Distributed Control of Linear Quadratic Mean Field Social Systems with Heterogeneous Agents

Consider a large-population system with $K\in\mathbb{R}$ clusters $\mathcal{C}_{q}$, $q\in\mathcal{V}_{K}\triangleq\{1,2,...,K\}$, where the cluster $C_{q}$ contains $N_{q}$ homogeneous agents with the same dynamics, cost functions and communication capabilities. In this scenario, the large-population system contains a total of $N\triangleq\sum_{q\in\mathcal{V}_{K}}N_{q}$ heterogeneous agents $\mathcal{A}_{i}$, $i\in\mathcal{I}\triangleq\{1,2,...,N\}$, and the agents are partitioned into the $K$ disjoint clusters $\mathcal{C}_{q}$, $q\in\mathcal{V}_{K}$. The set of agents belonging to the cluster $\mathcal{C}_{q}$ is denoted by $\mathcal{I}_{q}\subseteq\mathcal{I}$ with $|\mathcal{I}_{q}|=N_{q}$, $\cup_{q\in\mathcal{V}_{K}}\mathcal{I}_{q}=\mathcal{I}$, and $\cap_{q\in\mathcal{V}_{K}}\mathcal{I}_{q}=\emptyset$. In this paper, we also list the agents as $\mathcal{A}_{i}$, $i\in\mathcal{I}_{q}$, $q\in\mathcal{V}_{K}$.

The agents in different clusters communicate with each other through a network whose topology is modeled as a directed graph $\mathcal{G}_{K}=\{\mathcal{V}_{K},\mathcal{E}_{K},\mathcal{M}_{K}\}$, where the clusters and the communication channels between clusters are represented by the node set $\mathcal{V}_{K}$ and the edge set $\mathcal{E}_{K}\subseteq\mathcal{V}_{K}\times\mathcal{V}_{K}$, respectively. An edge denoted by the pair $(j,i)$ represents a communication channel from cluster $\mathcal{C}_{j}$ to cluster $\mathcal{C}_{i}$. The neighbor set of cluster $\mathcal{C}_{q}$ is denoted by $\mathcal{N}_{q}\triangleq\{\ p\ |\ p\in\mathcal{V}_{K},\ (p,q)\in\mathcal{E}_{K}\}$. Let $E_{K}=[e_{ij}]\in\mathbb{R}^{K\times K}$ be the communication matrix, where $e_{ij}=1$ if $(j,i)\in\mathcal{E}_{K}$, $e_{ij}=0$ if $(j,i)\notin\mathcal{E}_{K}$. The weighted adjacency matrix of $\mathcal{G}_{K}$ is denoted by $\mathcal{M}_{K}=[m_{ij}]\in\mathbb{R}^{K\times K}$. The agents are coupled with each other according to the adjacency matrix.

Let $x_{i}\in\mathbb{R}^{n}$ and $u_{i}\in\mathbb{R}^{m}$ be the state and control of agent $\mathcal{A}_{i},i\in\mathcal{I}_{q},q\in\mathcal{V}_{K}$. The local cluster mean field term of cluster $\mathcal{C}_{q}$ is defined as the empirical mean of the states of all agents in the cluster, i.e.,

$\displaystyle x^{K}_{q}(t)\triangleq\frac{1}{N_{q}}\sum_{i\in\mathcal{I}_{q}}x_{i}(t),\quad q\in\mathcal{V}_{K}.$

(1)

Therefore the global cluster mean field term of the large-population system is stacked as $x^{K}\triangleq\mathrm{vec}(x^{K}_{1},x^{K}_{2},$ $...,x^{K}_{K})$. By the adjacency matrix $\mathcal{M}_{K}$, the influence of the global cluster mean field term $x^{K}$ on the agents in cluster $C_{q}$ is denoted as $z^{K}_{q}$, where

$\displaystyle z^{K}_{q}(t)\triangleq\frac{1}{K}\sum_{p\in\mathcal{V}_{K}}m_{qp}x^{K}_{p}(t),\quad q\in\mathcal{V}_{K}.$

(2)

Coupled by the term $z^{K}_{q}$, the dynamics of agent $\mathcal{A}_{i},i\in\mathcal{I}_{q},q\in\mathcal{V}_{q}$, is given by the following stochastic differential equation

$\displaystyle dx_{i}(t)=$

$\displaystyle\big{[}A_{q}x_{i}(t)+B_{q}u_{i}(t)+G_{q}z_{q}^{K}(t)\big{]}dt+\Sigma_{q}dw_{i}(t),\quad x_{i}(0)=x_{i0},$

(3)

where $A_{q},B_{q},G_{q},\Sigma_{q}$ are constant matrices with compatible dimensions and $\{w_{i},i\in\mathcal{I}_{q}\}$ are $1$-dimensional independent standard Brownian motions defined on a complete filtered probability space $(\Omega,\mathcal{F},\mathbb{P})$.

Let $\mathbf{x}\triangleq\mathrm{vec}(x_{1},x_{2},...,x_{N})$ and $\mathbf{u}\triangleq\mathrm{vec}(u_{1},u_{2},...,u_{N})$ be the state and control of the large-population system, respectively. Then the cost function of agent $\mathcal{A}_{i},i\in\mathcal{I}_{q},q\in\mathcal{V}_{K}$, is given by

$$\begin{split}J_{i}(\mathbf{u})=&\mathbb{E}\int_{0}^{T}\big{[}|x_{i}(t)-\Gamma_{q}z_{q}^{K}(t)|^{2}_{Q_{q}}+|u_{i}(t)|^{2}_{R_{q}}\big{]}dt+\mathbb{E}|x_{i}(T)-\Gamma_{q}z_{q}^{K}(T)|^{2}_{H_{q}},\end{split}$$

where $\Gamma_{q},Q_{q}\geq 0,R_{q}>0,H_{q}\geq 0$ are constant matrices with compatible dimensions. The social cost function is defined as

$\displaystyle J_{\rm soc}^{\rm(N)}(\mathbf{u})=\sum_{q\in\mathcal{V}_{K}}\sum_{i\in\mathcal{I}_{q}}J_{i}(\mathbf{u}).$

(4)

Let $M_{q}$, $q\in\mathcal{V}_{K}$, be the $q$-th row of $\mathcal{M}_{K}$ and

$$\bar{G}_{q}\triangleq\frac{M_{q}}{K}\otimes G_{q}.$$

Then the individual dynamics (3) can be rewritten as

$$\begin{split}dx_{i}(t)=&\big{[}A_{q}x_{i}(t)+B_{q}u_{i}(t)+\bar{G}_{q}x^{K}(t)\big{]}dt+\Sigma_{q}dw_{i}(t),\quad x_{i}(0)=x_{i0}.\end{split}$$

(5)

Similarly, let

$$\bar{\Gamma}_{q}\triangleq\frac{M_{q}}{K}\otimes\Gamma_{q}.$$

Thus the social cost (4) can be rewritten as

$$\begin{split}J_{\rm soc}^{\rm(N)}(\mathbf{u})=&\sum_{q\in\mathcal{V}_{K}}\sum_{i\in\mathcal{I}_{q}}\Big{\{}\mathbb{E}\int_{0}^{T}\big{[}|x_{i}(t)-\bar{\Gamma}_{q}x^{K}(t)|^{2}_{Q_{q}}+|u_{i}(t)|^{2}_{R_{q}}\big{]}dt+\mathbb{E}|x_{i}(T)-\bar{\Gamma}_{q}x^{K}(T)|^{2}_{H_{q}}\Big{\}}.\end{split}$$

(6)

In this paper, we apply the direct approach to sequentially study the mean field social control problem under centralized and distributed information patterns. So we first give the definitions of filtration used in this paper. Denote

	$\displaystyle\mathcal{F}_{t}^{i}\triangleq$	$\displaystyle\sigma(x_{i}(s),w_{i}(s),s\leq t),$
	$\displaystyle\mathcal{F}_{t}\triangleq$	$\displaystyle\sigma(\cup_{i\in\mathcal{I}_{q},q\in\mathcal{V}_{K}}\mathcal{F}_{t}^{i}),$
	$\displaystyle\mathbb{F}\triangleq$	$\displaystyle\{\mathcal{F}_{t}\}_{0\leq t\leq T},$
	$\displaystyle\mathcal{H}_{t}^{q,K}\triangleq$	$\displaystyle\sigma(\cup_{j\in\mathcal{I}_{p},p\in\mathcal{N}_{q}}\mathcal{F}^{j}_{t}),$
	$\displaystyle\mathcal{H}_{t}^{i}\triangleq$	$\displaystyle\sigma(\mathcal{H}_{t}^{q,K},\mathcal{F}_{t}^{i}),$
	$\displaystyle\mathbb{H}_{i}\triangleq$	$\displaystyle\{\mathcal{H}^{i}_{t}\}_{0\leq t\leq T}.$

Then the centralized and distributed admissible control sets of agent $\mathcal{A}_{i},i\in\mathcal{I}_{q},q\in\mathcal{V}_{K}$, are given by $\mathcal{U}_{c}[0,T]$ and $\mathcal{U}_{d}^{i}[0,T]$, respectively, where

	$\displaystyle\mathcal{U}_{c}[0,T]=$	$\displaystyle\big{\{}u(\cdot)\in L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\big{\}},$
	$\displaystyle\mathcal{U}_{d}^{i}[0,T]=$	$\displaystyle\big{\{}u(\cdot)\in L^{2}_{\mathbb{H}_{i}}(0,T;\mathbb{R}^{n})\big{\}}.$

As stated in the Introduction, to our knowledge, there is currently no general method for designing optimal distributed controllers. Thus we consider asymptotically optimal distributed controllers based on the mean field approximation methodology. To characterize the asymptotic optimality of distributed controllers, the following definition is introduced:

Therefore, in this paper we first solve Problem (P1) to obtain optimal centralized controllers, and then further study Problem (P2) to design distributed asymptotically optimal controllers based on mean field approximations. The following assumption on the distribution of initial states is imposed.

(A1) The initial states $x_{i0}$, $i\in\mathcal{I}_{q}$, $q\in\mathcal{V}_{K}$, are mutually independent with $\mathbb{E}[x_{i}(0)]=\bar{m}_{q}$, $i\in\mathcal{I}_{q}$, $q\in\mathcal{V}_{K}$, and there exists a finite constant $C_{0}$ independent of $N$ such that $\max_{1\leq i\leq N}\mathbb{E}|x_{i}(0)|^{2}\leq C_{0}$. Let $\bar{m}^{K}\triangleq\mathrm{diag}(\bar{m}_{1},\bar{m}_{2},...,\bar{m}_{K})$.

We make some denotations for convenience of the derivation of the main result. Denote $\pi_{q}\triangleq N_{q}/N$, $q\in\mathcal{V}_{K}$. Then $\pi=(\pi_{1},\pi_{2},...,\pi_{K})$ is a probability vector which gives the empirical distribution of the $K$-type agents. Denote

	$\displaystyle\Pi^{K}\triangleq$	$\displaystyle\ \mathrm{diag}(I_{n}\otimes\pi_{1},I_{n}\otimes\pi_{2},...,I_{n}\otimes\pi_{K}),$
	$\displaystyle N^{K}\triangleq$	$\displaystyle\ \mathrm{diag}(I_{n}\otimes N_{1},I_{n}\otimes N_{2},...,I_{n}\otimes N_{K}),$
	$\displaystyle G^{K}\triangleq$	$\displaystyle\ \mathrm{rows}(\bar{G}_{1},\bar{G}_{2},...,\bar{G}_{K}),$
	$\displaystyle Q^{K}\triangleq$	$\displaystyle\mathrm{diag}(Q_{1},Q_{2},...,Q_{K}),$
	$\displaystyle\Gamma^{K}\triangleq$	$\displaystyle\ \mathrm{rows}(\bar{\Gamma}_{1},\bar{\Gamma}_{2},...,\bar{\Gamma}_{K}),$
	$\displaystyle H^{K}\triangleq$	$\displaystyle\mathrm{diag}(H_{1},H_{2},...,H_{K}),$
	$\displaystyle D^{K}\triangleq$	$\displaystyle\ \mathrm{cols}(D_{1},D_{2},...,D_{K})$
	$\displaystyle=$	$\displaystyle\ N^{K}G^{K}(N^{K})^{-1},$
	$\displaystyle\bar{Q}^{K}\triangleq$	$\displaystyle\ \mathrm{rows}(\bar{Q}_{1},\bar{Q}_{2},...,\bar{Q}_{K})$
	$\displaystyle=$	$\displaystyle\ Q^{K}\Gamma^{K}+(N^{K})^{-1}(\Gamma^{K})^{T}Q^{K}N^{K}-(N^{K})^{-1}(\Gamma^{K})^{T}N^{K}Q^{K}\Gamma^{K},$
	$\displaystyle\bar{H}^{K}\triangleq$	$\displaystyle\ \mathrm{rows}(\bar{H}_{1},\bar{H}_{2},...,\bar{H}_{K})$
	$\displaystyle=$	$\displaystyle\ H^{K}\Gamma^{K}+(N^{K})^{-1}(\Gamma^{K})^{T}H^{K}N^{K}-(N^{K})^{-1}(\Gamma^{K})^{T}N^{K}H^{K}\Gamma^{K}.$

Note that we have $N^{K}=N\Pi^{K}$. By variational analysis, the necessary and sufficient condition is obtained for the solvability of Problem (P1) as follows.

Based on the open-loop centralized optimal controller (9), we next obtain its feedback representation by introducing two Riccati equations. Moreover, we also make the following denotations for convenience of discussions.

	$\displaystyle A^{K}$	$\displaystyle\triangleq\mathrm{diag}(A_{1},A_{2},...,A_{K}),$
	$\displaystyle B^{K}$	$\displaystyle\triangleq\mathrm{diag}(B_{1},B_{2},...,B_{K}),$
	$\displaystyle R^{K}$	$\displaystyle\triangleq\mathrm{diag}(R_{1},R_{2},...,R_{K}),$
	$\displaystyle\Sigma^{K}$	$\displaystyle\triangleq\mathrm{diag}(\Sigma_{1},\Sigma_{2},...,\Sigma_{K}).$

We propose the following Riccati equations

	$\displaystyle\begin{split}0=&\dot{P}^{K}(t)+P^{K}(t)A^{K}+(A^{K})^{T}P^{K}(t)+Q^{K}-P^{K}(t)B^{K}(R^{K})^{-1}(B^{K})^{T}P^{K}(t),\end{split}$			(10a)
	$\displaystyle\begin{split}0=&\dot{K}^{K}(t)+[A^{K}+N^{K}G^{K}(N^{K})^{-1}]^{T}K^{K}(t)+K^{K}(t)[A^{K}+G^{K}]+P^{K}(t)G^{K}\cr&+(N^{K})^{-1}(G^{K})^{T}N^{K}P^{K}(t)-\bar{Q}^{K}-K^{K}(t)B^{K}(R^{K})^{-1}(B^{K})^{T}[P^{K}(t)+K^{K}(t)]\cr&-P^{K}(t)B^{K}(R^{K})^{-1}(B^{K})^{T}K^{K}(t),\end{split}$			(10b)
	$\displaystyle P^{K}$	$\displaystyle(T)=H^{K},\quad K^{K}(T)=-\bar{H}^{K}.$

Let $P_{q}$ be the $q$-th diagonal element of $P^{K}$ and $\bar{K}_{q}$ be the $q$-th row of $K^{K}$, $q\in\mathcal{V}_{K}$. Then the Riccati equations (10a)-(10b) can be rewritten as

	$\displaystyle\begin{split}0=&\dot{P}_{q}(t)+P_{q}(t)A_{q}+A_{q}^{T}P_{q}(t)+Q_{q}-P_{q}(t)B_{q}R_{q}^{-1}B_{q}^{T}P_{q}(t),\end{split}$			(11a)
	$\displaystyle\begin{split}0=&\dot{\bar{K}}_{q}(t)+A_{q}^{T}\bar{K}_{q}(t)+D_{q}^{T}K^{K}(t)+\bar{K}_{q}(t)(A^{K}+G^{K})+P_{q}(t)\bar{G}_{q}+D_{q}^{T}P^{K}(t)\cr&-\bar{K}_{q}(t)B^{K}(R^{K})^{-1}(B^{K})^{T}[P^{K}(t)+K^{K}(t)]-\bar{Q}_{q}-P_{q}(t)B_{q}R_{q}^{-1}B_{q}^{T}\bar{K}_{q}(t),\end{split}$			(11b)
	$\displaystyle P_{q}$	$\displaystyle(T)=H_{q},\quad\bar{K}_{q}(T)=-\bar{H}_{q},\quad q\in\mathcal{V}_{K}.$

Therefore the result of feedback representation is given as follows.

We introduce the following notations to simplify the expression of closed-loop systems.

	$\displaystyle\tilde{A}_{q}\triangleq$	$\displaystyle\ A_{q}-B_{q}R_{q}^{-1}B_{q}^{T}P_{q},$		(14)
	$\displaystyle\tilde{G}_{q}\triangleq$	$\displaystyle\ \bar{G}_{q}-B_{q}R_{q}^{-1}B_{q}^{T}\bar{K}_{q},$		(15)
	$\displaystyle\tilde{A}^{K}\triangleq$	$\displaystyle\ \mathrm{diag}(\tilde{A}_{1},\tilde{A}_{2},...,\tilde{A}_{K}),$		(16)
	$\displaystyle\tilde{G}^{K}\triangleq$	$\displaystyle\ \mathrm{rows}(\tilde{G}_{1},\tilde{G}_{2},...,\tilde{G}_{K}).$

Under the feedback controller (12), the closed-loop systems of $\hat{x}_{i},\hat{x}^{K}_{q}$ and $\hat{x}^{K}$, $i\in\mathcal{I}_{q},q\in\mathcal{V}_{K}$ are given by

	$\displaystyle\begin{split}d\hat{x}_{i}(t)=&\ [\tilde{A}_{q}(t)\hat{x}_{i}(t)+\tilde{G}_{q}(t)\hat{x}^{K}(t)]dt+\Sigma_{q}dw_{i}(t),\quad\hat{x}_{i}(0)=x_{i0},\end{split}$			(17a)
	$\displaystyle\begin{split}d\hat{x}_{q}^{K}(t)=&\ [\tilde{A}_{q}(t)\hat{x}_{q}^{K}(t)+\tilde{G}_{q}(t)\hat{x}^{K}(t)]dt+\Sigma_{q}dw_{q}^{K}(t),\quad\hat{x}_{q}^{K}(0)=x^{K}_{q0},\end{split}$			(17b)
	$\displaystyle\begin{split}d\hat{x}^{K}(t)=&\ [\tilde{A}^{K}(t)+\tilde{G}^{K}(t)]\hat{x}^{K}(t)dt+\Sigma^{K}dw^{K}(t),\quad\hat{x}^{K}(0)=x^{K}_{0}.\end{split}$			(17c)

Note that the optimal centralized feedback controller (12) contains the individual state and the global cluster mean field term. Moreover, the ability of each agent to acquire information is different according to the communication matrix $E_{K}$. Therefore, the main idea of the design of distributed controllers is that each cluster makes a local estimation of the global cluster mean field term according to the network topology.

Therefore we design cluster mean filed estimators for agents in each cluster to estimate the global cluster mean filed term. In the following context we denote the local estimation of agent $\mathcal{A}_{i}$ in cluster $\mathcal{C}_{q},q\in\mathcal{V}_{K}$, for the global cluster mean field term $\check{x}^{K}$ by

$$\bar{x}^{K,q}\triangleq\mathrm{vec}(\bar{x}^{K,q}_{1},\bar{x}^{K,q}_{2},...,\bar{x}^{K,q}_{K}),$$

where $\bar{x}^{K,q}_{p}$ denotes the estimation of the local cluster mean field term $\check{x}^{K}_{p}$ by the agents in cluster $\mathcal{C}_{q}$.

For agents in cluster $\mathcal{C}_{q}$, if $p\in\mathcal{N}_{q}$, then agents can obtain the local cluster mean field term $\check{x}^{K}_{p}$ by communication, i.e,

$$\bar{x}^{K,q}_{p}=\check{x}^{K}_{p},$$

(18)

else agents should make their own local estimation by mean field approximations according to (17b), i.e,

$$\begin{split}&d\bar{x}^{K,q}_{p}(t)=[\tilde{A}_{q}(t)\bar{x}^{K,q}_{p}(t)+\tilde{G}_{q}(t)\bar{x}^{K,q}(t)]dt,\quad\bar{x}_{p}^{K}(0)=\bar{m}_{p}.\end{split}$$

(19)

In order to represent the dynamics of the cluster mean field estimator more compact, the following notation is introduced for $q,p\in\mathcal{V}_{K}$:

	$\displaystyle E_{qp}$	$\displaystyle\triangleq 1_{n}\otimes e_{ij},$
	$\displaystyle E^{K}_{q}$	$\displaystyle\triangleq\mathrm{rows}(E_{q1},E_{q2},...,E_{qK}),$
	$\displaystyle\bar{E}_{qp}$	$\displaystyle\triangleq 1_{n}-E_{qp},$
	$\displaystyle\bar{E}^{K}_{q}$	$\displaystyle\triangleq\mathrm{rows}(\bar{E}_{q1},\bar{E}_{q2},...,\bar{E}_{qK}),$
	$\displaystyle Z^{K}$	$\displaystyle\triangleq\mathrm{diag}(B_{1}R_{1}^{-1}B_{1}^{T}\bar{K}_{1},...,B_{K}R_{K}^{-1}B_{K}^{T}\bar{K}_{K}).$

By (18) and (19), the dynamics of the designed cluster mean field estimator for agent $\mathcal{A}_{i},i\in\mathcal{I}_{q},q\in\mathcal{V}_{K}$, is given by

$\displaystyle d\bar{x}^{K,q}(t)=$

$\displaystyle E^{K}_{q}\odot d\check{x}^{K}(t)+\bar{E}^{K}_{q}\odot[\tilde{A}^{K}(t)+\tilde{G}^{K}(t)]\bar{x}^{K,q}(t)dt,\quad\bar{x}^{K,q}(0)=E^{K}_{q}\odot x^{K}_{0}+\bar{E}^{K}_{q}\odot\bar{m}^{K}.$

(20)

Based on the above discussion, we propose the following distributed controller for agent $\mathcal{A}_{i},i\in\mathcal{I}_{q},q\in\mathcal{V}_{K}$:

$\displaystyle\check{u}_{i}(t)=$

$\displaystyle-R_{q}^{-1}B_{q}^{T}P_{q}(t)\check{x}_{i}(t)-R_{q}^{-1}B_{q}^{T}\bar{K}_{q}(t)\bar{x}^{K,q}(t),$

(21)

where $\bar{x}^{K,q}$ is the estimated global cluster mean field term of cluster $\mathcal{C}_{q}$ given by the cluster mean field estimator (20). Under this distributed controller, the average distributed control of cluster $\mathcal{C}_{q}$ is given by

$\displaystyle\check{u}_{q}^{K}(t)=$

$\displaystyle-R_{q}^{-1}B_{q}^{T}P_{q}(t)\check{x}_{q}^{K}(t)-R_{q}^{-1}B_{q}^{T}\bar{K}_{q}(t)\bar{x}^{K,q}(t).$

(22)

Let

$$\bar{x}^{K}\triangleq\mathrm{vec}(\bar{x}^{K,1},\bar{x}^{K,2},...,\bar{x}^{K,K})$$

be the estimated global cluster mean field term of all clusters in the large-population system. Therefore the closed-loop systems under the distributed controller are given by

	$\displaystyle\begin{split}d\check{x}_{i}(t)=&[\tilde{A}_{q}(t)\check{x}_{i}(t)+\bar{G}_{q}(t)\check{x}^{K}(t)-B_{q}R_{q}^{-1}B_{q}^{T}\bar{K}_{q}(t)\bar{x}^{K,q}(t)]dt+\Sigma_{q}dw_{i}(t),\quad\check{x}_{i}(0)=x_{i0},\end{split}$			(23a)
	$\displaystyle\begin{split}d\check{x}_{q}^{K}(t)=&[\tilde{A}_{q}(t)\check{x}_{q}^{K}(t)+\bar{G}_{q}\check{x}^{K}(t)-B_{q}R_{q}^{-1}B_{q}^{T}\bar{K}_{q}(t)\bar{x}^{K,q}(t)]dt+\Sigma_{q}dw_{q}^{K}(t),\quad\check{x}_{q}^{K}(0)=x^{K}_{q0},\end{split}$			(23b)
	$\displaystyle\begin{split}d\check{x}^{K}(t)=&\big{\{}[\tilde{A}^{K}(t)+G^{K}]\check{x}^{K}(t)-Z^{K}(t)\bar{x}^{K}(t)\big{\}}dt+\Sigma^{K}dw^{K}(t),\quad\check{x}^{K}(0)=x^{K}_{0}.\end{split}$			(23c)

Next we will show the asymptotic optimality property of the designed distributed controller. We first define two kinds of estimation errors of the designed estimator (20) with respect to the realized cluster mean field terms under the centralized controller (12) and distributed controller (21), respectively. For $q,p\in\mathcal{V}_{K}$, denote

	$\displaystyle\check{\xi}^{K,q}_{p}$	$\displaystyle\triangleq\check{x}^{K}_{p}-\bar{x}^{K,q}_{p},$
	$\displaystyle\hat{\xi}^{K,q}_{p}$	$\displaystyle\triangleq\hat{x}^{K}_{p}-\bar{x}^{K,q}_{p},$
	$\displaystyle\check{\xi}^{K,q}$	$\displaystyle\triangleq\mathrm{vec}(\check{\xi}^{K,q}_{1},\check{\xi}^{K,q}_{2},...,\check{\xi}^{K,q}_{K}),$
	$\displaystyle\hat{\xi}^{K,q}$	$\displaystyle\triangleq\mathrm{vec}(\hat{\xi}^{K,q}_{1},\hat{\xi}^{K,q}_{2},...,\hat{\xi}^{K,q}_{K}),$
	$\displaystyle\check{\xi}^{K}$	$\displaystyle\triangleq\mathrm{vec}(\check{\xi}^{K,1},\check{\xi}^{K,2},...,\check{\xi}^{K,K}),$
	$\displaystyle\hat{\xi}^{K}$	$\displaystyle\triangleq\mathrm{vec}(\hat{\xi}^{K,1},\hat{\xi}^{K,2},...,\hat{\xi}^{K,K}).$