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Set reachability and observability of probabilistic Boolean networks. (English) Zbl 1429.93034

Summary: In this paper, the set reachability and observability of probabilistic Boolean networks (PBNs) are investigated. Using a parallel extension technique, we proved that the observability problem of a PBN can be recast as a set reachability problem of an interconnected PBN. For set reachability analysis, we designed a random logic dynamical system (RLDS) from the PBN under consideration by reconstructing the state transfer graph (STG). We proved that, for a PBN, a target subset is reachable from an initial subset if and only if all solutions to the corresponding RLDS starting from the initial subset converge to the zero state. Based on the STG reconstruction technique and using the largest invariant subset algorithm, the necessary and sufficient conditions for finite-time set reachability with probability one and asymptotical set reachability in distribution were obtained. All the results are expressed in terms of the transition probability matrix between non-zero states of the RLDS. Further, the results related to set reachability were applied to the observability problem of PBNs. The necessary and sufficient conditions for finite-time observability with probability one and asymptotical observability in distribution were obtained. Finally, examples were presented to illustrate the effectiveness of the proposed method.

MSC:

93B03 Attainable sets, reachability
93B07 Observability
93C29 Boolean control/observation systems
93B70 Networked control
93E03 Stochastic systems in control theory (general)
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