The Pontryagin maximum principle is apllied to the Obukhov continuous hydrodynamic shell model in one space dimension restricted only to three shells. It is shown that many information can be extracted by simply reasoning with the conditions of the maximum principle, and, on the other hand, that this effort can become cumbersome. Thus, there are limits to what can be achieved in an efficient way in terms of computing the solution to the optimal control problem. The numerical computation of solutions to this optimal control problem based on a 'brute-force' type of shooting method reveals the existence of switching points in the control. © 2018 IEEE.