Regular extremal

non-singular extremal

An extremal $ y ( x) $ at all points of which the following condition holds:

$$ \tag{1 } F _ {y ^ \prime y ^ \prime } ( x , y ( x) , y ^ \prime ( x) ) \neq 0 , $$

where $ F ( x , y , y ^ \prime ) $ is the integrand appearing in a functional

$$ J = \int\limits _ { x _ {1} } ^ { {x _ 2 } } F ( x , y , y ^ \prime ) d x $$

which is to be minimized. Like any extremal, a regular extremal is, by definition, a smooth solution of the Euler equation

$$ F _ {y} - \frac{d}{dx} F _ {y ^ \prime } = 0 . $$

The points of an extremal at which (1) holds are called regular points. It is known that at every regular point, an extremal $ y ( x) $ has a continuous second-order derivative $ y ^ {\prime\prime} ( x) $. On a regular extremal, the second-order derivative $ y ^ {\prime\prime} ( x) $ is continuous. For a regular extremal the Euler equation

$$ F _ {y} - F _ {y ^ \prime x } - F _ {y ^ \prime y } y ^ \prime - F _ {y ^ \prime y ^ \prime } y ^ {\prime\prime} = 0 $$

can be written in the following form (that is, solved for the highest derivative):

$$ y ^ {\prime\prime} = f ( x , y , y ^ \prime ) . $$

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:

$$ F _ {y ^ \prime y ^ \prime } ( x , y ( x) , y ^ \prime ( x) ) < 0 . $$

Essential use is made of regularity when proving that an extremal $ y ( x) $ 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 $ n $ unknown functions,

$$ J = \int\limits _ { x _ {1} } ^ { {x _ 2 } } F ( x , y _ {1} \dots y _ {n} , y _ {1} ^ \prime \dots y _ {n} ^ \prime ) d x , $$

a regular extremal is an extremal for which at every point the $ n $- th order determinant

$$ \tag{2 } | F _ {y _ {i} ^ \prime y _ {j} ^ \prime } | \neq 0 . $$

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 $ F $ one must put the Lagrange function $ L $.

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).

References

[1]	G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947)
[2]	M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)

Comments

A family of curves in a domain $ D $ is called a field of curves if for every point of $ D $ there is exactly one member of the family passing through it. For an account of the role of field theory in the calculus of variations and fields of extremals cf. [a2] and Extremal field.

References

[a1]	L. Cesari, "Optimization - Theory and applications" , Springer (1983)
[a2]	Yu.P. Petrov, "Variational methods in optimum control theory" , Acad. Press (1968) pp. Chapt. IV (Translated from Russian)