Switching systems with dwell time: computing the maximal Lyapunov exponent

Nonlinear Analysis: Hybrid Systems. 2021. Vol. 40. Article 101021.

Chitour Y., Guglielmi N., Protasov V., Sigalotti M.

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 timeduration.

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.

Keywords: Lyapunov exponent, Linear switching systems, Control of constrained systems

Publication based on the results of: Uncertainty quantification in machine learning algorithms (2021)