Input Reconstructibility for Linear Dynamics. Ordinary Differential Equations

The paper deals with the standard input-output observation scheme for a dynamic system governed by a linear ordinary differential equation. The initial problem is to reconstruct the actually working time-varying input, given a state observation result. Normally, the problem has no solution: observation is too poor to select the real input from the collection of "possible" ones. It is proposed to turn the problem as follows: what information of the real input is reconstructible precisely? The dual setting: what information of the real input is totally non-reconstructible? The question of aftereffect arises naturally: does accumulation of observation results lead to the informational jump -- from nonreconstructibility to complete reconstructibility -- in the past? Posing and answering these questions is the goal of the present study