Exponential stability of singularly perturbed systems with mixed impulses (Q2061282)

In this paper, some sufficient conditions have been derived to ensure the global uniform exponential stability of hybrid systems, where the flow dynamics is unstable and exhibits a two-time-scale feature, while the jump dynamics contains both destabilizing and stabilizing impulses simultaneously. The mathematical framework is given by the singular perturbation theory and the system is described as \[ \begin{bmatrix} \dot x(t) \\ \varepsilon\dot z(t) \end{bmatrix} = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} x(t) \\ z(t) \end{bmatrix},\ t\in [t_k,t_{k+1}), \] \[ \begin{bmatrix} x(t_k) \\ z(t_k) \end{bmatrix} = \begin{bmatrix} D_{11} & D_{12} \\ D_{21} & D_{22} \end{bmatrix} \begin{bmatrix} x(t_k^-) \\ z(t_k^-) \end{bmatrix},\ k\in \mathbb{N}, \] where \(x(t)\in\mathbb{R}^{n_x}\) is the slow state vector, \(z(t)\in\mathbb{R}^{n_z}\) is the fast state vector, \(\varepsilon\) is a small parameter that indicates the degree of the fast and slow dynamics separation. \(A_{ij},\) \(D_{ij},\) \(i = 1, 2,\) \(j = 1,2\) are constant real matrices with appropriate dimensions and the matrix \(A_{22}\) is assumed to be Hurwitz. A new kind of impulse-dependent vector Lyapunov function has been constructed to explore the positive effect of the stabilizing impulse and describe the relationship between two consecutive impulses.