Functional Equations in the Theory of Dynamic Programming--VI, A Direct Convergence Proof

(b) Ri (x, y) < O. i = 1, 2, **, K. Here x and c are N-dimensional vectors, while y is an M-dimensional vector function. The maximization is over y. Since solutions of problems of this type are only in rare instances obtainable in explicit form, recourse must be had to some type of approximate solution if we are interested in numerical results. A method going back to Euler consists of approximating to the integral in (1) by a sum of the form (3) J1 (y) = EZ=oAF (x (k), y (k) ), and to the relations in (2) by the relations (a) x (k + 1) = x (k) + AG (x (k), y (k) ), x (O) = c, k = 0, 1, , n, (b) Ri (x (k), y (k) ) < 0O where (a) A = T/n, (5) (b) x (k) _ x (kT/n), y (k) _ y (kT/n). Under various assumptions concerning the functions F, G and R, it can be shown that (6) limbos Maxly} J1 (y) = MaxJ (y), or (7) limo. Maxiy) J1 (y) = Supy J (y). Here the maximization on the left-hand side is over all (n + 1) -dimensional sequences y (O), y (l), , y (N). 215