Sufficient Conditions for a Minimum

  • John L. Troutman
Part of the Undergraduate Texts in Mathematics book series (UTM)


As we have noted repeatedly, the equations of Euler-Lagrange are necessary but not sufficient to characterize a minimum value for the integral function
$$ F(Y) = \int_a^b {f(x,Y(x),Y'(x))dx = } \int_a^b {f[Y(x)]dx} $$
on set such as D = {YC1([a,b])d: Y(a) = A, Y(b) = B}, since they are only conditions for the stationary of F. However, in the presence of [strong] convexity of f(x,f) these conditions do characterize [unique] minimization. [Cf.§3.2, Problem 3.33 et seq.] Not all such functions are convex, but we have also seen in §7.6 that a minimizing function Y0 must necessarily satisfy the Weierstrass condition ℰ(x, Y0(x),\( {{Y'}_0} \)(x),W) ≥ 0, ∀ W ∈ ℝd, x ∈ [a,b], where ℰ(x, Y, Z, W)
$$ \mathop = \limits^{def} $$
f(x, Y, W) - f(x, Y, Z)- fz(x, Y, Z)⋅(W - Z), (1) and this is recognized as a convexity statement for f(x, Y, Z) along a trajectory in ℝ2d+1 defined by Y0.


Stationary Function Central Field Transversal Condition Stationary Trajectory Isoperimetric Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • John L. Troutman
    • 1
  1. 1.Department of MathematicsSyracuse UniversitySyracuseUSA

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