The stability by linear approximation of discrete-time nonlinear singular systems

Considering a discrete-time nonlinear descriptor system, we construct the structural form and prove a local existence theorem for solutions. The assumptions of the theorem guarantee that the first-approximation system has a left-invertible linear operator transforming the system to the structural form convenient for analysis. We obtain sufficient conditions for the stability of the nonlinear system by linear approximation under the assumptions that the corresponding part of the first-approximation system is reducible or regular. Also, we address the reducibility and regularity of linear discrete descriptor systems.