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)

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.