On the stability of planar randomly switched systems. (English) Zbl 1288.93090
A planar linear ordinary differential equation is considered wherein the coefficient matrix switches randomly between two Hurwitz matrices according to a continuous time Markov chain. Under the assumption that some convex combination of the two matrices has one stable and one unstable eigenvalue, it is shown that the system has a well defined Lyapunov exponent that is either strictly positive or strictly negative depending on a threshold for the jump rate of the Markov chain, implying either almost sure convergence to zero or almost sure blowing up (instability) accordingly. Two explicit examples and an application to matrix products are given.
Reviewer: Vivek S. Borkar (Mumbai)
MSC:
| 93E15 | Stochastic stability in control theory |
| 34F05 | Ordinary differential equations and systems with randomness |
| 60B20 | Random matrices (probabilistic aspects) |
| 15B52 | Random matrices (algebraic aspects) |
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