Geodesic
Riemann integrability of an optimal control in linear time-optimal problems
Izvestiya. Mathematics , Tome 25 (1985) no. 1, pp. 183-192.

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A linear time-optimal problem is considered. It is shown that the set of initial states for which a three-dimensional time-optimal problem at the origin, with constant coefficients, has no solution in the class of Riemann integrable controls can fill up a given ball to an arbitrary extent. For a large class of multidimensional systems it is shown that this set does not contain isolated points. Sufficient conditions for the existence of a Riemann integrable control are studied for problems with variable coefficients. Bibliography: 8 titles. 
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D. B. Silin. Riemann integrability of an optimal control in linear time-optimal problems. Izvestiya. Mathematics , Tome 25 (1985) no. 1, pp. 183-192. http://geodesic.mathdoc.fr/item/IM2_1985_25_1_a8/

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