\section{Proof of Claim \ref{claim:cost_optimal_NC}}
\label{Cost_of_the_Strategies}
\red{Start here}
We want to show that \eqref{eq:costforgstar_NC} holds. To this end, we show that 
\begin{align}
\ee [ c(X_t,U_t^*) | H_t^0]=  \tr(\Lambda_*)  + \ee [V_t - V_{t+1} | H_t^0].
\label{main_goal_NC}
\end{align}
where $\Lambda_* =  \sum_{n=1}^N \big( (1-p^n) [P_{*}^0]_{n,n}+p^n P_{*}^n \big)$.
Then, taking the expectation of the above equation and summing it from $0$ to $T$, the correctness of \eqref{eq:costforgstar_NC} is concluded as follows,
\begin{align}
J_{T}(g^*) 
= & \sum_{t=0}^T \ee [ c(X_t,U_t^*) ]
\notag\\
=&(T+1) \tr(\Lambda_*) 
+ \ee [V_0 - V_{T+1} ]
\notag\\
=&(T+1) \tr(\Lambda_*) 
- \ee [V_{T+1} ],
\end{align}
where the last equality holds because $V_0 = 0$. 

Now, to show that \eqref{main_goal_2C} holds, first note that $\ee[V_t | H_t^0] = V_t$ which is given by \eqref{V_t_2C}. Hence, we only need to calculate $\ee [ c(X_t,U_t^*) | H_t^0]$ and $\ee[V_{t+1} | H_t^0]$. 

\subsection{Calculating $\ee [ c(X_t,U_t^*) | H_t^0]$}
Note that 
\begin{small}
\begin{align}
&\ee [ c(X_t,U_t^*) | H_t^0]=  \underbrace{\ee [X_t^\tp Q X_t | H_t^0]}_{\mathbb{T}_4} + \underbrace{\ee[U_t^{*\tp} R U_t^* | H_t^0]}_{\mathbb{T}_5}. 
\label{cost_given_h}
\end{align}
\end{small}
We can simplify $\mathbb{T}_4$ as follows,
\begin{small}
\begin{align}
\mathbb{T}_4 &=  \hat X_t^\tp Q\hat X_t + \ee \Big[(X_t-\hat X_t)^\tp Q (X_t-\hat X_t)|H^0_t\Big] \notag \\
& =  \hat X_t^\tp Q\hat X_t + \sum_{n=1}^N \tr\big([Q]_{n,n}\Sigma_t^n \big) \notag \\
& =  \hat X_t^\tp Q(0)\hat X_t + \sum_{n=1}^N \tr\big([Q(n)]_{n,n}\Sigma_t^n \big),
\end{align}
\end{small}
where the second equality is correct because according to \cite[Lemma 3]{asghari_optimal_2017_arXiv},
given $H_t^0$, $X_t^n$ is independent of $X_t^m$ for $m \neq n$ and
the last equality is correct because from \eqref{Q_mj}, $Q(0) = Q$ and $[Q(n)]_{n,n} = [Q]_{n,n}= Q^{nn}$.
Similarly, since from \eqref{eq:opt_U_NC_fixed} we have $U_t^* = K_*^0 \hat X_t + \sum_{n=1}^N \tilde K_*^n (X_t - \hat X_t)$ where we have defined $\tilde K_*^n = \mathcal{L}_{zero}(K_*^0, K_*^n,n+1,n)$, we can simplify $\mathbb{T}_5$ as follows,

\vspace{-3mm}
\begin{small}
\begin{align}
\mathbb{T}_5 &=(K_*^0 \hat X_t)^\tp R K_*^0 \hat X_t +  \sum_{n=1}^N \tr \big( [(\tilde K_*^n)^{\tp} R \tilde K_*^n]_{n,n} \Sigma_{t}^n \big )
\notag\\
&= (K_*^0 \hat X_t)^\tp R(0) K_*^0 \hat X_t  


+ \sum_{n=1}^N \tr \big( [(\tilde K_*^n)^{\tp} R(n) \tilde K_*^n]_{n,n} \Sigma_{t}^n \big ),
 \label{cost_on_U_t}
\end{align}
\end{small}
where the last equality is correct because from \eqref{R_mj}, $R(0) = R$ and $[(\tilde K_*^n)^{\tp} R \tilde K_*^n]_{n,n} = 
[(\tilde K_*^n)^{\tp} R(n) \tilde K_*^n]_{n,n} = (K_*^n)^{\tp} R^{nn} K_*^n$.

\subsection{Calculating $\ee[V_{t+1}|H^0_t]$}
For $n \in \mathcal{N}$, we define
\begin{align}
&\hat X_{t+1|t}^n := \ee [X_{t+1}^n|H^0_{t}]
\\
&\Sigma_{t+1|t}^n := \cov(X_{t+1}^n|H^0_{t})
 \\
&\Sigma_t^n := \cov(X_{t}^n|H^0_{t}).
\label{sigma_t}
\end{align}
Recall that $\hat{X}_t^n = \ee [X_{t}^n|H^0_{t}]$.  Note that under the strategies $g^*$ defined in \eqref{eq:costforgstar_NC} and from \eqref{Model:system}, we have 

\vspace{-2mm}
\begin{small}
\begin{align}
X_{t+1} &= AX_t + B \Big[ 
K_*^0\hat X_t +  \sum_{n=1}^N \tilde K_*^n (X_t - \hat X_t)
 \Big] + W_t \notag \\
& = A_s(0) \hat X_t + \sum_{n=1}^N A_s(n)  ( X_t - \hat X_t) + W_t.
\label{X_dynamics_proof_NC}
\end{align}
\end{small}
In the above equation, we have defined $A_s(0) = A(0) + B(0) K_*^0$ and $A_s(n) = A(n) + B(n) \tilde K_*^n$ for $n \in \mathcal{N}$ where  matrices $A(n),B(n)$, $n \in \overline{\mathcal{N}}$ are as defined in \eqref{A_mj}-\eqref{B_mj}. Note that the last equality of \eqref{X_dynamics_proof_NC} is correct because from \eqref{A_mj}-\eqref{B_mj}, $A(0) = A$, $\sum_{n=1}^N A(n) = A$, $B(0) =B$, and $B\tilde K_*^n = B(n) \tilde K_*^n$ for $n \in \mathcal{N}$. Now using \eqref{X_dynamics_proof_2C}, we can calculate $\hat X_{t+1|t}$ and $\Sigma^n_{t+1|t}$ as follows, 
\begin{align}
\label{estimation_X_NC}
&\hat X_{t+1|t} = A_s(0) \hat X_t \\
&\Sigma_{t+1|t}^n =  \mathbf I + [A_s(n)]_{n,n} \Sigma_t^n [A_s(n)]_{n,n}^\tp.
\label{Sigma_t_NC}
\end{align}
From the definition of $V_{t+1}$ in \eqref{V_t_NC} and considering \eqref{sigma_t}, we have

\vspace{-2mm}
\begin{small}
\begin{align}
& \ee[V_{t+1}|H^0_t]= \underbrace{\ee\Big[
\hat X_{t+1}^\tp P_*^0 \hat X_{t+1} \Big| H^0_t
\Big]}_{\mathbb{T}_6}

+ \underbrace{\ee\Big[
\sum_{n=1}^N \tr \big(P_*^n \Sigma_{t+1}^n \big )
\Big| H^0_t
\Big]}_{\mathbb{T}_7}.
\end{align}
\end{small}

Note that  when $\Gamma_{t+1}^n = 1$, $\hat X_{t+1}^n = X_{t+1}^n$ and hence $\Sigma_{t+1}^n = 0$. Further,  when $\Gamma_{t+1}^n = 0$, $\hat X_{t+1}^n = \hat X_{t+1|t}^n$ and $\Sigma_{t+1}^n = \Sigma_{t+1|t}^n$. Let $\Delta$ be a block diagonal matrix whose diagonal are block matrices $\Gamma_{t+1}^1 \mathbf{I}_{d_X^1}, \ldots, \Gamma_{t+1}^N \mathbf{I}_{d_X^N}$, that is,
\begin{align}
\Delta := \begin{bmatrix}
   \Gamma_{t+1}^1 \mathbf{I}_{d_X^1} & & \text{\huge0}\\
          & \ddots & \\
     \text{\huge0} & & \Gamma_{t+1}^N \mathbf{I}_{d_X^N}
\end{bmatrix}.
\label{Delta}
\end{align}
Note that $\Delta$ is a random matrix that depends on $\Gamma_{t+1}^1,\ldots,\Gamma_{t+1}^N$.
Then, we can write 
\begin{align}
\label{estimation_state}
&\hat X_{t+1} = \Delta X_{t+1} +(1-\Delta) \hat X_{t+1|t}, \\
 &\Sigma_{t+1}^n  = (1-\Gamma_{t+1}^n) \Sigma_{t+1|t}^n, \quad n \in \mathcal{N}.
\label{estimation_covariance}
\end{align}
Now, we use \eqref{estimation_state} and \eqref{estimation_covariance} to calculate $\mathbb{T}_6$ and $\mathbb{T}_7$.

\subsubsection{Calculating $\mathbb{T}_6$}
Note that from \eqref{estimation_state}, $\hat X_{t+1}$ can be written as $ \hat X_{t+1}= \hat X_{t+1|t} + \Delta (X_{t+1} -\hat X_{t+1|t})$. Considering this, $\mathbb{T}_6$ can be written as

\vspace{-2mm}
\begin{small}
\begin{align}
&\mathbb{T}_6= 
\ee\Big[
\hat X_{t+1|t} ^\tp P_*^0 \hat X_{t+1|t} \Big| H^0_t \Big] \notag \\
&+ \ee\Big[
(X_{t+1} -\hat X_{t+1|t})^\tp  \Delta^{\tp} P_*^0 \Delta (X_{t+1} -\hat X_{t+1|t}) \Big| H^0_t \Big] \notag \\ & = \hat X_{t+1|t} ^\tp P_*^0 \hat X_{t+1|t} + \sum_{n=1}^N (1-p^n) \tr \big ([P_*^0]_{n,n}  \Sigma_{t+1|t}^n  \big).
\label{T_4_1}
\end{align}
\end{small}

The last equality of \eqref{T_4_1} is correct because

\vspace{-2mm}
\begin{small}
\begin{align}
& \ee\Big[
(X_{t+1} -\hat X_{t+1|t})^\tp  \Delta^{\tp} P_*^0 \Delta (X_{t+1} -\hat X_{t+1|t}) \Big| H^0_t \Big] \notag \\ 
&= 
\ee\Big[ \ee[
(X_{t+1} -\hat X_{t+1|t})^\tp  \Delta^{\tp} P_*^0 \Delta (X_{t+1} -\hat X_{t+1|t}) 


| \Delta,H_t^0] \Big| H^0_t \Big] \notag \\ 
&= 
\ee\Big[ \sum_{n=1}^N
(X_{t+1}^n -\hat X_{t+1|t}^n)^\tp  [\Delta^{\tp} P_*^0 \Delta]_{n,n} (X_{t+1}^n -\hat X_{t+1|t}^n) 


\Big| H^0_t \Big] \notag \\ 
&= 
\sum_{n=1}^N \ee\Big[ 
(X_{t+1}^n -\hat X_{t+1|t}^n)^\tp (\Gamma_{t+1}^n)^2  [P_*^0]_{n,n} (X_{t+1}^n -\hat X_{t+1|t}^n) 


\Big| H^0_t \Big] \notag \\ 
&=  \sum_{n=1}^N (1-p^n) \tr \big ([P_*^0]_{n,n}  \Sigma_{t+1|t}^n  \big),
\label{T_4_1_1}
\end{align}
\end{small}
where the first equality is correct because of tower property and the second equality is correct because according to \cite[Lemma 3]{asghari_optimal_2017_arXiv}, given the common information $H_t^0$, $X_{t+1}^n$ is independent of $X_{t+1}^m$ for $m \neq n$. The third equality is correct because $[\Delta^{\tp} P_*^0  \Delta]_{n,n} = (\Gamma_{t+1}^n)^2  [P_*^0]_{n,n}$ which can be verified through straightforward algebraic manipulations by using \eqref{Delta}. Finally, the last equality is correct because each term of the summation is only non-zero when $\Gamma_{t+1}^n =1$ which happens with probability $1-p^n$.

\subsubsection{Calculating $\mathbb{T}_7$}
Considering \eqref{estimation_covariance}, it is straightforward to see that 

\vspace{-2mm}
\begin{small}
\begin{align}
&\mathbb{T}_7= \sum_{n=1}^N p^n \tr \big (P_*^n \Sigma_{t+1|t}^n  \big).
\end{align}
\end{small}

\subsubsection{Simplifying $\ee[V_{t+1}|H^0_t]$}
Now, using $\mathbb{T}_6$ and $\mathbb{T}_7$ calculated above, $\ee[V_{t+1}|H^0_t]$ can be written as,
\begin{align}
\ee[V_{t+1}|H^0_t]&= 
\hat X_{t+1|t} ^\tp \Pi(\tilde P_*^0,0) \hat X_{t+1|t} \notag \\
&+ \sum_{n=1}^N \tr \big ([\Pi(\tilde P_*^n,n)]_{n,n}  \Sigma_{t+1|t}^n  \big),
\label{V_t_1}
\end{align}
where we have defined $\tilde P_*^0 = P_*^0$ and $\tilde P_*^n = \mathcal{L}_{zero}(P_*^0, P_*^n,n,n)$. Further, we have used the fact that $[\tilde P_*^n]_{n,n} = P_*^n$ and the definition of $\Pi(\cdot)$ in \eqref{Pi_matrix} where $\Theta$ is as defined in \eqref{eq:MJLS_theta}.

Using \eqref{estimation_X_NC}, we can write $\hat X_{t+1|t} ^\tp \Pi(\tilde P_*,0) \hat X_{t+1|t}$ as follows,

\begin{small}
\begin{align}
& \hat X_t ^\tp A_s(0)^\tp \Pi(\tilde P_*,0) A_s(0) \hat X_t
\notag\\
&= \hat X_t ^\tp \Big(\Phi(\tilde P_*,K_*^0,0) - Q(0) - (K_*^0)^\tp R(0) K_*^0 \Big)\hat X_t
\notag\\
&= \hat X_t^\tp \tilde P_*^0\hat X_{t}
 - \hat X_t^\tp Q(0) \hat X_{t}- (K_*^0\hat X_t)^\tp R(0) K_*^0\hat X_t,
\label{eq:Vtp1_mean_NC}
\end{align}
\end{small}
where the first equality is true from \eqref{Phi_def} in Lemma \ref{lm:KPrelation}. Furthermore, the last equality is correct because

\vspace{-2mm}
\begin{small}
\begin{align}
\Phi(\tilde P_*,K_*^0,0) = \Phi(\tilde P_*,\Psi^{\diamond}(\tilde P_*,0),0) = \Omega^{\diamond}(\tilde P_*,0) = \tilde P_*^0,
\end{align}
\end{small}
where the first equality is true because from \eqref{eq:K_finite_2C_fixed}, $K_*^0 = \Psi^{\diamond}(\tilde P_*,0)$, the second equality is true from \eqref{eq:relation1} in Lemma \ref{lm:KPrelation}, and the last equality is true because of definition operator $\Omega^{\diamond}(\cdot)$ in \eqref{Omega_MJ} and \eqref{eq:P_finite_2C_fixed}.

Similarly, using \eqref{Sigma_t_NC} and by defining $\hat P_*^n = \Pi(\tilde P_*^n,n)$, $\tr \big ([\Pi(\tilde P_*^n,n)]_{n,n}  \Sigma_{t+1|t}^n  \big)$ can be written as
\begin{small}
\begin{align}
&\tr \big ([\Pi(\tilde P_*^n,n)]_{n,n}  \Sigma_{t+1|t}^n  \big)
\notag\\
&=\tr \big ([\hat P_*^n]_{n,n} \big) + \tr \big ([A_{s}(n)]_{n,n}^{\tp} [\hat P_*^n]_{n,n} [A_{s}(n)]_{n,n} \Sigma_t^n \big)
\notag\\
&= \tr \big ([\hat P_*^n]_{n,n} \big)  
+\tr(\Phi(\tilde P_*,\tilde K_*^n,n) \Sigma_t^n)
\notag\\
&-\tr \big( [Q(n)]_{n,n} \Sigma_t^n \big)-\tr \big( [(\tilde K_*^n)^\tp R(n) \tilde K_*^n]_{n,n} \Sigma_t^n \big)
\notag\\
&= \tr \big ([\hat P_*^n]_{n,n} \big)  
+\tr([\tilde P_*^n]_{n,n} \Sigma_t^n)
\notag\\
&-\tr \big( [Q(n)]_{n,n} \Sigma_t^n \big)- \tr \big( [(\tilde K_*^n)^\tp R(n) \tilde K_*^n]_{n,n} \Sigma_t^n \big).
\label{eq:Vtp1_variance_NC}
\end{align}
\end{small}
Putting \eqref{eq:Vtp1_mean_NC} and \eqref{eq:Vtp1_variance_NC} into \eqref{V_t_1} and using the fact that $[\tilde P_*^n]_{n,n} = P_*^n$, we get
\begin{small}
\begin{align}
& \ee[V_{t+1}|H^0_t]
\notag\\
& =  \hat X_t^\tp P_*^0\hat X_{t}
 - \hat X_t^\tp Q(0) \hat X_{t}- (K_*^0\hat X_t)^\tp R(0) K_*^0\hat X_t
 \notag \\
& + \sum_{n=1}^N \tr \big ([\hat P_*^n]_{n,n} \big) +\sum_{n=1}^N \tr(P_*^n \Sigma_t^n)
\notag\\
&- \sum_{n=1}^N \tr \big( [Q(n)]_{n,n} \Sigma_t^n \big)-  \sum_{n=1}^N \tr \big( [(\tilde K_*^n)^\tp R(n) \tilde K_*^n]_{n,n} \Sigma_t^n \big)
 \notag \\
&=\sum_{n=1}^N \tr \big ([\hat P_*^n]_{n,n} \big) + \underbrace{ \hat X_t^\tp P_*^0\hat X_{t} + \sum_{n=1}^N \tr(P_*^n \Sigma_t^n)}_{=V_t} \notag \\
& - \underbrace{[\hat X_t^\tp Q(0) \hat X_{t} + \sum_{n=1}^N \tr \big( [Q(n)]_{n,n} \Sigma_t^n \big)]}_{=\mathbb{T}_4} \notag \\
& - \underbrace{ [(K_*^0\hat X_t)^\tp R(0) K_*^0\hat X_t + \sum_{n=1}^N \tr \big( [(\tilde K_*^n)^\tp R(n) \tilde K_*^n]_{n,n} \Sigma_t^n \big)]
}_{=\mathbb{T}_5}.
\label{eq:Vtp1_t}
\end{align}
\end{small}

Note that from \eqref{cost_given_h}, we have $\ee[c(X_t,U_t^*)|H^0_t] = \mathbb{T}_4 + \mathbb{T}_5$ and furthermore, 

\vspace{-3mm}
\begin{small}
\begin{align}
&\sum_{n=1}^N [\hat P_*^n]_{n,n} = \sum_{n=1}^N [\Pi(\tilde P_*,n)]_{n,n} \notag \\
&= \sum_{n=1}^N \big( (1-p^n) [\tilde P_*^0]_{n,n} + p^n  [\tilde P_*^n]_{n,n} \big) \notag \\
&=
\sum_{n=1}^N \big( (1-p^n) [P_{*}^0]_{n,n}+p^n P_{*}^n \big) = \Lambda_*.
\end{align}
\end{small}

Therefore,
\begin{small}
\begin{align}
& \ee[V_{t+1}|H^0_t]
= \Lambda_*
+ V_t - \ee[c(X_t,U_t^*)|H^0_t],
\end{align}
\end{small}
which concludes the correctness of \eqref{main_goal_NC}.
{"config": "arxiv", "file": "1806.06497/old_proof_claim2.tex"}