The dynamical discrepancy method in problems of reconstructing unknown characteristics of a second-order system

Abstract

This paper considers two problems of dynamical reconstruction of unknown characteristics of a system of nonlinear equations describing the process of innovation diffusion through inaccurate measurements of phase states. A dynamical variant for solving these problems is designed. The system is assumed to operate on a given finite time interval. The evolution of the system's phase state, i.e., the solution of the system, is determined by an unknown input. A precise reconstruction of the real input (acting on the system) is, generally speaking, impossible due to inaccurate measurements. Therefore, some approximation to this input is constructed which provides an arbitrary smallness to the real input if the measurement errors and the step of incoming information are sufficiently small. Based on the dynamical version of the discrepancy method, two algorithms for solving the problems in question are specified. One of them is oriented to the case of measuring all coordinates of the phase vector, and the other, to the case of incomplete measurements. The algorithms suggested are stable with respect to informational noises and computational errors. Actually, they are special regularizing algorithms from the theory of dynamic inverse problems.