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**The Pontryagin maximum principle 50 years later.**
*(English.
Russian original)*
Zbl 1122.49001

Maksimov, V. I. (ed.), Dynamical systems: modeling, optimization, and control. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica, Pleiades Publishing/distrib. by Springer. Proc. Steklov Inst. Math. 2006, Suppl. 1, S4-S12 (2006); translation from Tr. Inst. Mat. Mekh. 12, No. 1, 6-14 (2006).

From the introduction: Exactly 50 years ago the Pontryagin maximum principle was formulated by V. G. Boltyanskii, R. V. Gamkrelidze and L. S. Pontryagin [Dokl. Akad. Nauk SSSR 110, No. 1, 7–10 (1956; Zbl 0071.18203)], which heralded the emergence of the mathematical theory of optimal control. The elapsed half-century witnessed a successful development of the theory by many researchers in the field; a prominent position certainly should be credited to the scientist to whom this volume is dedicated. We thought it therefore appropriate to present here our vision of the maximum principle and of some of its consequences as we see them today.

Formulation in the mid-1950s of an optimal control problem 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 a new type of “extremality conditions”. It was also clear that the real reason for the difficulties 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 time-optimal problems for linear second-order differential equations with a closed interval on the real line as an admissible set of control seemed to be completely nonamenable, the most general control problem with an open set for the admissible values of the control parameter was trivially reduced to the classical Lagrange problem, hence the differential equations of the extremals for the problem were readily at hand.

The maximum principle was a mathematical answer to the challenge posed by modern technologies and was fully endorsed by engineers right after its publication as an adequate solution to the problem, though, in the beginning, mathematical communities involved in related fields were quite reluctant, with very few exceptions, to accept the result as something completely new.

For the entire collection see [Zbl 1116.37003].

Formulation in the mid-1950s of an optimal control problem 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 a new type of “extremality conditions”. It was also clear that the real reason for the difficulties 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 time-optimal problems for linear second-order differential equations with a closed interval on the real line as an admissible set of control seemed to be completely nonamenable, the most general control problem with an open set for the admissible values of the control parameter was trivially reduced to the classical Lagrange problem, hence the differential equations of the extremals for the problem were readily at hand.

The maximum principle was a mathematical answer to the challenge posed by modern technologies and was fully endorsed by engineers right after its publication as an adequate solution to the problem, though, in the beginning, mathematical communities involved in related fields were quite reluctant, with very few exceptions, to accept the result as something completely new.

For the entire collection see [Zbl 1116.37003].

### MSC:

49-03 | History of calculus of variations and optimal control |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

01A60 | History of mathematics in the 20th century |

49K15 | Optimality conditions for problems involving ordinary differential equations |

### Citations:

Zbl 0071.18203
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\textit{A. A. Agrachev} and \textit{R. V. Gamkrelidze}, in: Dynamical systems: modeling, optimization, and control. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica, Pleiades Publishing/distrib. by Springer. S4--S12 (2006; Zbl 1122.49001); translation from Tr. Inst. Mat. Mekh. 12, No. 1, 6--14 (2006)