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The “Pontryagin derivative” in optimal control. (English. Russian original) Zbl 1148.49015

Dokl. Math. 77, No. 3, 329-331 (2008); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 420, No. 1, 11-13 (2008).
Summary: Characteristic features of the maximum principle, such as its universality and simplicity of formulation, immediately reveal the logical structure of the principle consisting of two main interacting parts, namely, the native Hamiltonian format expressed as a Hamiltonian system with parameters that is invariantly connected to the optimal problem and the maximum condition, which “dynamically” eliminates the parameters in the course of solving the initial value problem for the Hamiltonian system as we proceed along the trajectory, thus providing the extremals of the problem.
Much has been said about the maximum condition since its discovery in 1956 and all the achievements in the field were mainly credited to it, whereas the Hamiltonian format of the maximum principle was always taken for granted and was never discussed seriously.
Meanwhile, the very possibility of formulating the maximum principle is intimately connected with its native Hamiltonian format and with the introduction of the control parameters contained in the definition of the optimal problem.
These starting steps were made by L. S. Pontryagin in 1956 on a completely “empty spot”, which led to the discovery of the maximum principle. It appeared as “Deus ex machina” and heralded the birth of a new mathematical discipline – the optimal control theory.

MSC:

49Kxx Optimality conditions
49-03 History of calculus of variations and optimal control
01A70 Biographies, obituaries, personalia, bibliographies
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