We study asymptotic stability of continuous-time systems with mode-dependent guaranteed dwell time. These systems are reformulated as special cases of a general class of mixed (discrete–continuous) linear switching systems on graphs, in which some modes correspond to discrete actions and some others correspond to continuous-time evolutions. Each discrete action has its own positive weight which accounts for its time-duration. We develop a theory of stability for the mixed systems; in particular, we prove the existence of an invariant Lyapunov norm for mixed systems on graphs and study its structure in various cases, including discrete-time systems for which discrete actions have inhomogeneous time durations. This allows us to adapt recent methods for the joint spectral radius computation (Gripenberg’s algorithm and the Invariant Polytope Algorithm) to compute the Lyapunov exponent of mixed systems on graphs.

我们研究具有模式依赖保证停留时间的连续时间系统的渐近稳定性。这些系统被重新构造为图上混合（离散-连续）线性切换系统的一般类的特殊情况，其中某些模式对应于离散动作，另一些模式对应于连续时间演化。每个离散动作都有自己的正权重，这决定了其持续时间。我们为混合系统建立了一个稳定性理论。特别是，我们证明了图上混合系统存在不变的Lyapunov范数，并在各种情况下研究其结构，包括离散时间系统，其中离散动作的持续时间不均匀。