Minimizing sequences for a linear-quadratic control problem with three-tempo variables under weak nonlinear perturbations

The paper deals with the construction of minimizing sequences for the problem of minimizing a weakly nonlinearly perturbed quadratic performance index on trajectories of a weakly nonlinear system with threetempo state variables. For this purpose, the so-called direct scheme for constructing an asymptotic solution is used, which consists in immediate substituting the postulated asymptotic expansion of the solution into the problem conditions and constructing a series of optimal control problems (in the case under consideration, linear-quadratic ones), the solutions of which are terms of the asymptotic expansion of the solution of the original nonlinear control problem. An estimate is obtained for the proximity of the optimal trajectory to the trajectory of the equation of state when some asymptotic approximation to the optimal control is used as a control. An example is given that illustrates in detail the proposed scheme for constructing minimizing sequences.