Euler-Lagrange and Hamiltonian formalisms in dynamic optimization

Author:
Alexander Ioffe

Journal:
Trans. Amer. Math. Soc. **349** (1997), 2871-2900

MSC (1991):
Primary 49K24, 49K15; Secondary 34A60, 34H05

DOI:
https://doi.org/10.1090/S0002-9947-97-01795-9

MathSciNet review:
1389779

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Abstract: We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler-Lagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler-Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set-valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler-Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one.

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Additional Information

**Alexander Ioffe**

Affiliation:
Department of Mathematics, The Technion, Haifa 32000, Israel

Email:
ioffe@math.technion.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-97-01795-9

Keywords:
Optimal control,
calculus of variations,
differential inclusion,
Lagrangian,
Hamiltonian,
maximum principle,
nonsmooth analysis,
approximate subdifferential

Received by editor(s):
January 23, 1995

Received by editor(s) in revised form:
January 17, 1996

Additional Notes:
The research was supported by the USA–Israel BSF grant 90–00455, by the Fund of Promotion of Science at the Technion grant 100-954 and in later stages, by the NSF grant DMS 9404128

Article copyright:
© Copyright 1997
American Mathematical Society