Degenerate problem of stabilization of a specified trajectory

Consideration is given to the problem of stabilization of a specified (supporting) mode as a degenerate problem of minimization of the root-mean-square deviation from it where there are no deviations of control actions. This formulation makes it possible to completely use available limited control resources for stabilization. We propose a method of approximate optimal synthesis in the neighborhood of a turnpike manifold that is obtained as a solution in the form of optimal synthesis of linearly quadratic problem with unbounded linear control. The investigation is performed for a linearized discrete model of the controlled system; however, the obtained stabilizing control can be applied in the initial nonlinear system directly or after the correction by regular iterations.