Improper integrals for stability theory of multidimensional phase systems
DOI:
https://doi.org/10.26577/jmmcs-2018-1-483Keywords:
nonsingular transformation, properties of solutions, improper integrals, dynamical system, countable equilibrium positionAbstract
A class of ordinary differential equations describing the dynamics of multidimensional phase systems
with a countable equilibrium position with periodic nonlinear functions from a given set is
considered. Such uncertainty of right side of the differential equation gives rise to the nonuniqueness
of the solution, which leads to the study of the properties of solutions of equations with
differential inclusions. A completely new approach to the study of the properties of the solution of
dynamical systems with a countable equilibrium position with incomplete information on nonlinearities
is proposed. By a nonsingular transformation, the original system is reduced to a special
kind, consisting of two parts. The first part of the differential equations is solvable with respect to
the components of the periodic function and the second part does not contain nonlinear functions.
The properties of solutions are studied, estimates for the solutions of the original system and the
transformed system are obtained and their boundedness is proved. Identities with respect to the
components of the nonlinear function are obtained and their relation to the phase variables is
established. The properties of quadratic forms with respect to phase variables and derivatives are
studied. The estimates of improper integrals along the solution of the system are obtained for two
cases: when the values of the integrals of the components of the nonlinear function in the period
are zero; When the values of the integrals in the period are different from zero. These results can
be used to obtain conditions for global asymptotic stability of multidimensional phase systems.
References
[2] Andronov A. A., Vitt A. Haykin S. E. Teoriya kolebaniya [Theory of oscillation], (M.: Fizmatgiz, 1959) : 600.
[3] Bakaev Yu.N. Nekotoryie voprosyi nelineynoy teorii fazovyih sistem [Some questions of the nonlinear theory of phase
systems], (M.: Trudyi VIL im. Zhukovskogo, 1959) : 105–110.
[4] Bakaev Yu. N., Guzh A. A. Optimalnyiy priem signalov chastotnoy modulyatsii v usloviyah effekta Dopplera [An optimal
reception of frequency modulation signals under the conditions of Doppler effect], Radiotehnika i elektronika, T. 10, No 1
(1965) : 36–46.
[5] Barbashin E. A., Tabueva V. A. Dinamicheskie sistemyi s tsilindricheskimi fazovyim prostranstvom [Dynamic systems
with cylindrical phase space], (M.: Nauka, 1969) : 305.
[6] Fazovaya sinhronizatsiya [Phase synchronization], Pod red. V.V. Shahgildyana i L.N. Belyustinoy, (M.: Svyaz, 1975) :
401.
[7] Leonov G. A. «Ustoychivost i kolebaniya fazovyih sistem» [Stability and oscillations of phase systems], Sibirskiy matem.
zhurnal, No 5 (1975) : 7–15.
[8] Leonov G. A. Ob ogranichennosti resheniy fazovyih sistem [Stability and oscillations of phase system], Vestnik LGU, No 1
(1976) : 10–15.
[9] Leonov G. A. «Ob odnom klasse dinamicheskih sistem s tsilindricheskim fazovyim prostranstvom» [On a class of dynamical
systems with a cylindrical phase space], Sibirskiy matema. zhurnal, No 1 (1976) : 10–17.
[10] Leonov G. A., Smirnova V. B., «Asimptotika resheniy sistemyi integro-differentsialnyih uravneniy s periodicheskimi nelineynyimi
funktsiyami» [Asymptotics of solutions of a system of integro-differential equations with periodic nonlinear
functions], Sibirskiy matem. zhurnal, No 4 (1978) : 115–124.
[11] Primenenie metoda funktsiy Lyapunova v energetike [Application of the Lyapunov function method in the engineering],
Pod red. Tagirova M.A., (Novosibirsk: Nauka, Sib. otdelenie, 1975) : 301.
[12] Triomi F. «Integrazione di uniquzione differenziale presentatasi in electrotechnica», Annali della Roma schuola Normale
Superiore de Pisa Scienza Physiche Matematiche, Vol 2, No 2 (1933) : 3–10.
