Time-optimal feedback control of a discontinuous dynamic object

The purpose of this paper is to present the solution of time-optimal problem of the controlled object u x f y y x + = = ) ( , ! ! , u ≤ 1 and motion resistance function f x x f x A x ( ) , ( ) = ≤ = − > 0 0 0 if if where 0 1 ≤ < A . In the formula of f x ( ) there exists a value A= b A = 2 2 − that plays an essential role. Namely, if A≤ b A then the time-optimal control process is typical. For the case A > b A two singular phenomena has been exhibited. If the first phenomenon is a case then, the first branch of the switching curve is formed classically by the time-optimal solution. None of the solution forms the second its branch. If the second singular phenomenon is a case then, there exists a set from which there starts two trajectories reaching the target in the same minimal time. Some suggestions as to practical applications have been given too.