ABSTRACT
We turn now to the problem of obtaining necessary conditions for local minimizers. As in the previous chapters, letX = PWS(t0, t1) denote the space of all real-valued piecewise smooth functions defined on [t0, t1]. For each PWS function x : [t0, t1] → R, define the functional J : X → R (a “function of a function”) by
J(x(·)) = t1∫
f (s, x(s), x˙ (s)) ds. (4.1)
Assume that the points [t0 x0] T and [t1 x1]
T are given and define the subset Θ of PWS(t0, t1) by
Θ = {x(·) ∈ PWS(t0, t1) : x (t0) = x0, x (t1) = x1} . (4.2)
Observe that J : X → R is a real valued function on X. The Simplest Problem in the Calculus of Variations
(the fixed endpoint problem) is the problem of minimizing J(·) on
such that
J(x∗(·)) = t1∫
f(s, x∗(s), x˙∗ (s))ds ≤ J(x(·))
=
f(s, x(s), x˙ (s))ds,
for all x (·) ∈ Θ. The basic goal in this chapter is the development of the classical
necessary conditions for local minimizers. We begin with a review of the basic definitions.
