Qualitative Analysis of Nonregular Differential-Algebraic Equations and the Dynamics of Gas Networks
DOI:
https://doi.org/10.15407/mag19.04.719Анотація
Одержано умови існування, єдиності та обмеженості глобальних розв'язків, а також граничної обмеженості розв'язків, та умови руйнування розв'язків нерегулярних напівлінійних диференціально-алгебраїчних рівнянь. Розглянуто приклад, який демонструє застосування одержаних результатів. В якості застосувань наводяться ізотермічні моделі газових мереж.
Mathematical Subject Classification 2020: 34A09, 34A12, 34C11, 34D23,
15A22
Ключові слова:
нерегулярне диференцiально-алгебраїчне рiвняння, вироджене диференціальне рівняння, сингулярний жмуток, глобальна розв'язність, обмеженість розв'язків, руйнування, дисипативністьПосилання
T.P. Azevedo-Perdicoúlis and G. Jank, Modelling aspects of describing a gas network through a DAE system, IFAC Proceedings Volumes 40 (2007), No. 20, 40--45. https://doi.org/10.3182/20071017-3-BR-2923.00007
P. Benner, S. Grundel, C. Himpe, C.Huck, T. Streubel, and C. Tischendorf, Gas Network Benchmark Models, Applications of Differential-Algebraic Equations: Examples and Benchmarks. Differential-Algebraic Equations Forum (Eds. S. Campbell, A. Ilchmann, V. Mehrmann, and T. Reis), Springer, Cham, 2018, 171--197. https://doi.org/10.1007/11221_2018_5
V.F. Chistyakov and E.V. Chistyakova, Application of the least squares method to solving linear differential-algebraic equations, Numer. Analys. Appl. 6(2013), 77-90. https://doi.org/10.1134/S1995423913010102
S.M. Chuiko, On a reduction of the order in a differential-algebraic system, J. Math. Sci. 235 (2018), No. 1, 2--14. https://doi.org/10.1007/s10958-018-4054-z
P. Domschke, B. Hiller, J. Lang, V. Mehrmann, R. Morandin, and C. Tischendorf, Gas Network Modeling: An Overview, Technische Universität Darmstadt, Darmstadt, 2021.
D.K. Faddeev, Lectures on algebra, Nauka, Moscow, 1984 (Russian).
M.S. Filipkovskaya, Continuation of solutions of semilinear differential-algebraic equations and applications in nonlinear radiotechnics, Bull. of V. Karazin Kharkiv National University. Series Math. Model. Inform. Tech. Automat. Control Syst. 19 (2012), No. 1015, 306--319 (Russian).
M.S. Filipkovska, Lagrange stability and instability of irregular semilinear differential-algebraic equations and applications, Ukrain. Math. J. 70 (2018), No. 6, 947--979. https://doi.org/10.1007/s11253-018-1544-6
M.S. Filipkovska (Filipkovskaya), A block form of a singular pencil of operators and a method of obtaining it, Visnyk of V.N. Karazin Kharkiv National University. Ser. ''Mathematics, Applied Mathematics and Mechanics'' 89 (2019), 33--58 (Russian). https://doi.org/10.26565/2221-5646-2019-89-04
M.S. Filipkovska, Lagrange stability of semilinear differential-algebraic equations and application to nonlinear electrical circuits, J. Math. Phys. Anal. Geom. 14 (2018), No. 2, 169--196. https://doi.org/10.15407/mag14.02.169
M. Filipkovskaya, Global solvability of singular semilinear differential equations and applications to nonlinear radio engineering, Chall. Modern Technology. 6 (2015), No. 1, 3--13.
M. Filipkovska (Filipkovskaya), Existence, boundedness and stability of solutions of time-varying semilinear differential-algebraic equations, Global and Stochastic Analysis 7 (2020), No. 2, 169--195.
F.R. Gantmacher, The theory of matrices, Vol. I, II, Amer. Math. Soc., Providence, RI, 2000.
M. Gugat and S. Ulbrich, Lipschitz solutions of initial boundary value problems for balance laws, Math. Models Methods Appl. Sci. 28 (2018), No. 5, 921--951. https://doi.org/10.1142/S0218202518500240
C. Huck, Perturbation analysis and numerical discretisation of hyperbolic partial differential algebraic equations describing flow networks, Dissertation, Humboldt Universität zu Berlin, 2018.
T. Kreimeier, H. Sauter, S.T. Streubel, C. Tischendorf, and A. Walther, Solving Least-Squares Collocated Differential Algebraic Equations by Successive Abs-Linear Minimization - A Case Study on Gas Network Simulation, Humboldt-Universität zu Berlin, preprint, 2022.
P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution, European Mathematical Society, Zurich, 2006. https://doi.org/10.4171/017
R. Riaza, Differential-Algebraic Systems: Analytical Aspects and Circuit Applications, World Scientific, Hackensack, NJ, 2008. https://doi.org/10.1142/6746
J. La Salle and S. Lefschetz, Stability by Liapunov's direct method with applications, Academic Press, New York, 1961.
L. Schwartz, Analyse Mathématique, I. Hermann, Paris, 1967 (French).
L. Schwartz, Analyse Mathématique, II, Hermann, Paris, 1967 (French).
A.G. Rutkas, Cauchy problem for the equation $Ax'(t) +Bx(t) = f(t)$, Differ. Uravn. 11 (1975), No. 11, 1996--2010 (Russian).
A.G. Rutkas, Solvability of semilinear differential equations with singularity, Ukrain. Math. J. 60 (2008), 262--276. https://doi.org/10.1007/s11253-008-0057-0
Rutkas A.G. and Filipkovskaya (Filipkovska) M.S. Extension of solutions of one class of differential-algebraic equations. J. Comput. Appl. Math. 1 (2013), 135--145 (Russian).
L.A. Vlasenko, Evolution Models with Implicit and Degenerate Differential Equations, System Technologies, Dnipropetrovsk, Ukraine, 2006 (Russian).
T. Yoshizawa, Stability theory by Liapunov's second method, The Mathematical Society of Japan, Tokyo, 1966.
