Investigation of some control models

Three simplified models of quarrying are considered. The first and third models are extremal, and the second is described in terms of a differential game. For each of the models, sufficient conditions on the parameters of the problem are obtained that guarantee a simple form of the optimal control. Pontrayagin's maximum principle and the theory of positional differential games, elaborated by N. N. Krasovskii and his school, are used.