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\section{Introduction}  We consider the problem of moment stabilization of a dynamical system where the estimated state is transmitted for control over a time-varying communication channel, as depicted in Fig.~\ref{fig:scheme}.  This problem has been studied extensively in the context of networked control systems and discussed in several special issue journals dedicated to the topic~\cite{Baillieul--Antsaklis20071, Baillieul--Antsaklis20072, Franceschetti-Kumar-Mitter-Teneketzis-2008}. topic~\cite{Baillieul--Antsaklis20071,Baillieul--Antsaklis20072,Franceschetti-Kumar-Mitter-Teneketzis-2008}.  Recently, it gained renewed attention due to its relevance for the design of cyberphysical systems~\cite{Kim-Kumar2012}. A tutorial review of the problem with extensive references appears in~\cite{Franceschetti-Minero-2014}.   The notion of Shannon capacity is in general not sufficient to characterize the trade-off between the entropy rate production of the plant, expressed by the growth of the state space spanned in open loop, and the communication rate required for its stabilization. A large Shannon capacity is useless for stabilization if it cannot be used in time for control. For the control signal to be effective, it must be appropriate to the current state of the system. Since decoding the wrong codeword implies applying a wrong signal and driving the system away from stability, applying an effective control signal depends on the history of whether previous codewords were decoded correctly or not. In essence, the stabilization problem is an example of \emph{interactive communication}, where two-way communication occurs through the feedback loop between the plant and the controller. Error correcting codes developed independently in this context~\cite{Forney1974, Schulman1996,Ostrovsky-etal-2009} context~\cite{Forney1974,Schulman1996,Ostrovsky-etal-2009}  have a natural tree structure representing past history and are natural candidates to be used for control. Alternative capacity notions with stronger reliability constraints than simply having a vanishing probability of error, and requiring these type of coding schemes have been proposed in the context of control, including the zero-error capacity~\cite{matveed-savkin2007b}, originally introduced by Shannon~\cite{Shannon1956}, and the anytime capacity proposed by Sahai~\cite{Sahai2001}, \cite{Como-Fagnani-Zampieri2010, Sahai-Mitter2006, simsek-jain-varaija2004, Xu2005, Sukhavasi-Hassibi2011a}. \cite{Como-Fagnani-Zampieri2010,Sahai-Mitter2006,simsek-jain-varaija2004,Xu2005,Sukhavasi-Hassibi2011a}.  Within this general framework, we focus on the $m$th moment stabilization of an unstable scalar system whose state is communicated over a rate-limited channel capable of supporting $R_k$ bits at each time step and evolving randomly in a Markovian fashion. The rate process is known casually to both encoder and decoder. Many variations of this ``bit-pipe'' model have been studied in the literature   ~\cite{ball1,ball2,Wong-Brocket1999, Brocket-Liberzon2000, Liberzon2003, Delchamps1990, Martins-Dehleh-Elia2006, Nair-Evans-2004, Imer-etal-2006, Tatikonda-Mitter2004, Tatikonda-Mitter2004b, Gupta-Martins2010,Yuksel2010,Yuksel-Basar2011, Borkar-Mitter-1997, Minero-etal-2009, Coviello-etal2012, Gupta-etal-2007,Gupta-etal-2009,Schenato-etal-2007,Huang-Dey-2007, Elia2005,Mitter-Elia2001}, ~\cite{ball1,ball2,Wong-Brocket1999,Brocket-Liberzon2000,Liberzon2003,Delchamps1990,Martins-Dehleh-Elia2006,Nair-Evans-2004,Imer-etal-2006,Tatikonda-Mitter2004,Tatikonda-Mitter2004b,Gupta-Martins2010,Yuksel2010,Yuksel-Basar2011,Borkar-Mitter-1997,Minero-etal-2009,Coviello-etal2012,Gupta-etal-2007,Gupta-etal-2009,Schenato-etal-2007,Huang-Dey-2007,Elia2005,Mitter-Elia2001},  including the case of fixed rate channel; the erasure channel where the rate process can assume value zero; and the packet loss channel, where the rate process can oscillate randomly between zero and infinity, allowing a real number with infinite precision to be transported across the channel in one time step. Connections between the rate limited and the packet loss channel have been pointed out in~\cite{Minero-etal-2009, Coviello-etal2012}, in~\cite{Minero-etal-2009,Coviello-etal2012},  showing that results for the latter model can be recovered by appropriate limiting arguments. The additive white Gaussian channel has been considered in~\cite{Tatikonda-etal2004, Braslavsky-Middleton-Freudenberg2007,Middleton-Roja2009,Freudenberg-Middleton-Solo2010, Goodwin-Quevedo2010} in~\cite{Tatikonda-etal2004,Braslavsky-Middleton-Freudenberg2007,Middleton-Roja2009,Freudenberg-Middleton-Solo2010,Goodwin-Quevedo2010}  and in this case the Shannon capacity is indeed sufficient to express the rate needed for stabilization. Extensions to the additive colored Gaussian channel~\cite{Ardestanizadeh--franceschetti2012} show that the maximum ``tolerable instability'' --- expressed by the sum of the logarithms of the unstable eigenvalues of the system that can be stabilized by a linear controller with a given power constraint over a stationary Gaussian channel--- corresponds to the Shannon feedback capacity~\cite{Kim2010}, that assumes the presence of a noiseless feedback link between the output and the input of the channel and that is subject to the same power constraint. This result suggests a duality between the problems of control and communication in the presence of feedback, and indeed it has been shown that efficient feedback communication schemes can be obtained by solving a corresponding control problem~\cite{Elia2004,Ardestanizadeh-etal-2012}. The major contribution of this paper is the introduction of a stability threshold function of  the channel's parameters and of the moment stability number $m$ that converges to the Shannon capacity for $m \rightarrow 0$, to the zero-error capacity for $m \rightarrow \infty$, and it provides a parametric characterization of the anytime capacity for the remaining values of $m$. This function yields a novel anytime capacity formula in the special case of the $r$-bit Markov erasure channel. To prove our results, we require some novel extensions of the theory Markov Jump Linear Systems (MJLS), that are of independent value. On the technical side, the sufficient condition for stability is obtained exploiting the idea of subsampling, while the necessary condition is based on the maximum entropy theorem and the entropy power inequality. In passim, although we do not deal with the case of vector systems directly, we point out that our results can be extended to this case exploiting usual bit-allocation techniques outlined in~\cite{Minero-etal-2009, Coviello-etal2012}, in~\cite{Minero-etal-2009,Coviello-etal2012},  at the expense of a more technical treatment that does not add much to the engineering insight and that we wish to avoid here. The rest of the paper is organized as follows. Some preliminary results on Markov Jump Linear Systems, necessary for our derivations are presented in Section~\ref{sec:mjls}. Section~III describes the system and channel model and introduces the stability threshold function, illustrating some of its properties. Section~IV describes relationships with the anytime capacity, and provides some representative examples. Section~V provides the formula for the anytime capacity of the Markov erasure channel. 
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\sup_k \E[|X_k|^m]<\infty,  \end{equation}  where the expectation is taken with respect to the random initial condition $x_0$, the additive disturbance $v_k$, and the rate process $R_k$.   %We make the usual assumption on the tail distribution of the disturbance to have bounded support. This is done for ease of presentation. Extensions to unbounded support can be obtained using standard adaptive schemes~\cite{Nair-Evans-2004,Minero-etal-2009, Coviello-etal2012}. schemes~\cite{Nair-Evans-2004,Minero-etal-2009,Coviello-etal2012}.  The following Theorem establishes the equivalence between the $m$-th moment stability of~\eqref{eq:scalsys} and the weak moment stability of a suitably defined MJLS.  
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%\end{figure}  \medskip  The proof is given in the appendix in the case of state feedback and assuming the disturbance has bounded support. These assumptions are made for ease of presentation and to compare our results to the ones on the anytime capacity that only apply to plants with bounded disturbance~\cite{Sahai2001}. The extension to unbounded disturbance can be easily obtained using standard, but more technical, adaptive encoding schemes described in~\cite{Nair-Evans-2004,Minero-etal-2009, Coviello-etal2012}. in~\cite{Nair-Evans-2004,Minero-etal-2009,Coviello-etal2012}.  Notice that~\eqref{eq:Rm} is obtained from by replacing $\bar{a}_i$ with $|\lambda|/2^{\bar{r}_i}$ in~\eqref{eq:sss} and by simplifying the resulting expression. We now mention several properties of the threshold function $R(m)$, whose proofs are given in the appendix.  \medskip