Geodesic
Suboptimal Regimes in the Fuller Problem
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 226-229.

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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. 
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O. E. Maikova. Suboptimal Regimes in the Fuller Problem. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 226-229. http://geodesic.mathdoc.fr/item/TM_2002_236_a23/

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