Ukrainian Mathematical Journal

, Volume 70, Issue 6, pp 947–979 | Cite as

Lagrange Stability and Instability of Irregular Semilinear Differential-Algebraic Equations and Applications

  • M. S. Filipkovskaya

We consider an irregular (singular) semilinear differential-algebraic equation \( \frac{d}{dt}\left[ Ax\right]+ Bx=f\left(t,x\right) \) and prove the theorems on Lagrange stability and instability. These theorems give sufficient conditions for the existence, uniqueness, and boundedness of the global solution to the Cauchy problem for a semilinear differential-algebraic equation and sufficient conditions for the existence and uniqueness of the solution with finite escape time for the analyzed Cauchy problem (this solution is defined on a finite interval and unbounded). The proposed theorems do not contain constraints similar to the global Lipschitz condition. This enables us to use them for the solution of more general classes of applied problems. Two mathematical models of radio-engineering filters with nonlinear elements are studied as applications.


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  1. 1.
    P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution, European Mathematical Society, Zürich (2006).CrossRefGoogle Scholar
  2. 2.
    R. Lamour, R. März, and C. Tischendorf, Differential-Algebraic Equations: a Projector Based Analysis, Springer, New York (2013).CrossRefGoogle Scholar
  3. 3.
    R. Riaza, Differential-Algebraic Systems: Analytical Aspects and Circuit Applications, World Scientific, Hackensack (2008).CrossRefGoogle Scholar
  4. 4.
    P. J. Rabier and W. C. Rheinboldt, Nonholonomic Motion of Mechanical Systems from a DAE Viewpoint, Society for Industrial and Applied Mathematics, Philadelphia (2000).CrossRefGoogle Scholar
  5. 5.
    L. Dai, Singular Control Systems, Springer, Berlin (1989).CrossRefGoogle Scholar
  6. 6.
    T. Stykel, “On criteria for asymptotic stability of differential-algebraic equations,” J. Appl. Math. Mech., 82, No. 3, 147–158 (2002).MathSciNetzbMATHGoogle Scholar
  7. 7.
    S. L. Campbell and V. H. Linh, “Stability criteria for DAEs with multiple delays and their numerical solutions,” Appl. Math. Comput., 208, 397–415 (2009).MathSciNetzbMATHGoogle Scholar
  8. 8.
    C. Tischendorf, “On the stability of solutions of autonomous index-1 tractable and quasilinear index-2 tractable DAEs,” Circ. Syst. Signal Process, 13, No. 2–3, 139–154 (1994).MathSciNetCrossRefGoogle Scholar
  9. 9.
    N. H. Du, V. H. Linh, V. Mehrmann, and D. D. Thuan, “Stability and robust stability of linear time-invariant delay differentialalgebraic equations,” SIAM J. Matrix Anal. Appl., 34, No. 4, 1631–1654 (2013).MathSciNetCrossRefGoogle Scholar
  10. 10.
    R. März, “Practical Lyapunov stability criteria for differential algebraic equations,” Banach Center Publ., 29, No. 1, 245–266 (1994).MathSciNetCrossRefGoogle Scholar
  11. 11.
    V. Tuan and P. V. Viet, “Stability of solutions of a quasilinear index-2 tractable differential algebraic equation by the Lyapunov second method,” Ukr. Mat. Zh., 56, No. 10, 1321–1334 (2004); English translation: Ukr. Math. J., 56, No. 10, 1574–1593 (2004).Google Scholar
  12. 12.
    R. Riaza, “Stability loss in quasilinear DAEs by divergence of a pencil eigenvalue,” SIAM J. Math. Anal., 41, No. 6, 2226–2245 (2010).MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. La Salle and S. Lefshetz, Stability by Lyapunov’s Direct Method, Academic Press, New York (1961).zbMATHGoogle Scholar
  14. 14.
    A. M. Samoilenko and G. P. Pelyukh, “Solutions of systems of nonlinear functional-differential equations bounded on the entire real axis and their properties,” Ukr. Mat. Zh., 46, No. 6, 737–747 (1994); English translation: Ukr. Math. J., 46, No. 6, 799–811 (1994).Google Scholar
  15. 15.
    A. Wu and Zh. Zeng, “Lagrange stability of memristive neural networks with discrete and distributed delays,” IEEE Trans. Neural Networks Learn. Systems, 25, No. 4, 690–703 (2014).MathSciNetCrossRefGoogle Scholar
  16. 16.
    F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).Google Scholar
  17. 17.
    A. G. Rutkas and M. S. Filipkovskaya, “Extension of a solution of one class of differential-algebraic equations,” Zh. Obchyslyuv. Prykl. Mat., No. 1 (111), 135–145 (2013).Google Scholar
  18. 18.
    M. Filipkovskaya, “Global solvability of singular semilinear differential equations and applications to nonlinear radio-engineering,” Challeng. Mod. Technol., 6, No. 1, 3–13 (2015).Google Scholar
  19. 19.
    A. G. Rutkas, “Solvability of semilinear differential equations with singularity,” Ukr. Mat. Zh., 60, No. 2, 225–239 (2008); English translation: Ukr. Math. J., 60, No. 2, 262–276 (2008).Google Scholar
  20. 20.
    P. R. Halmos, Finite-Dimensional Vector Spaces, Van Nostrand, Princeton (1958).Google Scholar
  21. 21.
    A. G. Rutkas and L. A. Vlasenko, “Existence, uniqueness, and continuous dependence for implicit semilinear functional differential equations,” Nonlin. Anal., 55, No. 1-2, 125–139 (2003).MathSciNetCrossRefGoogle Scholar
  22. 22.
    L. Schwartz, Analyse Math´ematique, I, Hermann, Paris (1967).Google Scholar

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Authors and Affiliations

  • M. S. Filipkovskaya
    • 1
  1. 1.Verkin Institute for Low Temperature Physics and Engineering, Ukrainian National Academy of SciencesKarazin Kharkov National UniversityKharkovUkraine

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