Error analysis on the initial state reconstruction problem

In this paper, we propose a method to estimate the initial state of linear dynamical system from noisy observation based on Kalman Filters and Optimal Smoothing techniques. The allows user have estimations in real-time, that is, new estimation for each observation. Moreover, at step, covariance matrix error is found, one minimizes variance after applying filter. We analyze stability when there no noise dynamics. This hypothesis usually seen sampling articles, differs classical framework control theory which plays fundamental role. particular, prove dynamic estimator always Lyapunov stable. Also, necessary sufficient conditions are given guarantee asymptotic dynamics time-invariant system, itself time-variant system.