Mathematics & Physics 2016, 9(2), 353–363 УДК 517.926, 517.977.1 Local R-observability of Differential-algebraic Equations Pavel S.Petrenko∗ Matrosov Institute for System Dynamics and Control Theory of SB RAS Lermontov, 134, Irkutsk, 664033 Russia Received 21.12.2015, received in revised form 05.02.2016, accepted 01.05.2016 A nonlinear system of first order ordinary differential equations is considered. <...> The system is unresolved with respect to the derivative of the unknown function and it is identically degenerate in the domain. <...> Analysis is carried out under assumptions that ensure the existence of a global structural form that separates "algebraic" and "differential" subsystems. <...> Local R-observability conditions are obtained by linear approximation of the system. <...> Such systems are called differential-algebraic equations (DAEs). <...> The measure of unresolvability for the DAEs with respect to the derivative is an integer value r: 0 r n, which is called index [1, pp. 16–17]. <...> Analysis is carried out under assumptions that function F has the property F(t, 0, 0) = 0 ∀t ∈ I. (1.3) Algebraic conditions of the full observability and R-observability based on the reduction to the Kronecker canonical form were obtained for the linear DAEs with constant coefficients and regular matrix pencil [2, pp. 29–44]. <...> The local observability based on normalization of the DAEs by so-called extended system is investigated in [4]. <...> The analysis is carried out under assumptions that ensure the existence of a global structural form that separates "algebraic" and "differential" subsystems. <...> All rights reserved c – 353 – Pavel S.Petrenko Local R-observability of Differential-algebraic Equations 2. <...> If rankΓr,x(αr) = n(r+1) then system (2.1) satisfies all the assumptions of the implicit function theorem [7, c. 66]. <...> Since matrix Γr,x has dimensions n(r+1)Чn(r+2) then in general case the non-special minor of n(r + 1) order of matrix Γr,x(αr) is not unique. <...> The implicit function determined in (2.2) depends on this minor. <...> We find the minor in the following way. <...> These n(r + 1) linearly independent columns represent <...>