A maximum-minimum principle for bang-bang systems

Abstract

Consider the problem of a rocket in vertical flight in a uniform gravitational field, the aerodynamic drag being neglected. The thrust is provided by several chemical rocket engines working in parallel that can be separated and dropped according to some optimal sequence in order to provide a maximum payload for a given total thrust at departure and a prescribed velocity gain.

The mathematical formulation provides the possibility of a continuous reduction in thrust, that is for the limiting case of an infinity of infinitesimal propulsion units. In this case it is known that, if the velocity performance is set high enough, the optimal sequence consists of a constant thrust arc during which no engines are dropped, followed by a continuous reduction in thrust that keeps the acceleration constant. There is however another type of extremal representing the separation from a finite amount of thrust. The real technical problem involves only this type of extremal and the constant thrust extremal. The optimization problem is then of the bang-bang type, the continuous acceleration type of extremal representing a “chattering” of the control.

It is remarkable that optimal bang-bang solutions, each corresponding to a prescribed number of engine separations, are found by applying a minimum principle for the Hamiltonian (instead of the usual maximum principle) during a portion of the trajectory. More precisely the optimal bang-bang trajectories imply the use of the maximum principle up to the first reversal in the sign of the switching function, then of the minimum principle with a finite number of sign reversals, then of the maximum principle again to the end. Eventually the first or last part (or both) are missing.

The optimality of such bang-bang solutions is established by the analysis of the second variation.