the case of the differentiability with respect to $m$, we use the concept of Gâteaux differentiability. Indeed, $F:L^{2}(\mathbb{R}^{n})\rightarrow \mathbb{R}$ is said to be Gâteaux differentiable if there uniquely exists $\frac{\partial F}{\partial m}(m)\in L^{2}(\mathbb{R}^{n})$, such that $$\frac{d}{d\theta}F(m+\theta\tilde{m})|_{\theta=0}=\int_{\mathbb{R}^{n}}\frac{\partial F}{\partial m}(m)(\xi)\tilde{m}(\xi)d\xi.$$ The second order derivative is a linear map from $L^{2}(\mathbb{R}^{n})$ into itself , defined by $$\frac{d}{d\theta}\frac{\partial F}{\partial m}(m+\theta\tilde{m})(\xi)|_{\theta=0}=\int_{\mathbb{R}^{n}}\frac{\partial^{2}F}{\partial m^{2}}(m)(\xi,\eta)\tilde{m}(\eta)d\eta.$$

We can state the second order Taylor’s formula as

$$F(m+\tilde{m})=F(m)+\int_{\mathbb{R}^{n}}\frac{\partial F}{\partial m}(m)(\xi)\tilde{m}(\xi)d\xi+\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\partial^{2}F}{\partial m^{2}}(m)(\xi,\eta)\tilde{m}(\xi)\tilde{m}(\eta)d\xi d\eta.$$

Consider a probability space $(\Omega,\mathcal{A},\mathbb{P})$ on which various Wiener processes are defined. We define first a standard Wiener process $w(t)$ in $\mathbb{R}^{n}$. To avoid cumbersome of notation, if there is no ambiguity, we shall suppress the arguments of functions in the rest of this paper.

We first introduce the classical mean field type control problem in which the state dynamic is represented by the stochastic differential equation of McKean-Vlasov type: $$\label{eq:1.1}
\left\{
\begin{split}
    &dx=g(x,m_{v(\cdot)},v(x))dt+\sigma(x)dw,\\
    &x(0)=x_{0};\\
\end{split}
\right.$$ in which $v(x)$ is a feedback and $m_{v(\cdot)}(x,t)$ is the probability density of the state $x(t)$. The initial value $x_{0}$ is a random variable independent of the Wiener process $w(\cdot)$. This density is well-defined provided that there is invertibility of $a(x)=\sigma(x)\sigma^{*}(x)$. We define the second order differential operator

$$\mathcal{A}\varphi(x)=-\frac{1}{2}\sum_{i,j}a_{ij}(x)\frac{\partial^{2}\varphi(x)}{\partial x_{i}\partial x_{j}}$$ and 