## List of issues > Series «Mathematics». 2019. Vol. 28

##
The Index and Split Forms of Linear
Differential-Algebraic Equations

We consider linear systems of ordinary differential equations (ODE) with rectangular matrices of coefficients, including the case when the matrix before the derivative of the desired vector function is not full rank for all argument values from the domain. Systems of this type are usually called differential-algebraic equations (DAEs). We obtained criteria for the existence of nonsingular transformations splitting the system into subsystems, whose solution can be written down analytically using generalized inverse matrices. The resulting solution formula is called a generalized split form of a DAE and can be viewed as a certain analogue of the Weierstrass-Kronecker canonical form. In particular, it is shown that arbitrary DAEs with rectangular coefficient matrices are locally reducible to a generalized split form. The structure of these forms (if it is defined on the integration segment) completely determines the structure of general solutions to the systems. DAEs are commonly characterizes by an integer number called index, as well as by the solution space dimension. The dimension of the solution space determines arbitrariness of the the general solution manifold. The index determines how many times we should differentiate the entries on which the solution to the problem depends. We show the ways of calculating these main characteristics.

Mikhail Bulatov, Dr. Sci. (Phys.–Math.), Matrosov Institute for System Dynamics and Control Theory of SB RAS, Post Box 292, 134, Lermontov st., Irkutsk, 664033, Russian Federation; tel: (3952)427100, e-mail: mvbul@icc.ru

Victor Chistyakov, Dr. Sci. (Phys.–Math.), Institute of System Dynamics and Control Theory of SB RAS, 134, Lermontov st., Irkutsk, 664033, Russian Federation; tel.: 8(3932)453029, e-mail: chist@icc.ru

Bulatov M.V., Chistyakov V.F. The Index and Split Forms of Linear Differential-algebraic Equations. *The Bulletin of Irkutsk State University. Series Mathematics*, 2019, vol. 28, pp. 21-35. https://doi.org/10.26516/1997-7670.2019.28.21

- Belov A.A., Kurdyukov A.P.
*Descriptor systems and control problems.*Moscow, Fizmatlit Publ., 2015, 272 p. (in Russian). - Boyarintsev Y.E.
*Regular and singular systems of linear ordinary differential equations.*Novosibirsk, Nauka Publ., 1980, 222 p. - Boyarintsev Yu.E., Chistyakov V.F.
*Algebro-differencial’nye sistemy. Metody resheniya i issledovaniya*[Algebraic Differential Systems. Methods for Investigation and Solution]. Novosibirsk, Nauka Publ., 1998, 224 p. (in Russian) - Brenan K.E., Campbell S.L., Petzold L.R.
*Numerical solution of initial-value problems in differential-algebraic equations (classics in applied mathematics; 14).*Philadelphia, SIAM, 1996. - Bulatov M.V., Chistyakov V.F. Odin metod chislennogo resheniya lineinich singu- larnich sistem ODU indeksa vishe edinitsi.
*Chislenie metodi analiza i ix prilozeniya.*Irkutsk, SEI SO AN SSSR Publ., 1987, pp. 100-105. (in Russian). - Bulatov M.V., Chistyakov V.F. A numerical method for solving differential- algebraic equations.
*Comput. Math. Math. Phys.*, 2002, vol. 42, no. 4, pp. 439–449. - Gantmacher F. R.
*The Theory of Matrices.*AMS Chelsea Publishing: Reprinted by American Mathematical Society, 2000, 660 p. - Hairer E., Lubich C., Roche M. The numerical solution of differential-algebraic system by Runge - Kutta methods, Report CH-1211, Dept. de Mathematiques, Universite de Geneve, Switzerland, 1988.
- Kunkel P., Mehrmann V. Canonical forms for linear differential-algebraic equations with variable coefficients.
*J. Comput. Appl. Math*, 1995, vol. 56, pp. 225-251. https://doi.org/10.1016/0377-0427(94)90080-9 - Kurina G.A. On regulating by descriptor systems in an innite interval.
*Izvestija RAN, Tekhnicheskaja Kibernetika*, 1993, no. 6, pp. 33–38. (in Russian). - Lamour R., Marz R., Tischendorf C. Differential-Algebraic Equations: A Projector Based Analysis. Springer-Verlag, 2013. https://doi.org/10.1007/978-3-642-27555-5
- Sidorov N. A. The Cauchy problem for a certain class of differential equations.
*Differ. Equations*, 1972, vol. 8, no. 8. pp. 1521–1524 (in Russian). - Sidorov N. A. The branching of the solutions of differential equations with a degeneracy.
*Differ. Equations*, 1973, vol. 9, no. 8. pp. 1464–1481. (in Russian) - Sidorov N. A. A study of the continuous solutions of the Cauchy problem in the neighborhood of a branch point.
*Soviet Math.*, 1976, vol. 20, no. 9, pp. 77–87. (in Russian). - Chistyakov V.F.
*Algebraic differential operators with finite-dimensional kernel*. Novosibirsk, Nauka Publ., 1996, 278 p. (in Russian).