Generalized Solutions to Continuous-Time Allocation Processes
针对连续时间资源分配问题，在可能不存在最优解的情况下，提出并刻画了广义解的概念，适用于初始资源禀赋的跨期消费效用最大化问题。
Here q E R n, the n-dimensional Euclidean space, x (t) is a R n-valued function, and u (x, t) Rn x [0, oo) -> R, is a nonnegative real function normalized such that u (0, t) = 0 for every t. Inequalities between vectors are understood coordinatewise. We shall be concerned mainly with the case where (L) might not have optimal solutions. We shall develop, characterize, and apply the notion of a generalized solution of this optimization problem. This variational problem arises in several contexts. See Aumann and Shapley [4, Chapter VI], Yaari [11], and the references therein. We shall adopt here the following interpretation. The coordinates of the vector q represent initial endowments of several exhaustible resources. The function x = x (t) is the rate at which the resources al-e consumed. The constraints mean that there is no re-filling (x (t) ? 0) and that the overall consumption is bounded by q. If the rate x (t) is consumed at t, it contributes the rate u(x(t), t) to the total utility. The latter is represented as the integral which has to be maximized. It is our economic interpretation that has led us to the choice of [0, xo) as the domain of integration. There is no mathematical significance in this choice. The analysis shows however that the behavior at t finite might be different from the behavior at t = oo; the difference is caused by the fact that a finite t has a neighborhood with finite Lebesgue measure, in contrast to t = 00, which has only neighborhoods of infinite Lebesgue measure. We shall refer to the role of this difference in the text.