Necessary conditions for an extremum

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Maxima and minima

Convex analysis and extremal problems, Nauka, Moscow in Russian. Google Scholar. Aubin J. On inverse function theorems for self-valued maps. Necessary conditions of extremum, M. Neustadt L.

A necessary condition for extremum of a functional

Optimization: A Theory of Necessary Conditions. See also Weierstrass—Erdmann corner conditions , for necessary conditions at a corner of an extremal. Log in. Namespaces Page Discussion. Views View View source History. Jump to: navigation , search. Weierstrass' necessary condition: For the functional to attain a strong local minimum on the extremal , it is necessary that the inequality where is the Weierstrass -function , be satisfied for all , , and all.

This condition may be expressed in terms of the function cf. Weierstrass' sufficient condition: For the functional to attain a strong local minimum on the vector function , it is sufficient that there exists a vector-valued field slope function geodesic slope cf.

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Hilbert invariant integral in a neighbourhood of the curve , for which and for all and any vector. References [1] M.

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We arrive at this:. We continue as we did in section 2. In the end, we arrive at the new Jacobi equation,. We then solve the corresponding Riccati equation and employ it in 8 to obtain.

Consequently, we obtain this inequality:. We summarize what we found below.

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Sufficient conditions for extrema - proper field, central feild, extremal field in hindi.

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    It can be turned into the second order linear ODE below via the substitution which is called the Jacobi equation for J. It is easy to check that on [a, b], and, of course, u solves 6.

    Weierstrass conditions (for a variational extremum)

    Then we have which holds independently of how our smooth family of extremals was constructed. We arrive at this: We continue as we did in section 2. Content Details. Share with a friend.