State observation problem for general time reversible system and applications

This paper is concerned with a state observation problem for the general time reversible system, i.e., to recover the initial state of the given system from suitable observation data. An abstract unified framework is presented. By means of an iterative time reversal technique, we derive two asymptotic formulae of reconstructing the initial state approximately with explicit convergence rates. General results are then applied to the state observation problems for the Schrödinger equation and the wave equation.