An iterative procedure for computing time-optimal controls

An iterative procedure is derived for computing time controls for regulator systems described by vector differential equations of the form \dot{x}=Ax+bu where A and b are constant |u| \leq 1 . The iterative computational procedure is based on the parameterization of the minimal time control in terms of the initial conditions of the system's adjoint differential equation. A function is derived relating the initial state of the system to the adjoint system's initial conditions and to the time required by the optimal control to drive that state to zero. Mapping properties of this function, on which the control computation depends, are discussed. To demonstrate the results of this computational procedure, an example is included in which a time-optimal control for a fourth order system with two pairs of complex roots is computed.