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The paper presents theoretical results obtained by the authors for solving problems of synthesis of optimal control strategies for quasilinear dynamical stochastic systems of diffusion type with information constraints. Constraints are expressed in the fact that each component of the control strategy vector depends on its predetermined set of precisely measured components of the state vector. These results are used to solve the problem of calming the oscillations of a satellite with one elastic rod in a near-earth orbit. As a bar, a radio antenna or a beam of gravitational stabilization can act. In the problem of the satellite, various options of awareness are examined, according to which it is possible to judge what dynamic characteristics are worth measuring, and from which measurements one can refuse.