Reconstruction of an Unbounded Input of a System of Differential Equations

We consider the problem of reconstructing an unbounded nonsmooth input of a system of ordinary differential equations that are nonlinear in the state variables and linear in control. The problem has two features. First, we assume that the state coordinates of the system are measured (with error) at discrete instants of time. Second, we assume that the unknown input is an element of the space of functions with square integrable Euclidean norm, i.e., it may be nonsmooth and unbounded. Taking into account this feature of the problem, we construct an algorithm for solving it that is stable to information noise and computational errors. The algorithm is based on a combination of constructions of the theory of ill-posed problems and the well-known extremal shift method from the theory of positional differential games.