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Carathéodory equivalence, Noether theorems, and Tonelli full-regularity in the calculus of variations and optimal control. (English) Zbl 1084.49022

Summary: We study, in a unified way, the following questions related to the properties of Pontryagin extremals for optimal control problems with unrestricted controls:
(i) How do the transformations defining the equivalence of two problems transform the extremals?
(ii) How does one obtain quantities which are conserved along any extremal?
(iii) How does one verify that the set of extremals includes the minimizers predicted by the existence theory?
These questions are closely related with
(i) the Carathéodory method which establishes a correspondence between the minimizing curves of equivalent problems;
(ii) the interrelation between the concept of invariance and the theory of optimality conditions in optimal control, which are the concern of the Noether theorems;
(iii) the regularity conditions for the minimizers and the work pioneered by Tonelli.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49J15 Existence theories for optimal control problems involving ordinary differential equations
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49N60 Regularity of solutions in optimal control
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