On a class of inverse control problems

The paper considers the problem of describing the reachable set for a nonlinear control system under the condition that the values of the integral functional of control and the trajectory of the system are bounded from above by a given number. This problem can be considered as the inverse problem for the problem of optimal control. It is proved that any admissible control that steers the trajectory of the system to the boundary of reachable set is a local solution of some optimal control problem with a given integral functional (of the primal problem) under the condition of controllability of the linearized system.