# Study of a One-Dimensional Optimal Control Problem with a Purely State-Dependent Cost

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## Abstract

A one-dimensional optimal control problem with a state-dependent cost and a unimodular integrand is considered. It is shown that, under some standard assumptions, this problem can be solved without using the Pontryagin maximum principle, by simple methods of the classical analysis, basing on the Tchyaplygin comparison theorem. However, in some modifications of the problem, the usage of Pontryagin’s maximum principle is preferable. The optimal synthesis for the problem and for its modifications is obtained.

## Keywords

Optimal control One-dimensional state Unimodular integrand State-dependent cost Tchyaplygin comparison theorem Pontryagin maximum principle## Mathematics Subject Classification

49K21 49K15 90C30## Notes

### Acknowledgments

This research was partially supported by the Russian Foundation for Basic Research under grants 14-01-00784 and 16-01-00585.

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