Journal of Mathematical Sciences, Vol. 94, No. 3, 1999 PONTRYAGIN~S MAXIMUM PRINCIPLE IN OPTIMAL CONTROL THEORY A. V. Arutyunov UDC 517.977.52 Introduction Thirty-five years have elapsed since the publication of the monograph Mathematical Theory of Optimal Processes by L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko in 1961. It laid the foundation of the mathematical theory of optimal control and soon became a classical work. The Pon- tryagin maximum principle was first demonstrated in this monograph for the general optimal control problem described by a system of ordinary differential equations. Subsequently a large number of excellent textbooks, monographs, and papers [1-4, 15, 16, 17, 20, 21, 23, 26, 28, 30, 34, 37, 40] came out, all concerned with the mathematical theory of optimal control. But the Pontryagin maximum principle remains as before the central outcome of this theory. Nonetheless, certain aspects of the Pontryagin maximum principle have so far escaped the attention of researchers (for example, nondegeneracy conditions for problems with phase con- straints, problems with nonregular mixed constraints, application of the maximum principle for investigating particular classes of problems, etc.). In this paper, we shall study the Pontryagin maximum principle as applied to optimal control problems described by systems of ordinary differential equations under terminal, phase, and mixed constraints as well as geometric constraints imposed on the control. We shall describe two methods of proving the maximum principle. One of them is based on the penalty-function method and is applied in studying linear-convex problems and problems studied within the class of relaxed controls. The second is a combination of the finite~dimensional approximation technique and the penalty-function apparatus: it is used in studying the general problems. Central attention is paid to the nondegeneracy conditions of the maximum principle for problems with phase constraints. We have intentionally not touched upon the historical aspects leading to the discovery of the maximum principle. This question contains many disputable and obscure points, and different authors maintain different, often conflicting, viewpoints. Furthermore, the same author brings forward diametrically opposite arguments at different times (see, for example, [15, pp. 6-8; 44]). The list of publications on this topic given at the end is not complete; they can found to some extent in the review [18]. Acknowledgments. This work is partly supported by the Russian Foundation for Basic Research, project No. 96-01-00920. 1. Statement of the Problem We shall study the following general optimal control problem: = f(x, u,t), t e [tl,t~], tl <_ t2, = e u (t) R(z, ul, t) < O, cy,(~, t) <_ o, (1.1) (1.2) (1.3) (1.4) Translated from Itogi Nauki i Tekhaiki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 42, Dinamicheskie Sistemy-6, 1997. 1072-3374/99/9403-1311$22.00 9 Kluwer Academic/Plenum Publishers 1311