges and universities.

References:

  1. Averina T. A. Metod Monte-Karlo dlya analiza dinamiki nelineynykh sistem so sluchaynoy strukturoy [Monte Carlo method for analysis of dynamics of nonlinear systems with random structure] // Tr. konf. "Identifikatsiya sistem i zadachi upravleniya SICPRO'03" [Proceedings of the II International conference "System Identification and Control Problems SICPRO'03"]. - M. : IPU RAN [Moscow: Institute of Control Sciences, Russian Academy of Sciences], 2003. - P. 2106-2121
  2. Averina T. A. Statisticheskiy analiz sistem s peremennoy strukturoy upravleniya [Statistical analysis of systems with variable control structure] // Tr. konf. "Identifikatsiya sistem i zadachi upravleniya SICPRO'04" [Proceedings of the III International conference "System Identification and Control Problems SICPRO'04"]. - M. : IPU RAN [Moscow: Institute of Control Sciences, Russian Academy of Sciences], 2004. - P. 490-501
  3. Antosik P., Mikusinski J., Sikorski R. Theory of distributions. The sequential approach. - Elsevier Scientific, 1973
  4. Artem'ev V. M. Teoriya dinamicheskikh sistem so sluchaynymi izmeneniyami struktury [The theory of dynamical systems with random changes in the structure]. - Minsk: Vysheyshaya shkola [Minsk: Higher School], 1979
  5. Artem'ev V. M., Gane V. A., Stepanov V. L. Upravlenie v sistemakh s razdeleniem vremeni [Control systems with timesharing]. - Minsk: Vysheyshaya shkola [Minsk: Higher School], 1982
  6. Balakrishnan A. V. Applied functional analysis. - Springer-Verlag, 1976
  7. Bateman H., Erdelyi A. Higher transcendental functions. Vol. II. - McGraw-Hill, 1953
  8. Bortakovskiy A. S., Panteleev A. V. Lineynaya algebra v primerakh i zadachakh [Linear algebra: examples and problems]. - M. : Vysshaya shkola [Moscow: Higher School], 2005
  9. Bortakovskiy A. S., Panteleev A. V. Analiticheskaya geometriya v primerakh i zadachakh [Analytic geometry: examples and problems]. - M. : Vysshaya shkola [Moscow: Higher School], 2005
  10. Bukhalev V. A. Raspoznavanie, otsenivanie i upravlenie v sistemakh so sluchaynoy skachkoobraznoy strukturoy [Recognition, estimation and control in systems with random jump structure]. - M. : Fizmatlit [Moscow: Fizmatlit Publishing Company, Nauka Publishers], 1996
  11. Vinogradov V. I. SPEKTR - integrirovannaya problemno-orientirovannaya sistema modelirovaniya [SPECTR - integrated problem-oriented system for modeling] // Informatika. Avtomatizatsiya proektirovaniya [Information sciences. Computer-aided design]. - M. : VIMI [Moscow: VIMI], 1992. № 2-3. - P. 50-56
  12. Vulikh B. Z. Vvedenie v funktsional'nyy analiz [Introduction to functional analysis]. - M. : Nauka [Moscow: Nauka Publishers], 1967
  13. Galanina E. N. Instrumental'nye sredstva obespecheniya etapa algoritmizatsii rascheta sistem upravleniya spektral'nym metodom [Algorithmization tools for control systems calculation by spectral method] // Informatika. Avtomatizatsiya proektirovaniya [Information sciences. Computer-aided design]. - M. : VIMI [Moscow: VIMI], 1992. № 2-3. - P. 57-60
  14. Gantmakher F. R. Teoriya matrits [The theory of matrices]. - M. : Nauka [Moscow: Nauka Publishers], 1966
  15. Gane V. A., Kuklev E. A., Stepanov V. L. Sistemy upravleniya pri skachkoobraznykh vozdeystviyakh [Control systems with jump input actions]. - Minsk: Nauka i tekhnika [Minsk: Science and Technology], 1985
  16. Gardiner C. W. Handbook of stochastic methods for physics, chemistry and the natural sciences. - Springer, 1997
  17. Gnedenko B. V., Belyaev Yu. K., Solov'ev A. D. Matematicheskie metody v teorii nadezhnosti. Osnovnye kharakteristiki nadezhnosti i ikh statisticheskiy analiz [Mathematical methods in reliability theory. Main characteristics of reliability and statistical analysis]. - M. : Nauka [Moscow: Nauka Publishers], 1965
  18. Grishin V. N., Dyatlov V. A., Milov L. T. Modeli, algoritmy i ustroystva identifikatsii slozhnykh system [Models, algorithms and devices for identification of complex systems]. - L. : Energoatomizdat [Leningrad: Energoatomizdat], 1985
  19. Gurman V. I. Vyrozhdennye zadachi optimal'nogo upravleniya [Degenerate problems of optimal control]. - M. : Nauka [Moscow: Nauka Publishers], 1977
  20. Jackson D. Fourier series and orthogonal polynomials. - The Mathematical Association of America, 1941
  21. Emel'yanov S. V. Sistemy avtomaticheskogo upravleniya s peremennoy strukturoy [Automatic control systems with variable structure]. - M. : Nauka [Moscow: Nauka Publishers], 1967
  22. Ivanov V. K., Vasin V. V., Tanana V. P. Teoriya lineynykh nekorrektnykh zadach i ee prilozheniya [Theory of linear ill-posed problems and its applications]. - M. : Nauka [Moscow: Nauka Publishers], 1978
  23. Kazakov I. E. Statisticheskaya dinamika sistem s peremennoy strukturoy [Statistical dynamics of systems with variable structure]. - M. : Nauka [Moscow: Nauka Publishers], 1977
  24. Kazakov I. E. Stochastic systems with random change of structure (a survey) // Soviet Journal of Computer and Systems Sciences. - 1989. Vol. 27. № 4. - P. 78-93
  25. Kazakov I. E., Artem'ev V. M. Optimizatsiya dinamicheskikh sistem sluchaynoy struktury [Optimization of dynamic systems with random structure]. - M. : Nauka [Moscow: Nauka Publishers], 1980
  26. Kazakov I. Ye., Artem'ev V. M., Bukhalev V. A. Analiz sistem sluchaynoy struktury [Analysis of systems with random structure]. - M. : Fizmatlit [Moscow: Fizmatlit Publishing Company, Nauka Publishers], 1993
  27. Kazakov I. E., Mal'chikov S. V. Analiz stokhasticheskikh sistem v prostranstve sostoyaniy [Analysis of stochastic systems in the state space]. - M. : Nauka [Moscow: Nauka Publishers], 1983
  28. Kireev V. I., Panteleev A. V. Chislennye metody v primerakh i zadachakh [Numerical methods: examples and problems]. - M. : Vysshaya shkola [Moscow: Higher School], 2006
  29. Kolmogorov A. N., Fomin S. V. Elementy teorii funktsiy i funktsional'nogo analiza [Elements of the theory of functions and functional analysis]. - M. : Nauka [Moscow: Nauka Publishers], 1976
  30. Korn G. A., Korn T. M. Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and review. - McGraw-Hill, 1968
  31. Krasovskiy A. A. Fazovoe prostranstvo i statisticheskaya teoriya dinamicheskikh system [Phase space and the statistical theory of dynamical systems]. - M. : Nauka [Moscow: Nauka Publishers], 1974
  32. Cooke R. G. Infinite matrices and sequence spaces. - MacMillan and Co., 1950
  33. Ladyzhenskaya O. A., Solonnikov V. A., Ural'ceva N. N. Linear and quasilinear equations of parabolic type. - AMS, 1968
  34. Lapin S. V., Egupov N. D. Teoriya matrichnykh operatorov i ee prilozhenie k zadacham avtomaticheskogo upravleniya [The theory of matrix operators and its application to problems of automatic control]. - M. : Izdatel'stvo MGTU im. N. E. Baumana [Bauman Moscow State Technical University Press], 1997
  35. Lysenko L. N., Nguen Tang Kyong. Teoreticheskie i prikladnye aspekty sinteza mul'tistrukturnykh skhem rekurrentnoy obrabotki informatsii v navigatsionnykh sistemakh letatel'nykh apparatov [Theoretical and applied aspects of the synthesis multistructural schemes of recurrent information processing systems in the aircraft navigation] // Izvestiya RAN. Teoriya i sistemy upravleniya [Control Theory and Control Systems]. - 1997. № 6. - P. 38-48
  36. Miller B. M., Pankov A. R. Teoriya sluchaynykh protsessov [The theory of stochastic processes]. - M. : Fizmatlit [Moscow: Fizmatlit Publishing Company, Nauka Publishers], 2002
  37. Nikitin N. N. Digital modelling methods for stochastic differential equations when absorbing boundaries are present // Automation and Remote Control. - 1988. Vol. 49. - P. 605-612
  38. Panteleev A. V. Variatsionnoe ischislenie v primerakh i zadachakh [The calculus of variations: examples and problems]. - M. : Vysshaya shkola [Moscow: Higher School], 2006
  39. Panteleev A. V., Bortakovskiy A. S. Teoriya upravleniya v primerakh i zadachakh [The control theory: examples and problems]. - M. : Vysshaya shkola [Moscow: Higher School], 2003
  40. Panteleev A. V., Letova T. A. Metody optimizatsii v primerakh i zadachakh [Optimization methods: examples and problems]. - M. : Vysshaya shkola [Moscow: Higher School], 2005
  41. Panteleev A. V., Semenov V. V. Sintez optimal'nykh sistem upravleniya pri nepolnoy informatsii [Synthesis of optimal control systems with incomplete information]. - M. : Izdatel'stvo MAI [Moscow Aviation Institute Press], 1992
  42. Panteleev A. V., Sotskova I. L. Priblizhennyy metod sinteza optimal'nykh stokhasticheskikh sistem pri nepolnoy informatsii [Approximate method for the synthesis of optimal stochastic systems with incomplete information] // Problemy matematiki v fiziko-tekhnicheskikh i ekonomicheskikh zadachakh [Problems of mathematics in physics, engineering and economics]: Mezhved. sb. nauch. tr. [Scientific papers] - M. : MFTI [Moscow Institute of Physics and Technology Press], 1993. - P. 135-142
  43. Panteleev A. V., Yakimova A. S. Teoriya funktsiy kompleksnogo peremennogo i operatsionnoe ischislenie v primerakh i zadachakh [Theory of functions of a complex variable and operational calculus: examples and problems]. - M. : Vysshaya shkola [Moscow: Higher School], 2001
  44. Panteleev A. V., Yakimova A. S., Bosov A. V. Obyknovennye differentsial'nye uravneniya v primerakh i zadachakh [Ordinary differential equations: examples and problems]. - M. : Vysshaya shkola [Moscow: Higher School], 2001
  45. Pugachev V. S. Lektsii po funktsional'nomu analizu [Lectures on functional analysis]. - M. : Izdatel'stvo MAI [Moscow Aviation Institute Press], 1996
  46. Pugachev V. S., Sinitsyn I. N. Stokhasticheskie differentsial'nye sistemy. Analiz i fil'tratsiya [Stochastic differential systems. Analysis and filtering]. - M. : Nauka [Moscow: Nauka Publishers], 1990
  47. Rybakov K. A. Programmnoe obespechenie spektral'nykh preobrazovaniy Spectrum [Spectral method software] // Trudy MAI [Moscow Aviation Institute Proceedings]. - 2003. № 14. - http://www. mai. ru
  48. Rybakov K. A. Spektralnye kharakteristiki lineynykh funktsionalov i ikh prilozheniya k analizu i sintezu stokhasticheskikh sistem upravleniya [Spectral characteristics of linear functionals and their applications to stochastic control systems analysis and synthesis] // Trudy MAI [Moscow Aviation Institute Proceedings]. - 2005. № 18. - http://www. mai. ru
  49. Rybakov K. A., Sotskova I. L. An optimal control for random-structure nonlinear systems under incomplete state vector information // Automation and Remote Control. - 2006. Vol. 67. - P. 1070-1081
  50. Rybakov K. A., Sotskova I. L. Algoritm analiza mnogomernykh stokhasticheskikh nepreryvno-diskretnykh sistem spektral'nym metodom [Analysis algorithm of multidimensional continuous-discrete stochastic systems by the spectral method] // Teoreticheskie voprosy vychislitel'noy tekhniki i programmnogo obespecheniya [Theoretical questions of computer technology and software]: Mezhvuz. sb. nauch. tr. [Scientific papers] - M. : MIREA [Moscow State Institute of Radio Engineering, Electronics and Automation], 2003. - P. 81-89
  51. Rybakov K. A., Sotskova I. L. Algoritmicheskoe obespechenie spektral'nogo metoda analiza sistem upravleniya v neogranichennykh oblastyakh izmeneniya vremeni i fazovykh koordinat [Spectral method algorithms for control system analysis in unbounded areas of time and phase coordinates] // Trudy MAI [Moscow Aviation Institute Proceedings]. - 2004. № 16. - http://www. mai. ru
  52. Rybakov K. A., Sotskova I. L. Spektral'nyy metod analiza nelineynykh sistem so sluchaynoy strukturoy [The spectral method for analysis of nonlinear systems with random structure] // Aviakosmicheskoe priborostroenie [Aerospace Instrument-Making]. - 2006. № 2. - P. 12-16
  53. Rybin V. V. Printsipy postroeniya i semanticheskie osobennosti programmnoy sistemy spektral'nogo metoda analiza sistem upravleniya LA [Principles of construction and semantic features of a spectral analysis software for aircraft control systems] // Semanticheskoe programmirovanie v avtomatizirovannom proektirovanii sistem upravleniya letatel'nymi apparatami [Semantic programming in computer-aided design for aircraft control systems]: Tem. sb. nauch. tr. [Scientific papers] - M. : MAI [Moscow Aviation Institute Press], 1985. - P. 52-59
  54. Rybin V. V. Opisanie signalov i lineynykh nestatsionarnykh sistem upravleniya v bazisakh veyvletov i ikh analiz v vychislitel'nykh sredakh [Description of linear nonstationary signals and control systems by the wavelets and their analysis with computers]. - M. : Izdatel'stvo MAI [Moscow Aviation Institute Press], 2003
  55. Rybin V. V. Razrabotka i primenenie paketa rasshireniya Spektr_SM paketa Simulink CKM Matlab [Development and application of expansion package "Spektr_SM" for Simulink Matlab]. - M. : Izdatel'stvo MAI [Moscow Aviation Institute Press], 2004
  56. Rybin V. V. Razrabotka i primenenie paketa rasshireniya Spektr_SM sistemy VisSim+Mathcad [Development and application of the Spektr_SM extension package for Simulink CMS Matlab] // Trudy MAI [Moscow Aviation Institute Proceedings]. - 2004. № 17. - http://www. mai. ru
  57. Szego G. Orthogonal polynomials. - AMS, 1939
  58. Semenov V. V. Spektral'nyy analiz i sintez lineynykh sistem s peremennymi parametrami na konechnykh nestatsionarnykh intervalakh vremeni [Spectral analysis and synthesis of linear systems with variable parameters on the unsteady intervals] // Tekhnicheskaya kibernetika. Teoriya avtomaticheskogo regulirovaniya [Technical cybernetics. The theory of automatic control] / Pod red. V. V. Solodovnikova [Ed. by V. V. Solodovnikov]. Kn. 3, ch. 1. Teoriya nestatsionarnykh, nelineynykh i samonastraivayushchikhsya sistem avtomaticheskogo regulirovaniya [Vol. 3, part 1. The theory of nonstationary , nonlinear and adaptive control systems]. - M. : Mashinostroenie [Moscow: Mechanical engineering], 1969. - P. 136-196
  59. Semenov V. V. Spektral'nyy metod analiza nestatsionarnykh stokhasticheskikh sistem [Spectral method of nonstationary stochastic system analysis] // Izvestiya vuzov. Priborostroenie [Proceedings of the universities. Instrument-making]. - 1970. T. XIII [Vol. XIII]. № 2. - P. 37-42
  60. Semenov V. V. Uravnenie obobshchennoy kharakteristicheskoy funktsii vektora sostoyaniya sistem avtomaticheskogo upravleniya [Equation for generalized characteristic function of the state vector for automatic control systems] // Analiticheskie metody sinteza regulyatorov [Analytical methods for the regulator synthesis]: Mezhvuz. nauch. sb. [Scientific papers] - Saratov: SPI [Saratov Polytechnic Institute], 1977. Vyp. 2 [Issue 2]. - P. 3-36
  61. Semenov V. V. Sintez algoritmov upravleniya nelineynymi sistemami pri sluchaynykh vozdeystviyakh s ogranichennym sostavom tochnykh izmereniy [Synthesis of control algorithms for nonlinear stochastic systems with a limited amount of measurements] // Analiticheskie metody sinteza regulyatorov [Analytical methods for the regulator synthesis]: Mezhvuz. nauch. sb. [Scientific papers] - Saratov: SPI, 1978 [Saratov Polytechnic Institute]. Vyp. 3 [Issue 3]. - P. 3-20
  62. Semenov V. V. Printsipy organizatsii matematicheskogo obespecheniya SAPR dinamiki sistem upravleniya s dialogovymi formirovatelyami programm [CAD principles for dynamic control systems with interactive programs formers] // Pererabotka informatsii v zadachakh upravleniya [information processing in control problems]: Tem. sb. nauch. tr. [Scientific papers] - M. : MAI [Moscow Aviation Institute Press], 1980. - P. 4-14
  63. Semenov V. V., Panteleev A. V., Rudenko E. A., Bortakovskiy A. S. Metody opisaniya, analiza i sinteza nelineynykh sistem upravleniya [Methods of description, analysis and synthesis for nonlinear control systems]. - M. : Izdatel'stvo MAI [Moscow Aviation Institute Press], 1993
  64. Semenov V. V., Rybin V. V. Algoritmicheskoe i programmnoe obespechenie rascheta nestatsionarnykh nepreryvno-diskretnykh sistem upravleniya LA spektral'nym metodom [Algorithms and software for unsteady discrete-continuous aircraft control systems by the spectral method]. - M. : MAI [Moscow Aviation Institute Press], 1984
  65. Semenov V. V., Sivtsov V. I. Obobshchenie spektral'nogo metoda analiza nestatsionarnykh sistem na konechnykh intervalakh vremeni na nelineynye sistemy [A generalization of spectral method for analysis of nonstationary systems on finite time intervals to nonlinear systems] // Izvestiya vuzov. Priborostroenie [Proceedings of the universities. Instrument-making]. - 1969. T. XII [Vol. XII]. № 12. - P. 63-68
  66. Semenov V. V., Solodovnikov V. V. Spektral'nyy analiz lineynykh sistem s peremennymi parametrami na konechnykh nestatsionarnykh intervalakh vremeni [Spectral analysis of linear systems with variable parameters on finite unsteady time intervals] // Avtomatika i telemehanika [Automation and Remote Control]. - 1968. № 11. - P. 14-23
  67. Sklyarevich A. N. Nadezhnost' sistem s nakopleniem narusheniy [Reliability of systems with accumulation of violations]. - Riga: Zinatne, 1969
  68. Sklyarevich A. N. Vvedenie v statisticheskuyu dinamiku sistem s vozmozhnymi narusheniyami [Introduction to the statistical dynamics of systems with possible violations]. - Riga: Zinatne, 1973
  69. Sokolov N. P. Vvedenie v teoriyu mnogomernykh matrits [An introduction to the theory of multidimensional matrices]. - Kiev: Naukova dumka, 1972
  70. Solodovnikov V. V., Dmitriev A. N., Egupov N. D. Spektral'nye metody rascheta i proektirovaniya sistem upravleniya [Spectral methods for control systems calculation and design]. - M. : Mashinostroenie [Moscow: Mechanical engineering], 1986
  71. Solodovnikov V. V., Semenov V. V. Spektralnaya teoriya nestatsionarnykh sistem upravleniya [Spectral theory of nonstationary control systems]. - M. : Nauka [Nauka Publishers], 1974
  72. Solodovnikov V. V., Semenov V. V. Spektral'nyy metod rascheta nestatsionarnykh sistem upravleniya letatel'nymi apparatami [Spectral method for calculating unsteady aircraft control systems]. - M. : Mashinostroenie [Moscow: Mechanical engineering], 1975
  73. Solodovnikov V. V., Semenov V. V., Peshel' M., Nedo D. Raschet sistem upravleniya na TsVM: spektral'nyy i interpolyatsionnyy metody [Design of control systems on digital computers: spectral and interpolational methods]. - M. : Mashinostroenie [Moscow: Mechanical engineering], 1979
  74. Sotskova I. L. Semantichekaya model' analiza stokhasticheskikh sistem upravleniya LA // Semanticheskoe programmirovanie v avtomatizirovannom proektirovanii sistem upravleniya letatel'nymi apparatami [Semantic programming in computer-aided design for aircraft control systems]: Tem. sb. nauch. tr. [Scientific papers] - M. : MAI [Moscow Aviation Institute Press], 1985. - P. 59-68
  75. Sotskova I. L. Primenenie apparata obobshchennoy kharakteristicheskoy funktsii k analizu stokhasticheskikh sistem upravleniya LA [Application of generalized characteristic functions to analysis of stochastic control systems for aircrafts] // Zadachi stokhasticheskogo upravleniya [Stochastic control problems]: Tem. sb. nauch. tr. [Scientific papers] - M. : MAI [Moscow Aviation Institute Press], 1986. - P. 71-78
  76. Sotskova I. L. Issledovanie korrektnosti kraevykh zadach dlya uravneniya Fokkera-Planka-Kolmogorova v klasse obobshchennykh kharakteristicheskikh funktsiy [Studying the correctness of boundary value problems for the Fokker-Planck-Kolmogorov equation in the generalized characteristic function class] // Novye zadachi optimizatsii aviatsionnykh system [New aircraft systems optimization problem]: Tem. sb. nauch. tr. [Scientific papers] - M. : MAI [Moscow Aviation Institute Press], 1989. - P. 26-33
  77. Spravochnik po teorii avtomaticheskogo upravleniya [Handbook on the control theory] / Pod red. A. A. Krasovskogo [Ed. by A. A. Krasovsky]. - M. : Nauka [Moscow: Nauka Publishers], 1987
  78. Suetin P. K. Klassicheskie ortogonal'nye mnogochleny [Classical orthogonal polynomials]. - M. : Nauka [Moscow: Nauka Publishers], 1979
  79. Suetin P. K. Ortogonal'nye mnogochleny po dvum peremennym [Orthogonal polynomials in two variables]. - M. : Nauka [Moscow: Nauka Publishers], 1988
  80. Tablitsy i matematicheskoe obespechenie spektralnogo metoda teorii avtomaticheskogo upravleniya [Tables and mathware for the Spectral Method of Control Theory] / Pod red. V. V. Semenova [Ed. by V. V. Semenov]. - M. : MVTU im. N. E. Baumana [Bauman Moscow State Technical University Press], 1973
  81. Teoriya sistem s peremennoy strukturoy [Theory of systems with variable structure] / Pod red. S. V. Emel'yanova [Ed. by S. V. Emel'yanov]. - M. : Nauka [Moscow: Nauka Publishers], 1970
  82. Tikhonov A. N., Arsenin V. Y. Solution of ill-posed problems. - Winston & Sons, 1977
  83. Tikhonov V. I., Mironov M. A. Markovskie protsessy [Markov Processes]. - M. : Sovetskoe radio [Moscow: Soviet Radio], 1977
  84. Utkin V. I. Variable structure systems: Present and future // Automation and Remote Control. - 1983. Vol. 44. - P. 1105-1120
  85. Fedosov E. A., Insarov V. V., Selivokhin O. S. Sistemy upravleniya konechnym polozheniem v usloviyakh protivodeystviya sredy [Control systems for object terminal positioning under enviroment's counteraction]. - M. : Nauka [Moscow: Nauka Publishers], 1989
  86. Formalev V. F., Reviznikov D. L. Chislennye metody [Numerical methods]. - M. : Fizmatlit [Moscow: Fizmatlit Publishing Company, Nauka Publishers], 2004
  87. Shaikin M. E. A coordinate-free approach to the method of moments in the theory of multidimensional stochastic systems // Automation and Remote Control. - 2002. Vol. 63. - P. 766-776
  88. Di Paola M., Sofi A. Approximate solution of the Fokker-Planck-Kolmogorov equation // Probabilistic Engineering Mechanics. - 2002. Vol. 17. № 4. - P. 369-384
  89. Fok J., Guo B., Tang T. Combined Hermite spectral-finite difference method for the Fokker-Planck equation // Mathematics of computation. - 2001. Vol. 71. № 240. - P. 1497-1528
  90. Ghosh M. K., Arapostathis A., Marcus S. I. Ergodic control of switching diffusions // SIAM Journal on Control and Optimization. - 1997. Vol. 35. № 6. - P. 1952-1988
  91. Kontorovitch V. Pontryagin equations for non-linear dynamic systems with random structure // Nonlinear Analysis. - 2001. Vol. 47. № 3. - P. 1501-1512
  92. Liberzon D., Brockett R. W. Spectral analysis of Fokker-Planck and related operators arising from linear stochastic differential equations // SIAM Journal on Control and Optimization. - 2000. Vol. 38. № 5. - P. 1453-1467
  93. Risken H. The Fokker-Planck equation: Methods of solution and applications. - New York: Springer-Verlag, 1996
  94. Semenov V. V., Sotskova I. L. The spectral method for solving Fokker-Planck-Kolmogorov equation for stochastic control system analysis // Second IFAC symposium on stochastic control. Preprints, part 1. - 1986. - P. 131-136
  95. Ulyanov S. V., Feng M., Ulyanov V. S., Yamafuji K., Fukuda T., Arai F. Stochastic analysis of time-variant nonlinear dynamic systems // Probabilistic Engineering Mechanics. - 1998. Vol. 13. № 3. - P. 183-226
  96. Wei G. W. A unified approach for the solution of the Fokker-Planck equation // Journal of Physics A: Mathematical and General. - 2000. Vol. 33. - P. 4935-4953

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