AN EXTENSION OF PONTRYAGIN'S MAXIMUM PRINCIPLE.

The problem of optimum control of systems whose control signals belong to a prespecified class of functions is studied. The result is formulated in a maximum principle of the Pontryagin type. The extended maximum principle states that a necessary condition for the optimum control is that the first variation of the time integral of the Hamiltonian is zero if the control function can vary in two opposite directions, is non-positive if the control function can vary in only one direction. The conditions imposed on the admissible controls are i The control signals may be restricted to a prespecified class of functions which are piecewise continuous with a finite number of points of discontinuity, and which are continuous on the left at every instant under consideration has no isolated points on the time curve. ii The admissible class of controls admits infinitesimal variation. iii The control signals satisfy some boundedness constraints. The extended maximum principle is proved for the three classes of systems Class 1 The system is linear with respect to the control variables. Class 2 The control function of the system is a scalar or the finite variations of all components of the control vector occur in precisely the same intervals. Class 3 The points of discontinuity of the control function are not subject to variation.