Ordinary differential equations with finite relations not solvable for derivatives: stability of their solutions (Q1882010)

For a DAE system of the form \[ F(t, x,\dot x,y)= 0,\tag{1} \] where \(x\) is a point in the \(n_x\)-dimensional space \(\mathbb{R}^{n_x}\), \(y\) is a point in the \(n_y\)-dimensional space \(\mathbb{R}^{n_y}\), and \(F: \mathbb{R}^{1+ 2n_x+ n_y}\to \mathbb{R}^{n_x+ n_y}\) is a smooth mapping under which the point \(0\in \mathbb{R}^{n_x+ n_y}\) is regular, an auxiliary differential system of the form \[ pdt- dx= 0,\quad dF= 0,\tag{2} \] is constructed. The integral curves of system (2) are called the solutions of system (1). The author relates the Lyapunov stability of system (2) to the stability of system (1).