The Pontryagin derivative in optimal control. (English. Russian original) Zbl 1208.49021

Gamkrelidze, R. V.

Proc. Steklov Inst. Math. 268, 87-92 (2010); translation from Trudy Mat. Inst. Steklova 268, 94-99 (2010).

Honouring Pontryagin’s memory Gamkrelidze discusses in this interesting paper the Hamiltonian format of Pontryagin’s maximum principle. After historical remarks he compares in the first section of his paper the principle with the Euler-Lagrange method and shows – using Legrendre transformation and regular problems – that the extremals in the Euler-Lagrange method can be represented as a Hamilton flow and how on the contrary the family of extremals is handled in the maximum principle. In the second section leaning on the master Hamiltonian and the corresponding vector field he discusses the Hamiltonian format of the maximum principle (in invariant formulation) using Lie derivative, duality with help of conjugation and differential-geometric invariants and makes a proposal in relation to the title of the paper.

Reviewer: Alfred Göpfert (Halle)

MSC:

49K15	Optimality conditions for problems involving ordinary differential equations
34H05	Control problems involving ordinary differential equations
49S05	Variational principles of physics
49N60	Regularity of solutions in optimal control

Keywords:

Pontryagin’s maximum principle; Hamiltonian systems; duality; conjugation; Lie derivative