Estimation of states and parameters in continuous non-linear systems with discrete observations

We consider a class of continuous non-linear systems defined by the ordinary differential equation x = f(x, t) + g(x, t)u, where u is an unknown input representing noise or disturbances. The object is to estimate states and parameters in these systems by means of a fixed number of discrete observations y_i = h(x(t_i), t_i) + v_i, 1 ? i ? m, where the v_i represents unknown errors in the measurements y_i. No statistical assumptions are made concerning the nature of the unknown input u or the unknown measurement errors v_i. A weighted least squares criterion is defined as a measure of the optimal estimate. A result concerning the existence of solutions of the differential equation which minimize the criterion is presented. The necessary conditions for an optimal estimate, a set of Euler-Lagrange equations and multi-point discontinuous non-linear boundary conditions, are given. The multi-point problem is converted to an equivalent continuous two-point boundary value problem of larger dimension in the case in which the observations are assumed to be linear functions of the state. A pair of equivalent quasilinearization algorithms is defined for the two-point system and the multi-point system. Quadratic convergence for these algorithms is proved.