Problems of Optimization—A General View

Abstract

Here we are concerned with minima and maxima of functionals of the form

$$ I\left[ x \right] = \int_{{{{t}_{1}}}}^{{{{t}_{2}}}} {{{f}_{0}}(t,x(t),x'(t))dt,(')} = d/dt, $$

(1.1.1)

where we think of I[x] as dependent on an n-vector continuous function x(t) = (x¹, ... ,xⁿ), t ₁ ≤t ≤ t ₂, or continuous curve of the form C:x = x(t), t ₁ ≤ t ≤ t ₂, in R ⁿ⁺¹ ,in a suitable class. Actually the subject of our inquiry will go much farther than the mere analysis of minima and maxima of functionals.