An iterative procedure for the computation of fixed-time fuel-optimal controls

The Maximum Principle of Pontryagin is used to develop an iterative procedure for computing the fuel-optimal control which steers the state of a linear constant plant from some initial state to a target hypersphere, centered at the origin of the state space, in fixed time. The necessary conditions of the Maximum Principle are used to define a two point boundary value problem and this problem is in turn reduced to that of inverting a function which maps the final boundary conditions into the state space. It is shown that the mapping is continuously differentiable to all orders almost everywhere and that a corresponding linearized map has an inverse. The existence of the linearized inverse is used to establish an iterative procedure which is essentially a modified Newton\´s method. It is shown that the procedure converges rapidly (using a known "step-size") when the current guess is "close" and bounds are obtained on the behavior of the error sequence for the case when the guess is not close. Finally, some experimental results are reported which illustrate the usefulness of the technique.