Liapounov's Theorem on the Range of a I ctor Measure and Pontryagin's Maximum Principle

In this paper we study the necessary condition for the optimal control of a non-linear dynamical system. We assume that the system under consideration is described by a canonical system of ordinary differential equations for the state variables. The differential equations depend also on certain parameters called control variables. T h e problem is to choose the value of these control variables, which can be timedependent, in order to satisfy given initial and end-point conditions and to maximize a given functional. When there are no restrictions on the values which can be taken by the control variables, the problem is relatively simple and equivalent to the Problem 0f BOLZA of the calculus of variations. Such a situation, however, is very unlikely to occur in practice because of the physical limitations of the control devices. The best known theory of the general case is given by the Maximum Principle of PONTRYAGIN [1]: an extension of the multiplier rule and of the Hamiltonian approach to the calculus of variations. In this paper we give a new derivation of this Maximum Principle. This derivation is characterized by the three following elements: (t) the introduction of a new and more restrictive concept of optimal solution which greatly simplifies the subsequent mathematical t reatment; (2) a particular change of variables suggested by the concept of "set of possible events" and by an extension of HUVGr.~S' Principle [3]; (3). the application of a theorem of LxAPou~ov on the convexity of the range of a vector measure [2]. A great variety of problems can be put in the particular form considered in this paper. For the appropriate transformations the reader is referred to the paper by PONTRVAGtN and his associates [1], where he will also find all background material for the present study.