Section: New Results

New results: switched systems

• In [12] we study the phenomenon of polynomial instability of switched systems. Stability properties for continuous-time linear switched systems are at first determined by the (largest) Lyapunov exponent associated with the system, which is the analogue of the joint spectral radius for the discrete-time case. We provided a characterization of marginally unstable systems, i.e., systems for which the Lyapunov exponent is equal to zero and such that there exists an unbounded trajectory. We also analyzed the asymptotic behavior of their trajectories. Our main contribution consists in pointing out a resonance phenomenon associated with marginal instability. In the course of our study, we derived an upper bound of the state at time $t$, which is polynomial in $t$ and whose degree is computed from the resonance structure of the system. We also derived analogous results for discrete-time linear switched systems.

• The paper [13] is concerned with the stability of planar linear singularly perturbed switched systems of the type $\dot{x}\left(t\right)=\sigma \left(t\right)A_1^{ϵ}x\left(t\right)+(1-\sigma \left(t\right))A_2^{ϵ}x\left(t\right)$, where $\sigma :[0,+\infty )\to \{0,1\}$, $A_1^{ϵ}$ and $A_2^{ϵ}$ are real matrices which represent singularly perturbed modes. By $ϵ$ we denote here the parameter of singular perturbation. We propose a characterization of the stability properties of such singularly perturbed switched systems based on the results given in [47] . More generally, we study transitions as $ϵ$ varies and we restrict their number and nature. Finally, we compare the results obtained in this way with the Tikhonov-type results for differential inclusions obtained in the literature.