# Difference between revisions of "Regular extremal"

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''non-singular extremal'' | ''non-singular extremal'' | ||

− | An [[Extremal|extremal]] | + | An [[Extremal|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 | + | 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|Euler equation]] | which is to be minimized. Like any extremal, a regular extremal is, by definition, a smooth solution of the [[Euler equation|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 | + | 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): | 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|Legendre condition]] (in the strong form), according to which at all points of the extremal the following inequality holds: | The regularity property (1) is directly connected with the necessary [[Legendre condition|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 | + | 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 | + | 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 | + | 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|Bolza problem]]), a regular extremal is defined in a similar way, except that in (2) instead of | + | In some general problems of the calculus of variations on a conditional extremum (see [[Bolza problem|Bolza problem]]), a regular extremal is defined in a similar way, except that in (2) instead of $ F $ |

+ | one must put the [[Lagrange function|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|Optimal singular regime]]). | 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|Optimal singular regime]]). | ||

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====References==== | ====References==== | ||

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)</TD></TR></table> | ||

− | |||

− | |||

====Comments==== | ====Comments==== | ||

− | A family of curves in a domain | + | 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. [[#References|[a2]]] and [[Extremal field|Extremal field]]. | ||

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Cesari, "Optimization - Theory and applications" , Springer (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Yu.P. Petrov, "Variational methods in optimum control theory" , Acad. Press (1968) pp. Chapt. IV (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Cesari, "Optimization - Theory and applications" , Springer (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Yu.P. Petrov, "Variational methods in optimum control theory" , Acad. Press (1968) pp. Chapt. IV (Translated from Russian)</TD></TR></table> |

## Latest revision as of 08:10, 6 June 2020

*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) |

**How to Cite This Entry:**

Regular extremal.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Regular_extremal&oldid=48480