On the program character of trajectory control over observations of a moving target

We consider a Gaussian linear-quadratic control problem with incomplete data for a stochastic object (target) from a moving observer with nonlinear dynamics in continuous time. Characteristic features of this formulation include the fact that the stochastic dynamics equation of the observation object and the actual observation equations depend on the vector of the current state of the observer. The observer's control objective is to choose a rational trajectory of its movement in the sense of minimizing the expectation of a certain payoff functional (cost). In problems of this kind, optimal control over observations (in the considered case, control over the trajectory of the observer) is usually sought in the class of program controls, i.e., as a function of time and initial conditions. A natural question arises: what will change if we extend the class of program controls by the observer to the class of positional ones, i.e., to the class of controls that depend not only on time and initial conditions but also on the realizations of observations and the trajectory of the observer at a given time moment. We consider two meaningful special cases of the general formulation of the problem where such an extension does not improve the control quality. We give examples and proof of the theorem.