A Berger-Wang formula for impulsive switched systems

This paper addresses a class of impulsive systems defined by a mix of continuous-time and discrete-time switched linear dynamics. We first analyze a related class of weighted discrete-time switched systems for which we establish a Berger–Wang-type result. An analogous result is then derived for impulsive systems and subsequently used to characterize their exponential stability through a spectral approach, thereby extending existing results in switched-systems theory.

💡 Research Summary

The paper studies a class of hybrid impulsive systems that combine continuous‑time linear dynamics with discrete‑time switching and instantaneous state jumps. The authors first introduce weighted discrete‑time switched systems, where each mode is described by a pair (A, τ) consisting of a matrix A and a dwell‑time τ. For such systems they define two growth‑rate quantities: λ, based on the induced norm of the product of matrices, and μ, based on the spectral radius of the same product. Under the irreducibility assumption on the set of matrices, they prove a Berger–Wang type formula showing λ = μ. This result extends the classical Berger–Wang theorem (which corresponds to the unit‑weight case τ = 1) to the weighted setting and provides a uniform exponential bound on all trajectories.

Next, the authors map the impulsive switched system Σ_{Z,τ} (continuous dynamics ẋ = Z₁(tₖ)x on each interval