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Resonance and marginal instability of switching systems. (English) Zbl 1342.37029

Summary: We analyze the so-called Marginal Instability of linear switching systems, both in continuous and discrete time. This is a phenomenon of unboundedness of trajectories when the Lyapunov exponent is zero. We disprove two recent conjectures of Y. Chitour et al. [Syst. Control Lett. 61, No. 6, 747–757 (2012; Zbl 1250.93112)] stating that for generic systems, the resonance is sufficient for marginal instability and for polynomial growth of the trajectories. The concept of resonance originated with the same authors is modified. A characterization of marginal instability under some mild assumptions on the system is provided. These assumptions can be verified algorithmically and are believed to be generic. Finally, we analyze possible types of fastest asymptotic growth of trajectories. An example of a marginally unstable pair of matrices with non-polynomial growth is given.

MSC:

37C75 Stability theory for smooth dynamical systems
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
34A38 Hybrid systems of ordinary differential equations
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
39A30 Stability theory for difference equations

Citations:

Zbl 1250.93112

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References:

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