ABSTRACT
In what follows, we consider calculation of variations in the deterministic framework to introduce the main ideas of dynamic programming.
6.1.1.1 General problem of the calculus of variations
Very often, this problem takes the following form:
- Let [0, T ] be the time period. - Let U be a function w.r.t. three variables (t, x, y) with real values, assumed
to be continuously-differentiable on [0, T ]× Rd × Rd. - We search for a function x(.), continuously-differentiable on [0, T ] × Rd
solution of the optimization problem (P):
max U(x(.)) = ∫ T 0
U(t, x(t), · x(t))dt,
under : x(0) = x0 and x(T ) = xT , (6.1)
where · x(t) denotes the derivative of x(.) w.r.t. the current time, and x0 and
xT are given. In what follows, the general results are illustrated by some basic examples.
6.1.1.2 Standard consumption-saving problems
