The article discusses a number of problems of space cybernetics connected with the optimum process control of informational interaction of a spacecraft with the surface of the Earth. A spacecraft is regarded as an informational active mobile plant, i.e. as a complicated mobile system supplied with necessary devices for the realization of informational interaction with the enclosing physical medium and the corresponding necessary onboard resource. It is shown that these problems are reduced to the problems of optimum programmed control by some special differential dynamic system in a Hilbert space of conditions. For a solution of the specified problems, the paper uses L.S. Pontrjagin's expanded maximum principle and the general concept of Lagrange.
In the article the mathematical model of informational interaction of the spacecraft with a surface of the Earth is considered. In a basis of construction of model lies proposed of the author the concept of active movable object as the complicated movable system intended for informational, energy or material interaction with an ambient physical environment or with other similar systems. It is shown that the corresponding model can be presented in the form of Fredholm integral operator mapping set of elements of a Hilbert space of controls (a class of admissible control actions) in a Hilbert space of information states. Investigated the properties of this operator and the corresponding reachability sets in space of information states. Considered a simplified variant of the proposed mathematical model - for interaction with the discrete environment (the isolated sources of information).
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