Variational Calculus and Optimal Control pp 282-337 | Cite as

# Sufficient Conditions for a Minimum

Chapter

## Abstract

As we have noted repeatedly, the equations of Euler-Lagrange are necessary but not sufficient to characterize a minimum value for the integral function on set such as D = {

$$ F(Y) = \int_a^b {f(x,Y(x),Y'(x))dx = } \int_a^b {f[Y(x)]dx} $$

*Y*∈*C*^{1}([*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*Y*_{0}must necessarily satisfy the Weierstrass condition ℰ(*x,**Y*_{0}(*x*),\( {{Y'}_0} \)(*x),**W*) ≥ 0, ∀*W*∈ ℝ^{d},*x*∈ [*a,b*], where ℰ(*x, Y, Z, W*)$$ \mathop = \limits^{def} $$

*f*(*x, Y, W*) -*f*(*x, Y, Z*)-*f*_{z}(*x, Y, Z*)⋅(*W - Z*), (1) and this is recognized as a convexity statement for*f*(x, Y,*Z*) along a trajectory in ℝ^{2d+1}defined by*Y*_{0}.## Keywords

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