Optimal trajectories of infinite-horizon deterministic control systems

Abstract

We study the infinite-horizon deterministic control problem of minimizing∫ ^T₀ L(z, ż) dt, T→∞, whereL(z, ·) is convex inż for fixedz but not necessarily jointly convex in (z, ż). We prove the existence of a solution to the infinite-horizon Bellman equation and use it to define a differential inclusion, which reduces in certain cases to an ordinary differential equation. We discuss cases where solutions of this differential inclusion (equation) provide optimal solutions (in the overtaking optimality sense) to the optimization problem.

A quantity of special interest is the minimal long-run average-cost growth rate. We compute it explicitly and show that it is equal to min _x L(x, 0) in the following two cases: one is the scalar casen = 1 and the other is' when the integrand is in a separated form$l(x) + g(\dot x)$