АНАЛИЗ УСТОЙЧИВОСТИ ЖЕСТКИХ СИСТЕМ ОБЫКНОВЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ
Ключевые слова:
Устойчивость по Ляпунову, анализ устойчивости жестких систем, решение жестких систем
Аннотация
Предложен метод анализа устойчивости в смысле Ляпунова систем обыкновенных
дифференциальных уравнений. Метод базируется на критериях устойчивости в виде необхо-
димых и достаточных условий, полученных на основе векторно-матричных преобразований
разностных схем численного интегрирования. Представлены разновидности критериев в
мультипликативной, аддитивной и матричной форме. Конструкция критериев влечет воз-
можность их программной реализации. Для повышения достоверности анализа устойчиво-
сти приближения решения, входящего в конструкцию критериев, находятся на основе кусоч-
но-интерполяционной аппроксимации полиномами Лагранжа, преобразованными к форме с
числовыми коэффициентами. Проведен программный и численный эксперимент по анализу
устойчивости модели периодической реакции Белоусова-Жаботинского, относящейся к классу жестких систем, при заданных начальных условиях. Анализ выполняется на основе пред-
ставленных критериев и по результатам работы программы однозначно определяется ха-
рактер устойчивости в режиме реального времени. На основе результатов эксперимента
можно утверждать, что замена разностных приближений решения на кусочно-
интерполяционные приближения повышает достоверность анализа устойчивости, сокраща-
ет время исследования, позволяет определять асимптотические свойства решения. В целом
предложенный подход является альтернативой методам качественной теории дифференци-
альных уравнений и дает возможность в режиме реального времени достоверно установить
характер устойчивости жестких систем обыкновенных дифференциальных уравнений.
Литература
1. Mel'nikov G.I., Mel'nikov V.G., Dudarenko N.A., Talapov V.V. Ustoychivost' dvizheniya
nelineynykh dinamicheskikh sistem pri postoyanno deystvuyushchikh vozmushcheniyakh
[tability of nonlinear dynamical system motion under constantly acting perturbations],
Nauchno-tekhnicheskiy vestnik informatsionnykh tekhnologiy, mekhaniki i optiki [Scientific
and technical journal of information technologies, mechanics and optics], 2019, Vol. 19, No. 2,
pp. 216-221.
2. Mironov V.V., Mitrokhin Yu.S. Tekhnologicheskiy podkhod k issledovaniyu ustoychivosti
dinamicheskikh sistem: prikladnye voprosy [Constructive approach to the research of dynamic
systems stability: applied problems], Vestnik RGRTU [Vestnik of RSREU]. 2017, No. 59,
pp. 127-135.
3. Aleksandrov A.Yu., Zhabko A.P., Kosov A.A. Analiz ustoychivosti i stabilizatsiya nelineynykh
sistem na osnove dekompozitsii [Analysis of stability and stabilization of nonlinear systems
via decomposition], Sibirskiy matematicheskiy zhurnal [Siberian mathematical journal]. 2015,
Vol. 56, No. 6, pp. 1215-1233.
4. Hammarling S.J. Numerical solution of the stable, non-negative definite Lyapunov equation,
IMA J. of Num. Analysis, 1982, Vol. 2, No. 3, pp. 303-323.
5. Luyckx L., Loccufier M., Noldus E. Computational methods in nonlinear stability analysis:
stability boundary calculations, J. Comput. Appl. Math, 2004, Vol. 168, No. 12, pp. 289-297.
6. Giesl P., Hafstein S. Computation of Lyapunov functions for nonlinear discrete time systems
by linear programming, J. Difference Equ. Appl, 2014, Vol. 20, No. 4, pp. 610-640.
7. Olgac N., Sipahi R. A practical method for analyzing the stability of neutral type LTI-time
delayed systems, Automatica, 2004, Vol. 40, No. 5, pp. 847-853.
8. Hafstein S. A constructive converse Lyapunov theorem on asymptotic stability for nonlinear
autonomous ordinary differential equations, Dynamical Systems, 2005, Vol. 20, pp. 281-299.
9. Romm Ya.E. Komp'yuterno-orientirovannyy analiz ustoychivosti na osnove rekurrentnykh
preobrazovaniy raznostnykh resheniy obyknovennykh differentsial'nykh uravneniy [Computeroriented
stability analysis based on recurrent transformation of difference solutions of ordinary
differential equations], Kibernetika i sistemnyy analiz [Cybernetics and Systems Analysis],
2015, Vol. 51, No. 3, pp. 107-124.
10. Bulanov S.G. Analiz ustoychivosti sistem lineynykh differentsial'nykh uravneniy na osnove
preobrazovaniya raznostnykh skhem [Stability analysis of systems of linear differential equations
based on transformation of difference schemes], Mekhatronika, avtomatizatsiya,
upravlenie [Mekhatronika, Avtomatizatsiya, Upravlenie], 2019, Vol. 20, No. 9, pp. 542-549.
11. Romm Ya.E., Bulanov S.G. Programmnye kriterii ustoychivosti resheniya zadachi Koshi dlya
sistem lineynykh differentsial'nykh uravneniy na osnove raznostnykh skhem chislennogo
integrirovaniya [Program stability criteria for the solution of the Cauchy problem for systems
of linear differential equations based on difference numerical integration schemes], Izvestiya
vuzov. Severo-Kavkazskiy region. Tekhnicheskie nauki. Spetsial'nyy vypusk «Matematicheskoe
modelirovanie i komp'yuternye tekhnologii» [University news. North-caucasian region. Technical
sciences series. Special issue «Mathematical Modeling and Computer Technologies»],
2004, pp. 75-80.
12. Romm Ya.E., Bulanov S.G. Chislennyy eksperiment po komp'yuternomu analizu ustoychivosti
resheniy obyknovennykh differentsial'nykh uravneniy na osnove kriteriev matrichnogo vida
[Computer analysis of the stability of systems of linear differential equations with nonlinear
addition], Dep. v VINITI [Dep. in VINITI], 14.08.17, No. 89, 20 p.
13. Romm Ya.E., Bulanov S.G. Komp'yuternyy analiz ustoychivosti sistem lineynykh
differentsial'nykh uravneniy s nelineynoy dobavkoy [Computer analysis of the stability of systems
of linear differential equations with nonlinear addition], Dep. v VINITI [Dep. in VINITI],
11.03.10, No. 147, 33 p.
14. Romm Ya.E. Modelirovanie ustoychivosti po Lyapunovu na osnove preobrazovaniy
raznostnykh skhem resheniy obyknovennykh differentsial'nykh uravneniy [Modeling of stability
according to Lyapunov based on difference schemes transformations for solutions of ordinary
differential equations], Matematicheskoe modelirovanie [Mathematical Modeling], 2008,
Vol. 20, No. 12, pp. 105-118.
15. Romm Ya.E., Dzhanunts G.A. Kusochnaya interpolyatsiya funktsiy, proizvodnykh i integralov
s prilozheniem k resheniyu obyknovennykh differentsial'nykh uravneniy [Piecewise interpolation
of functions, derivatives and integrals with application to the solution of ordinary differential
equations], Sovremennye naukoemkie tekhnologii [Modern high technologies], 2020, No.
12 (part 2), pp. 291-316.
16. Dzhanunts G.A., Romm Ya.E. Var'iruemoe kusochno-interpolyatsionnoe reshenie zadachi
Koshi dlya obyknovennykh differentsial'nykh uravneniy s iteratsionnym utochneniem [The
varying piecewise interpolation solution of the Cauchy problem for ordinary differential equations
with iterative refinement], Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki
[Computational Mathematics and Mathematical Physics Journal], 2017, Vol. 57, No. 10,
pp. 1616-1634.
17. Romm Ya.E. Komp'yuterno-orientirovannyy analiz ustoychivosti resheniy differentsial'nykh
sistem [Computer-oriented stability analysis of solutions of differential systems], Sovremennye
naukoemkie tekhnologii [Modern high technologies], 2020, No. 4, pp. 42-63].
18. Bulanov S.G. Computer analysis of differential systems stability based on linearization and
matrix multiplicative criteria, Journal of Physics: Conf. Series, 2021, 1902 012101.
19. Khayrer E., Vanner G. Reshenie obyknovennykh differentsial'nykh uravneniy. Zhestkie i
differentsial'no-algebraicheskie zadachi [Solving ordinary differential equations. Rigid and differential-
algebraic problems]. Moscow: Mir, 1999, 685 p.
20. Doban A., Lazar M. Computation of Lyapunov functions for nonlinear differential equations
via a Yoshizawa-type construction, 10th IFAC Symp. on Nonlinear Control Systems NOLCOS:
IFAC-PapersOnLine, 2016, pp. 29-34.
21. Zhaolu T., Chuanqing G. A numerical algorithm for Lyapunov equations, J. Appl. Math.
Comput, 2008, Vol. 202, No. 1, pp. 44-53.
22. Xiao-Lin L., Yao-Lin J. Numerical algorithm for constructing Lyapunov functions of polynomial
differential systems, J. Appl. Math. Comput, 2009, Vol. 29, No. 1-2, pp. 247-262.
nelineynykh dinamicheskikh sistem pri postoyanno deystvuyushchikh vozmushcheniyakh
[tability of nonlinear dynamical system motion under constantly acting perturbations],
Nauchno-tekhnicheskiy vestnik informatsionnykh tekhnologiy, mekhaniki i optiki [Scientific
and technical journal of information technologies, mechanics and optics], 2019, Vol. 19, No. 2,
pp. 216-221.
2. Mironov V.V., Mitrokhin Yu.S. Tekhnologicheskiy podkhod k issledovaniyu ustoychivosti
dinamicheskikh sistem: prikladnye voprosy [Constructive approach to the research of dynamic
systems stability: applied problems], Vestnik RGRTU [Vestnik of RSREU]. 2017, No. 59,
pp. 127-135.
3. Aleksandrov A.Yu., Zhabko A.P., Kosov A.A. Analiz ustoychivosti i stabilizatsiya nelineynykh
sistem na osnove dekompozitsii [Analysis of stability and stabilization of nonlinear systems
via decomposition], Sibirskiy matematicheskiy zhurnal [Siberian mathematical journal]. 2015,
Vol. 56, No. 6, pp. 1215-1233.
4. Hammarling S.J. Numerical solution of the stable, non-negative definite Lyapunov equation,
IMA J. of Num. Analysis, 1982, Vol. 2, No. 3, pp. 303-323.
5. Luyckx L., Loccufier M., Noldus E. Computational methods in nonlinear stability analysis:
stability boundary calculations, J. Comput. Appl. Math, 2004, Vol. 168, No. 12, pp. 289-297.
6. Giesl P., Hafstein S. Computation of Lyapunov functions for nonlinear discrete time systems
by linear programming, J. Difference Equ. Appl, 2014, Vol. 20, No. 4, pp. 610-640.
7. Olgac N., Sipahi R. A practical method for analyzing the stability of neutral type LTI-time
delayed systems, Automatica, 2004, Vol. 40, No. 5, pp. 847-853.
8. Hafstein S. A constructive converse Lyapunov theorem on asymptotic stability for nonlinear
autonomous ordinary differential equations, Dynamical Systems, 2005, Vol. 20, pp. 281-299.
9. Romm Ya.E. Komp'yuterno-orientirovannyy analiz ustoychivosti na osnove rekurrentnykh
preobrazovaniy raznostnykh resheniy obyknovennykh differentsial'nykh uravneniy [Computeroriented
stability analysis based on recurrent transformation of difference solutions of ordinary
differential equations], Kibernetika i sistemnyy analiz [Cybernetics and Systems Analysis],
2015, Vol. 51, No. 3, pp. 107-124.
10. Bulanov S.G. Analiz ustoychivosti sistem lineynykh differentsial'nykh uravneniy na osnove
preobrazovaniya raznostnykh skhem [Stability analysis of systems of linear differential equations
based on transformation of difference schemes], Mekhatronika, avtomatizatsiya,
upravlenie [Mekhatronika, Avtomatizatsiya, Upravlenie], 2019, Vol. 20, No. 9, pp. 542-549.
11. Romm Ya.E., Bulanov S.G. Programmnye kriterii ustoychivosti resheniya zadachi Koshi dlya
sistem lineynykh differentsial'nykh uravneniy na osnove raznostnykh skhem chislennogo
integrirovaniya [Program stability criteria for the solution of the Cauchy problem for systems
of linear differential equations based on difference numerical integration schemes], Izvestiya
vuzov. Severo-Kavkazskiy region. Tekhnicheskie nauki. Spetsial'nyy vypusk «Matematicheskoe
modelirovanie i komp'yuternye tekhnologii» [University news. North-caucasian region. Technical
sciences series. Special issue «Mathematical Modeling and Computer Technologies»],
2004, pp. 75-80.
12. Romm Ya.E., Bulanov S.G. Chislennyy eksperiment po komp'yuternomu analizu ustoychivosti
resheniy obyknovennykh differentsial'nykh uravneniy na osnove kriteriev matrichnogo vida
[Computer analysis of the stability of systems of linear differential equations with nonlinear
addition], Dep. v VINITI [Dep. in VINITI], 14.08.17, No. 89, 20 p.
13. Romm Ya.E., Bulanov S.G. Komp'yuternyy analiz ustoychivosti sistem lineynykh
differentsial'nykh uravneniy s nelineynoy dobavkoy [Computer analysis of the stability of systems
of linear differential equations with nonlinear addition], Dep. v VINITI [Dep. in VINITI],
11.03.10, No. 147, 33 p.
14. Romm Ya.E. Modelirovanie ustoychivosti po Lyapunovu na osnove preobrazovaniy
raznostnykh skhem resheniy obyknovennykh differentsial'nykh uravneniy [Modeling of stability
according to Lyapunov based on difference schemes transformations for solutions of ordinary
differential equations], Matematicheskoe modelirovanie [Mathematical Modeling], 2008,
Vol. 20, No. 12, pp. 105-118.
15. Romm Ya.E., Dzhanunts G.A. Kusochnaya interpolyatsiya funktsiy, proizvodnykh i integralov
s prilozheniem k resheniyu obyknovennykh differentsial'nykh uravneniy [Piecewise interpolation
of functions, derivatives and integrals with application to the solution of ordinary differential
equations], Sovremennye naukoemkie tekhnologii [Modern high technologies], 2020, No.
12 (part 2), pp. 291-316.
16. Dzhanunts G.A., Romm Ya.E. Var'iruemoe kusochno-interpolyatsionnoe reshenie zadachi
Koshi dlya obyknovennykh differentsial'nykh uravneniy s iteratsionnym utochneniem [The
varying piecewise interpolation solution of the Cauchy problem for ordinary differential equations
with iterative refinement], Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki
[Computational Mathematics and Mathematical Physics Journal], 2017, Vol. 57, No. 10,
pp. 1616-1634.
17. Romm Ya.E. Komp'yuterno-orientirovannyy analiz ustoychivosti resheniy differentsial'nykh
sistem [Computer-oriented stability analysis of solutions of differential systems], Sovremennye
naukoemkie tekhnologii [Modern high technologies], 2020, No. 4, pp. 42-63].
18. Bulanov S.G. Computer analysis of differential systems stability based on linearization and
matrix multiplicative criteria, Journal of Physics: Conf. Series, 2021, 1902 012101.
19. Khayrer E., Vanner G. Reshenie obyknovennykh differentsial'nykh uravneniy. Zhestkie i
differentsial'no-algebraicheskie zadachi [Solving ordinary differential equations. Rigid and differential-
algebraic problems]. Moscow: Mir, 1999, 685 p.
20. Doban A., Lazar M. Computation of Lyapunov functions for nonlinear differential equations
via a Yoshizawa-type construction, 10th IFAC Symp. on Nonlinear Control Systems NOLCOS:
IFAC-PapersOnLine, 2016, pp. 29-34.
21. Zhaolu T., Chuanqing G. A numerical algorithm for Lyapunov equations, J. Appl. Math.
Comput, 2008, Vol. 202, No. 1, pp. 44-53.
22. Xiao-Lin L., Yao-Lin J. Numerical algorithm for constructing Lyapunov functions of polynomial
differential systems, J. Appl. Math. Comput, 2009, Vol. 29, No. 1-2, pp. 247-262.
