Weak*approximations for the solution of a dynamic reconstruction problem

Abstract

We consider the problem of the dynamic reconstruction of an observed state trajectory x() of an affine deterministic dynamic system and the control that has generated this trajectory. The reconstruction is based on current information about inaccurate discrete measurements of x(). A correct statement of the problem on the construction of approximations ul() of the normal control ul() generating x∗() is refined. The solution of this problem obtained using the variational approach proposed by the authors is discussed. Conditions on the input data and matching conditions for the approximation parameters (parameters of the accuracy and frequency of measurements of the trajectory and an auxiliary regularizing parameter) are given. Under these conditions, the reconstructed trajectories xl() of the dynamical system converge uniformly to the observed trajectory x∗() in the space of continuous functions C as l → ∞. It is proved that the proposed controls ul() converge weakly∗to u∗() in the space of summable functions L1. © 2021 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.