On singular phenomena in certain time-optimal problem

The purpose of this paper is to present the solution of time-optimal problem of the controlled object the dynamics of which is given by: (x) over dot = y, (y) over dot = f(x) +u, where \u\ less than or equal to 1 and motion resistance function f(x) = 0 if x less than or equal to 0, f(x) = -A if x > 0 where 0 less than or equal to A < 1. That model describes dynamics of a very important class of industrial installations. As the time-optimal problem will be understood a transfer of the initial state z(0) = (x(0), y(0)) is an element of R-2 to the target state z(1) = (x(1), 0), x(1) greater than or equal to 0 in a minimum time t* < infinity. There has been shown that in the formula defining resistance function f(x) there exists a value A = A(b) = 2 - root 2 that plays an essential role in time-optimal structure formation. Namely, if A less than or equal to A(b) then the time-optimal control process is typical, analogous as in classical case (x) double over dot = u, \u\ less than or equal to 1, i.e. there exists a switching curve formed by the trajectories of time-optimal solutions reaching the target state and the time-optimal process is formed by at most one switching operation. For the case A > A(b) We Will examine two following singular phenomena. (a) If the target state z(1) = (0, 0) then there exists the switching curve, dividing the state plane into two sets, however only one its branch is formed by the time-optimal solution reaching the target z(1) = (0, 0) and generated by the control u = -1. None of solution forms the second branch of switching curve. It is formed by a state-locus depending on the value of A only. In dependency of the starting state z(0) the time-optimal control process is generated by bang-bang control with none, one or two switching operations. This is the first singular phenomenon, because any small decrease of the value A over A(b) requires to change the structure which would be able to generate the time-optimal process. (b) The paper shows, that if the target state z(1) (x(1), 0), x(1) > 0 then there exists a set of the starting states from which there start two trajectories reaching the target in the same minimum time. This is the second phenomenon. Finally, some suggestions as to practical applications have been given too.