Synthesis of control in the vicinity of the approximate solution of the problem with partially fixed the right end

The article is devoted to the method of improving control in the neighborhood of the trajectory of the current approximation. The method is based on the replacement of a control problem with a partially fixed the right end on a simple linear-quadratic problem. For a new problem, the solution can be found in explicit analytical form, and the resulting control applied to the original model. We consider the application of the algorithm for correcting a simple approximation of the sliding mode.