 
Summary: The Pontryagin Maximum Principle 50 years
later
A. A. Agrachev and R.V. Gamkrelidze
1 Introductory remarks
Exactly 50 years ago the Pontryagin Maximum Principle was formulated in
[1], which heralded the emergence of the mathematical theory of optimal
control. The elapsed halfacentury witnessed a successful development of
the theory by many researchers in the field, among whom a prominent po
sition certainly should be credited to the scientist to whom this volume is
dedicated. We thought therefore appropriate to present here our vision of the
maximum principle and of some of its consequences as we see them today.
Formulation in mid fifties of the past century of an optimal control prob
lem with a closed set of admissible values for the control parameter was a new
type of an extremal problem not amenable to the existing methods. From
the very beginning it was obvious that an adequate mathematical treatment
of the problem would prompt new type of ``extremality conditions''. It was
also clear that the real reason of the di#culties was not the generality of the
problem, but rather its specific character related to the fact that, in typical
problems, the admissible set of values of the control parameter was closed.
Whereas the simplest timeoptimal problems for linear second order di#er
