Overview

Optimal control of a growth/consumption model

“…(The state trajectory x * of Theorem 1 is one such solution.) By a selection theorem, 5 there exists a control function u such that (x, u) is a process for (GC) t 0 ,x 0 . Now define t ′ :…”

“…These techniques involve construction a field of normal extremals by applying the Pontryagin maximum principle to a family of problems parameterized by the initial time and state and verifying that the resulting utility, as a function of the initial data, satisfies the Hamilton Jacobi equation in some guise. 5 Standard verification techniques are not directly applicable to the (GC) problem, owing to the presence of (non-Lipschitz) fractional singularity terms in the utility and dynamics. Our aim is to provide analytic tools for solving problems of this nature, based on verification functions that are locally Lipschitz continuous, but whose slope may be unbounded near the boundary of their domains, and illustrate their usefulness, by applying them to the (GC) problem.…”

L∞estimates on trajectories confined to a closed subset

“…We begin by recalling an important existence theorem with accompanying estimates, known as Filippov's Existence Theorem (see [1] or [21]), which is frequently invoked in our analysis.…”

“…By the Relaxation Theorem (which asserts the density, with respect to the L ∞ norm, of the set of F -trajectories with a fixed initial state in the set of co F -trajectories, with the same initial state; cf. [1] or [21]), there exists a sequence of F -trajectories x i (.) : [s i , t] → R n such that, for all integer i ≥ 2, we have x i (s i ) = x (s i ) and…”

“…Given a multifunction G : D ; R n and x ∈ D (where D is closed), we define the limit inferior (in the Kuratowski sense) of G at x to be (cf. [1] or [21])…”