How Floquet theory applies to index-1 differential algebraic equations. (English) Zbl 0903.34002
In this clear and well-writen article the authors develop a Floquet theory for index-1 differential algebraic equations (DAEs). The theorem of Floquet on the representation of the fundamental matrix [G. Floquet, Ann. Sci. Éc. Norm. Supér. 12, 47-89 (1883; JFM 15.0279.01)] and the reduction theorem of A. M. Lyapunov [The general problem of the stability of motion, London (1992; Zbl 0786.70001) (Originally: Kharkow, 1892) (Russian)], and, finally, the theorem on the stability of periodic solutions are generalized for DAEs. The authors show that the results for DAEs are as clear and simple as for regular ordinary differential equations (ODEs).
In contrast to regular ODEs, DAEs have only implicitly given dynamic parts. To work out and utilize this implicit structure, a decoupling technique is applied, which goes back to E. Griepentrog and R. März [Differential-algebraic equations and their numerical treatment, Teubner-Texte Math. 88, Leipzig: BSB B. G. Teubner (1986; Zbl 0629.65080)]. Via a proper decoupling, the classical procedures of ODE theory [see for example L. S. Pontrjagin, Gewöhnliche Differentialgleichungen, Mathematik für Naturwissenschaft und Technik. 11. Berlin: VEB Deutscher Verlag der Wissenschaften (1965; Zbl 0134.30201)] become applicable. The authors prove, that each linear DAE with periodic coefficients is periodically equivalent to a constant coefficient DAE in Kronecker normal form. The article contains a result on the stability of the trivial solution in the case of nonautonomous DAEs with a constant linear part and a small nonlinearity.
In contrast to regular ODEs, DAEs have only implicitly given dynamic parts. To work out and utilize this implicit structure, a decoupling technique is applied, which goes back to E. Griepentrog and R. März [Differential-algebraic equations and their numerical treatment, Teubner-Texte Math. 88, Leipzig: BSB B. G. Teubner (1986; Zbl 0629.65080)]. Via a proper decoupling, the classical procedures of ODE theory [see for example L. S. Pontrjagin, Gewöhnliche Differentialgleichungen, Mathematik für Naturwissenschaft und Technik. 11. Berlin: VEB Deutscher Verlag der Wissenschaften (1965; Zbl 0134.30201)] become applicable. The authors prove, that each linear DAE with periodic coefficients is periodically equivalent to a constant coefficient DAE in Kronecker normal form. The article contains a result on the stability of the trivial solution in the case of nonautonomous DAEs with a constant linear part and a small nonlinearity.
Reviewer: S.Siegmund (Augsburg)
MSC:
| 34A09 | Implicit ordinary differential equations, differential-algebraic equations |
| 34C25 | Periodic solutions to ordinary differential equations |
Keywords:
differential algebraic equations; Floquet theory; monodromy matrix; stability of periodic solutions; JFM 15.0279.01References:
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