Geodesic
Lagrange stability of semilinear differential-algebraic equations and application to nonlinear electrical circuits
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 2, pp. 169-196 
Maria S. Filipkovska. Lagrange stability of semilinear differential-algebraic equations and application to nonlinear electrical circuits. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 2, pp. 169-196. http://geodesic.mathdoc.fr/item/JMAG_2018_14_2_a4/
@article{JMAG_2018_14_2_a4,
     author = {Maria S. Filipkovska},
     title = {Lagrange stability of semilinear differential-algebraic equations and application to nonlinear electrical circuits},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {169--196},
     year = {2018},
     volume = {14},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2018_14_2_a4/}
}
TY  - JOUR
AU  - Maria S. Filipkovska
TI  - Lagrange stability of semilinear differential-algebraic equations and application to nonlinear electrical circuits
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2018
SP  - 169
EP  - 196
VL  - 14
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/JMAG_2018_14_2_a4/
LA  - en
ID  - JMAG_2018_14_2_a4
ER  - 
%0 Journal Article
%A Maria S. Filipkovska
%T Lagrange stability of semilinear differential-algebraic equations and application to nonlinear electrical circuits
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2018
%P 169-196
%V 14
%N 2
%U http://geodesic.mathdoc.fr/item/JMAG_2018_14_2_a4/
%G en
%F JMAG_2018_14_2_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

A semilinear differential-algebraic equation (DAE) is studied focusing on the Lagrange stability (instability). The conditions for the existence and uniqueness of global solutions (a solution exists on an infinite interval) of the Cauchy problem, as well as the conditions of the boundedness of the global solutions, are obtained. Furthermore, the obtained conditions of the Lagrange stability of the semilinear DAE guarantee that every its solution is global and bounded and, in contrast to the theorems on the Lyapunov stability, allow us to prove the existence and uniqueness of global solutions regardless of the presence and the number of equilibrium points. We also obtain the conditions for the existence and uniqueness of solutions with a finite escape time (a solution exists on a finite interval and is unbounded, i.e., is Lagrange unstable) for the Cauchy problem. The constraints of the type of global Lipschitz condition are not used which allows to apply efficiently the work results for solving practical problems. The mathematical model of a radio engineering filter with nonlinear elements is studied as an application. The numerical analysis of the model verifies theoretical studies.

[1] R. Andrzejewski, J. Awrejcewicz, Nonlinear Dynamics of a Wheeled Vehicle, Advances in Mechanics and Mathematics, 10, Springer, New York, NY, 2005 | Zbl

[2] U. M. Ascher, L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1998 | MR | Zbl

[3] A. Bacciotti, L. Rosier, “Liapunov and Lagrange Stability: Inverse Theorems for Discontinuous Systems”, Mathematics of Control, Signals and Systems, 11 (1998), 101–128 | DOI | MR | Zbl

[4] K. E. Brenan, S. L. Campbell, L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1996 | MR | Zbl

[5] L. Dai, Singular Control Systems, Lecture Notes in Control and Information Sciences, 118, Springer-Verlag, Berlin–Heidelberg, 1989 | DOI | MR | Zbl

[6] M. S. Filipkovska, “Lagrange stability and numerical method for solving semilinear descriptor equations”, Visn. Kharkiv. Nats. Univ. Mat. Model. Inform. Tekh. Avt. Syst. Upr., 26:1156 (2015), 152–167 (Russian)

[7] M. Filipkovskaya, “Global solvability of singular semilinear differential equations and applications to nonlinear radio engineering”, Challenges of Modern Technology, 6 (2015), 3–13

[8] C. W. Gear, L. R. Petzold, “ODE methods for the solution of differential/algebraic systems”, SIAM J. Numer. Anal., 21 (1984), 716–728 | DOI | MR | Zbl

[9] P. Kunkel, V. Mehrmann, Differential-Algebraic Equations. Analysis and Numerical Solution, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2006 | MR | Zbl

[10] J. La Salle, S. Lefschetz, Stability by Liapunov's Direct Method with Applications, Mathematics in Science and Engineering, 4, Academic Press, New York–London, 1961 | MR | Zbl

[11] R. Lamour, R. März, C. Tischendorf, Differential-Algebraic Equations: A Projector Based Analysis, Differential-Algebraic Equations Forum, Springer, Heidelberg, 2013 | MR | Zbl

[12] R. März, “Practical Lyapunov stability criteria for differential algebraic equations”, Numerical Analysis and Mathematical Modelling, Banach Center Publ., 29, Polish Acad. Sci. Inst. Math., Warsaw, 1994, 245–266 | DOI | MR | Zbl

[13] R. E. O'Malley, L. V. Kalachev, “Regularization of nonlinear differential-algebraic equations”, SIAM J. Math. Anal., 25 (1994), 615–629 | DOI | MR | Zbl

[14] P. J. Rabier, W. C. Rheinboldt, “Discontinuous solutions of semilinear differential-algebraic equations. II. $P$-consistency”, Nonlinear Anal., 27 (1996), 1257–1280 | DOI | MR | Zbl

[15] T. Reis, T. Stykel, “Lyapunov balancing for passivity-preserving model reduction of RC circuits”, SIAM J. Appl. Dyn. Syst., 10 (2011), 1–34 | DOI | MR | Zbl

[16] R. Riaza, Differential-Algebraic Systems. Analytical Aspects and Circuit Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008 | MR | Zbl

[17] A. G. Rutkas, “Cauchy problem for the equation $Ax'(t) +Bx(t) = f(t)$”, Differ. Uravn., 11 (1975), 1996–2010 (Russian) | MR | Zbl

[18] A. G. Rutkas, M. S. Filipkovska, “Extension of solutions of one class of differential-algebraic equations”, Zh. Obchysl. Prykl. Mat. (2013, no. 2, 135–145 (Russian) | MR

[19] A. G. Rutkas, L. A. Vlasenko, “Existence, uniqueness and continuous dependence for implicit semilinear functional differential equations”, Nonlinear Anal., 55 (2003), 125–139 | DOI | MR | Zbl

[20] L. Schwartz, Analyse Mathématique, v. I, Hermann, Paris, 1967 (French) | MR

[21] Differ. Equ., 40 (2004), 50–62 | DOI | MR | Zbl

[22] C. Tischendorf, “On the stability of solutions of autonomous index-1 tractable and quasilinear index-2 tractable DAEs”, Circuits Systems Signal Process., 13 (1994), 139–154 | DOI | MR | Zbl

[23] L. A. Vlasenko, Evolution models with implicit and degenerate differential equations, Sistemnye Tekhnologii, Dniepropetrovsk, 2006 (Russian)

[24] A. Wu, Z. Zeng, “Lagrange stability of memristive neural networks with discrete and distributed delays”, IEEE Trans. Neural Netw. Learn. Syst., 25 (2014), 690–703 | DOI