Abstract
One of the features that distinguish MJLS from linear systems is the fact that stability (instability) for each mode of operation does not guarantee the stability (instability) of the system as a whole. This chapter provides a broad account on mean-square stability (MSS) for continuous-time MJLS. We follow an operator-theoretical approach to deal with this subject, trying as much as possible to trace a parallel with the stability theory results for continuous-time linear systems. In this way the MSS of MJLS is studied via the spectrum of an augmented matrix or via the existence of a positive-definite solution for a set of coupled Lyapunov equations. Besides the homogeneous case, we consider two scenarios regarding additive disturbances: the one in which the disturbances are characterized via a Wiener process and the one characterized by any function in \(L^{m}_{2}\). Finally, we treat the concepts of mean-square stabilizability and detectability and make a brief incursion in the case with partial information.
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Notes
- 1.
For 1≤p<∞, L p (ℝ+) is the space of Lebesgue-measurable functions f from ℝ+ to ℝ such that \(\int_{0}^{\infty}|f(t)|^{p} \,dt < \infty\).
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Costa, O., Fragoso, M., Todorov, M. (2013). Mean-Square Stability. In: Continuous-Time Markov Jump Linear Systems. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34100-7_3
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DOI: https://doi.org/10.1007/978-3-642-34100-7_3
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