Ex.11 Data Assimilation

$$\begin{cases} \begin{aligned} \frac{\partial L}{\partial x_i} &= 0 & (i=1,\ldots,n+1)\\ \frac{\partial L}{\partial \lambda_i} &= 0 & (i=1,\ldots,n+1)\\ \frac{\partial L}{\partial\theta} &= 0 \end{aligned} \end{cases}$$

$$\begin{aligned} \frac{\partial L}{\partial x_i} &= \frac{\partial J_D\left(x_1,\ldots,x_n\right)}{\partial x_i} - \left(\lambda_i - \lambda_{i+1}\frac{\partial F\left(x_i,\theta\right)}{\partial x_i}\right) = 0\\ \therefore \lambda_i &= \lambda_{i+1}\frac{\partial F\left(x_i,\theta\right)}{\partial x_i} + \frac{\partial J_D\left(x_1,\ldots,x_n\right)}{\partial x_i} \end{aligned}$$

$$\begin{aligned} \frac{\partial L}{\partial x_{n+1}} &= \frac{\partial J_D\left(x_1,\ldots,x_n\right)}{\partial x_{n+1}} - \lambda_{n+1} = -\lambda_{n+1} = 0\\ \therefore \lambda_{n+1} &= 0 \end{aligned}$$

$$\begin{aligned} x_{i+1} &= F\left(x_i;\theta\right)\\ \frac{\partial x_{i+1}}{\partial\theta} &= \frac{\partial F\left(x_i;\theta\right)}{\partial\theta}\\ &= \frac{\partial F}{\partial x}\left(x_i;\theta\right)\frac{}{} \end{aligned}$$