Estimation of parameters of differential equations according to inaccurate observations

Abstract

In this paper, the problem of estimating the parameters of a system of ordinary differential equations of the fi rst order from inaccurate observations over a short time interval is solving. We are talking about systems of differential equations resolved with respect to the derivative (about normal systems), in which the number of parameters coincides with the number of equations, with given initial conditions. By analogy with linear regression analysis, a sufficiently large number of observations are selecting over the time under consideration and the values of the functions standing in the right parts of the normal system of equations and the values of their derivatives at the initial moment of time are estimating. By analogy with the method of moments, unknown parameters of the initial system of differential equations are determining by estimates of the values of functions and its derivatives. In the work, the properties of the obtained estimates are investigating and, under certain conditions imposed on the step of splitting the time interval, their asymptotic non-bias and consistency with an increase in the number of observations are proving. Computational experiments were carriing out for two special cases and their results are demonstrating in the work. The algorithm proposed in the paper for estimating the parameters of a system of ordinary differential equations by inaccurate deterministic observations, in contrast to classical optimization algorithms, allows us to estimate the rate of convergence of the obtained estimates to the estimated parameters. In addition, the consideration of a small interval of time observation makes it possible to build an experiment planning procedure. Along with systems of ordinary differential equations, the proposed algorithm can be applying to systems of partial differential equations, which is planning to be implemented by the authors in the future.