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Herein, we investigate the asymptotical controllability of continuous-time probabilistic logic control systems (CT-PLCSs) under piecewise constant and right continuous controls (PCRCCs). The theory of an invariant subset for a CT-PLCS controlled by a PCRCC is developed. Based on this theory, we prove that a state is asymptotically reachable from another state under a PCRCC if and only if it is asymptotically reachable under a simple periodic PCRCC; in other words, the control values are pairwise distinct within one period. Thereafter, a verifiable, necessary and sufficient condition for the asymptotical reachability between states is obtained, based on which verifiable, necessary and sufficient conditions for the asymptotical reachability and asymptotical controllability are derived. In particular, we prove that a CT-PLCS is asymptotically controllable if and only if each state can be asymptotically stabilized by PCRCCs. In addition, the proof of the proposed results is constructive, which naturally leads to the development of methods of designing PCRCCs. Finally, the examples are provided to illustrate the theoretical results.

This study focuses on the problem of asymptotical stabilization of logic dynamical systems (LDSs) under the situation where the only available measurable information is the real-time output. We propose a novel time-invariant control strategy called output-based random control (OBRC), which breaks through the limitation of time-invariant deterministic output feedback (TIDOF). The TIDOF can be regarded as a special type of the OBRC, but the OBRC is capable of stabilizing some LDSs that cannot be stabilized by any TIDOF. A necessary and sufficient condition for asymptotical stabilizability by the OBRC and a method of finding an optimal OBRC that minimizes a quadratic cost function are proposed. The proposed OBRC was applied to a reduced gene network for the lac operon in the bacterium Escherichia coli. The results of numerical computations and time-domain simulations indicate that the OBRC outperforms the TIDOF.

We propose a new concept, stability in distribution (SD) of a probabilistic Boolean network (PBN), which determines whether the probability distribution converges to the distribution of the target state (namely, a one-point distributed random variable). In a PBN, stability with probability one, stability in the stochastic sense, and SD are equivalent. The SD is easily generalized to subset stability, i.e., to set stability in distribution (SSD). We prove that the transition probability from any state to an invariant subset (or to a fixed point) is nondecreasing in time. This monotonicity is an important property in establishing stability criteria and in calculating or estimating the transient period. We also obtain a verifiable, necessary, and sufficient condition for SD of PBNs with independently and identically distributed switching. We then show that SD problems of PBNs with Markovian switching and PBN synchronizations can be recast as SSD problems of Markov chains. After calculating the largest invariant subset of a Markov chain in a given set by the newly proposed algorithm, we propose a necessary and sufficient condition for SSDs of Markov chains. The proposed method and results are supported by examples. [ABSTRACT FROM AUTHOR]