On the statistical stability of input identification problems for dynamic systems

An ill-posed problem of the reconstruction of input from measurements of output in a linear dynamic system is considered. Instead of the deterministic model of errors in the specification of input data, which is traditional for such problems, a stochastic model is proposed. More exactly, it is assumed that there is noise in the output measurement channel; the noise is modeled by realizations of a random element with a weak distribution in the space of outputs. To determine correctly the identification error, an extension of the space of outputs of the dynamic system is constructed. The notion of linear generalized solving procedure is introduced. Criteria for the statistic stability of the problem (the stability of the pseudo-inverse input-output operator of the system with respect to random errors in output measurements) are obtained.