# Regular extremal

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non-singular extremal

An extremal at all points of which the following condition holds: (1)

where is the integrand appearing in a functional which is to be minimized. Like any extremal, a regular extremal is, by definition, a smooth solution of the Euler equation The points of an extremal at which (1) holds are called regular points. It is known that at every regular point, an extremal has a continuous second-order derivative . On a regular extremal, the second-order derivative is continuous. For a regular extremal the Euler equation can be written in the following form (that is, solved for the highest derivative): The regularity property (1) is directly connected with the necessary Legendre condition (in the strong form), according to which at all points of the extremal the following inequality holds: Essential use is made of regularity when proving that an extremal can be included in a field of extremals surrounding it. If condition (1) is violated at even one point, the extremal cannot always be included in a field. This condition for including the extremal in a field is one of the sufficient conditions for being an extremal.

The above definition of a regular extremal is given for the simplest problem of the calculus of variations, which concerns functionals depending on one unknown function. For functionals depending on unknown functions, a regular extremal is an extremal for which at every point the -th order determinant (2)

In some general problems of the calculus of variations on a conditional extremum (see Bolza problem), a regular extremal is defined in a similar way, except that in (2) instead of one must put the Lagrange function .

An extremal for which the regularity condition ((1) or (2)) is violated at every point of some section is called a singular extremal, and the section is called a section of singular regime. For singular regimes there are necessary conditions supplementing the known classical necessary conditions for an extremum (see Optimal singular regime).

How to Cite This Entry:
Regular extremal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_extremal&oldid=17757
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article