On the optimality of the boundary control for the non-linear control system with controls appearing linearly†

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Abstract
In this paper, the optimization of a non-linear control system is studied, where both the system and the performance index are linear in the control variables. This study is concerned mainly with the optimality of the totally singular control of the maximum principle. The problem is reduced to the study of & non-regular case of the classical calculus of variations, for which anew criterion is found concerning the property of its extremals By means of this new criterion, it is proved that the totally singular control cannot be optimal for the higher dimensional case. In other words, the optimal control should be the boundary control

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