We consider linear time-varying system of first order ordinary differential equations with identically degenerate matrix of the derivative of the unknown function. Such systems are called differential-algebraic equations (DAE). The unsolvability measure with respect to the derivatives for some DAE is an integer that is called the index of the DAE. The analysis is carried out under the assumption of the existence of a structural form with separated differential and algebraic subsystems. This structural form is equivalent to the initial system in the sense of solution, and the operator which transformes the DAE into the structural form possesses the left inverse operator. The finding of the structural form is constructive and do not use a change of variables. In addition the problem of consistency of the initial data is solved automatically. The approach uses the concept of r-derivative array equations, where r is the unsolvability index of the DAE. The existence of a nonsingular minor of order n(r + 1) in the matrix describing derivative array equations is a necessary and sufficient condition for the existence of this structural form (n is the dimension of DAE system). We investigate robust controllability of non- stationary DAE with perturbations given by matrix norms (unstructured uncertainty), which are present in matrices with the unknown function and control function. The problem of the robust controllability is to find the conditions under which the perturbed system will remain completely or R-controllable on some interval in the presence of this property of the initial DAE system. It is constructed a structural form for the perturbed DAE system and based on it’s analysis sufficient conditions for robust complete and R-controllability of the DAE of the indeces 1 and 2 are obtained.

1. Gantmacher F.R. Teoriya matrits [The theory of matrices]. Moscow, Nauka Publ., 1988, 548 p. (in Russian).

2. Trenogin V.A. Funktsional’nyy analiz [Functional analysis]. Moscow, Nauka Publ., 1980 (in Russian).

3. Brenan K.E., Campbell S.L., Petzold L.R. Numerical solution of initial-value problems in differential-algebraic equations. SIAM, 1996, 251 p.

4. Campbell S.L., Griepentrog E. Solvability of general differential algebraic equations. SIAM J. Sci. Stat. Comp., 1995, no. 16, pp. 257-270. https://doi.org/10.1137/0916017

5. Chou J.H., Chen S.H., Fung R.F. Sufficient conditions for the controllability of linear descriptor systems with both time-varying structured and unstructured parameter uncertainties. IMA J. Math. Control Inform., 2001, vol. 18, no. 4, pp. 469-477. https://doi.org/10.1093/imamci/18.4.469

6. Chou J.H., Chen S.H., Zhang Q.-L. Robust controllability for linear uncertain descriptor systems. Linear Algebra Appl., 2006, vol. 414, no. 2-3, pp. 632-651. https://doi.org/10.1016/j.laa.2005.11.005

7. Dai L. Singular control system. Lecture notes in control and information sciences. Springer-Verlag, 1989, vol. 118.

8. Lin C., Wang J.L., Soh C.-B. Necessary and sufficient conditions for the controllability of linear interval descriptor systems. Automatica, 1998, vol. 34, no. 3, pp. 363-367. https://doi.org/10.1016/S0005-1098(97)00204-5

9. Lin C., Wang J.L., Soh C.-B. Robust C-controllability and/or C-observability for uncertain descriptor systems with interval perturbation in all matrices. IEEE Trans. Automat. Control., 1999, vol. 44, no. 9, pp. 1768-1773. https://doi.org/10.1109/9.788550

10. Lin C., Wang J.L., Soh C.-B., Yang G.H. Robust controllability and robust closed-loop stability with static output feedback for a class of uncertain descriptor systems. Linear Algebra Appl., 1999, vol. 297, no. 1-3, pp. 133-155. https://doi.org/10.1016/s0024-3795(99)00150-0

11. Mehrmann V., Stykel T. Descriptor systems: a general mathematical framework for modelling, simulation and control. Automatisierungstechnik, 2006, vol. 8, pp. 405-415. https://doi.org/10.1524/auto.2006.54.8.405

12. Petrenko P.S. Differential controllability of linear systems of differential-algebraic equations. Journal of Siberian Federal University. Mathematics & Physics, 2017, vol. 10, no. 3, pp. 320-329. https://doi.org/10.17516/1997-1397-2017-10-3-320-329

13. Petrenko P.S. Local R-observability of differential-algebraic equations. Journal of Siberian Federal University. Mathematics & Physics, 2016, vol. 9, no. 3, pp. 353- 363. https://doi.org/10.17516/1997-1397-2016-9-3-353-363.

14. Shcheglova A.A. Controllability of nonlinear algebraic differential systems. Automation and Remote Control, 2008, vol. 69, no. 10, pp. 1700-1722. https://doi.org/10.1134/s0005117908100068

15. Shcheglova A.A. The solvability of the initial problem for a degenerate linear hybrid system with variable coefficients. Russian Mathematics, 2010, vol. 54, no. 9, pp. 49-61. https://doi.org/10.3103/S1066369X10090057

16. Shcheglova A.A., Petrenko P.S. Stabilizability of solutions to linear and nonlinear differential-algebraic equations. Journal of Mathematical Sciences, 2014, vol. 196, no. 4, pp. 596-615. https://doi.org/10.1007/s10958-014-1679-4

17. Shcheglova A.A., Petrenko P.S. Stabilization of solutions for nonlinear differential-algebraic equations. Automation and remote control, 2015, vol. 76, no. 4, pp. 573- 588. https://doi.org/10.1134/s0005117915040037

18. Shcheglova A.A., Petrenko P.S. The R-observability and R-controllability of linear differential-algebraic systems. Russian Mathematics, 2012, vol. 56, no. 3, pp. 66-82. https://doi.org/10.3103/s1066369x12030097