# EKF smoothing for prediction at t=0 when no there is no measurement

I have a simple first-order reaction batch system for which I have some discrete measurements ($0<t_{k}\le t_{endbatchsample}$).

I have an initial guess for $x_0$ and $P_0$ and from here I perform the time update step (in a continuous-discrete EKF fashion) until $t_1$, where the measurement update kicks in. This process is repeated until $t_{endbatchsample}$ In this particular application it is useful for me two things:

1. Prediction from $t^+_k$ (posterior) until the $t_{endbatchsample}$ with covariance (in this case I just have one state, variance) propagation through the continuous Riccati equation. This is implemented.

2. Correction of $x_0$ and $P_0$ after the batch. For this, I have thought of the (Extended)RTS smoother, but I am not sure how to perform it since there was no measurement at $t_0$, hence no EKF estimate. Is there a conventional way to predict (extrapolate) the new $x_0$ and $P_0$ (at $t_0$) after a series of measurements are available?

Sorry for any confusion, this isn't really my field.

Best regards