Dynamic reconstruction of the control and parameters under uncertain information on the system (Q2703871)

The author considers a class of problems where one wishes to find what control was applied to the system, or what were the parameters, from uncertain observation data. His dynamic system is \(dx/dt=f(t,x(t),u(t))\), \(t\in T=[t_0, \vartheta]\), \(x(0)=t_0\). The state of the system \(x(t)\) is an \(n\)-dimensional vector, with \(-\infty <t_0< \vartheta< \infty\). The technique suggested here consists of the following steps: An a priori assumption is supposed to be supplied. Specifically, the observer has a priori knowledge of some states \(G(t)\) containing the current state \(x(t)\) in order to find the approximate control \(u(t)\). The observer restores these controls, inducing a pencil of trajectories agreeing with the observation. The presence of noise yields some sets different from \(G(t)\). Techniques similar to those suggested by Arsenin and Tikhonov for dealing with ill-posed problems are proposed. Following these introductory remarks, the author offers a lengthy and very complex statement of the problem. Then he proves a theorem stating sufficient conditions for the existence of solutions. A second theorem proves validity of an estimate concerning the reconstruction of the control problem.NEWLINENEWLINENEWLINEThis article condenses a lot of information into four and a half pages. Each major step suggested here probably deserves a separate publication.