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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 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 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 system, 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}~\cite{shannon1961}, 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. For this reason, usage of alternative capacity notions with stronger reliability constraints than a vanishing probability of error has been proposed, including the zero-error capacity~\cite{matveed-savkin2007b}, originally introduced by Shannon~\cite{Shannon1956}, and the anytime capacity of Sahai and Mitter~~\cite{Sahai2001}, \cite{Como-Fagnani-Zampieri2010, Sahai-Mitter2006, simsek-jain-varaija2004, Xu2005}. \cite{Como-Fagnani-Zampieri2010,Sahai-Mitter2006,simsek-jain-varaija2004,Xu2005}.  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 $k$ and evolving randomly in a Markovian fashion, see Fig.~\ref{fig:scheme}. 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, 
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%Borkar-Mitter-1997,   Minero-etal-2009, Coviello-etal2012, Gupta-etal-2007,Gupta-etal-2009,Schenato-etal-2007,  %Huang-Dey-2007,   Elia2005,Mitter-Elia2001, matveed-savkin2009}, 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 the two values of 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. Beside the bit-pipe model, stabilization over the additive white Gaussian channel has also 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 has been shown to be 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 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 the present 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, as well as novel formulas for memoryless rate processes including Binomial, Poisson, and Geometric distributions. It also provides a general strategy to compute the anytime capacity of an arbitrary memoryless channel with given rate distribution.  %Beside the memoryless erasure channel and the additive white Gaussian noise channel with input power constraint, these are the only cases where the anytime capacity is computed.   To prove our results, we require some extensions of the theory of 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, 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 provide additional engineering insight. 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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The proof is given in the appendix   %in the case of state feedback [THIS IS CLEAR FROM THE EQ OF THE SYSTEM]  assuming the disturbance has bounded support. This assumption is 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 mention several properties of the threshold function $R(m)$, whose proofs are given in the appendix. 
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Clearly, if a MJLS is strongly stable, then it is also weakly stable according to our definition. It can be verified that our proof of the necessary condition in Theorem~\ref{lemmamjls} extends to the case of strong moment stability, thereby establishing the necessity of~\eqref{eq:pp} for the strong moment stability of~\eqref{eq:mjls}. In contrast, the subsampling technique used in the direct part of Theorem~\ref{lemmamjls} is not suitable to prove strong moment stability because it cannot prevent undesirable fluctuations in the system dynamics between any two sampling periods.   In~\cite{minero-etal-2013} we presented a different proof technique based on the binomial expansion of $(a_k z_k+w_k)^m$ which can be used to prove that~\eqref{eq:pp} is a sufficient condition for the strong moment stability of~\eqref{eq:mjls}. This approach however requires $m$ to be an integer number, as opposed to the subsampling technique which can be applied to any $m>0$.   Finally, there is no difficulty, sic et simpliciter, but at the expense of a more technical treatment, to extend our results to vector systems using bit-allocation techniques outlined in~\cite{Minero-etal-2009, Coviello-etal2012}, in~\cite{Minero-etal-2009,Coviello-etal2012},  and to disturbances of unbounded support using standard adaptive schemes~\cite{Nair-Evans-2004,Minero-etal-2009, Coviello-etal2012}. schemes~\cite{Nair-Evans-2004,Minero-etal-2009,Coviello-etal2012}.  %\subsection{Error exponents}  %We now compare the the anytime error exponent to error exponents of block codes. Specifically, we can compare the curves in Fig.~\ref{fig:scheme} obtained by evaluating~\eqref{eq:anyMarkov} with the results by Como, Y\"{u}ksel, and Tatikonda in \cite[Section VI]{Comoetal2009}. \emph{to do}  %\begin{figure}