Optimal control of non-linear systems via quadratic criteria with bounded controls

The paper suggests a method of developing an optimal control of a single class of nonlinear systems via a quadratic criterion with a bounded type of inequality for the controls. This method is a further derivation from the method of successive approximations suggested in the earlier works of the group of authors, to which the authors of the current paper belong. By modifying the given method, the researchers have managed to state the existence of an optimal control of the problem in question and to synthesize the actual optimal control. The crucial issue of optimal control development is the problem of convergence of the method of successive approximations. Besides, the suggested scheme leads to a computational procedure that implies constructing a solution for a two-point boundary value problem. As known, it causes certain computational difficulties. In order to avoid those difficulties, the paper includes a modified scheme that converges and provides control which is close to an optimal one. It is demonstrated that the developed scheme reduces the initial problem to a sequence of Cauchy problems that can be easily solved using the simplest methods of numerical analysis. To illustrate the suggested method, the paper shows the results of a computational experiment on developing optimal control for a controlled system described with Van der Pol equation. In this case, it turned out that it is the modified scheme that gives the optimal control.