Suboptimal Regimes in the Fuller Problem

In certain optimal control problems, the optimization of a functional requires an infinite number of control switchings on a finite time interval. For such problems, a (suboptimal) control with a finite number of switchings is sought that possesses the following property: If, starting from a certain time moment, the optimal control is replaced by the suboptimal, then the loss in the value of the functional to be minimized is no greater than a given $\varepsilon>0$. The analysis is performed on the basis of the Fuller problem $\int _0^T x^2(t)dt \to \min$, where $\dot x=y$, $\dot y=u$, and $|u|\le 1$; as a suboptimal control, we take the optimal control from the time-optimal problem on the trajectories of the same system.