On the fragility of the basis on the Hamilton–Jacobi–Bellman equation in economic dynamics
通过一个最优增长模型的例子，展示了汉密尔顿-雅可比-贝尔曼方程存在无穷多解但值函数不满足该方程的现象，并给出了值函数成为唯一解的条件。
In this paper, we provide an example of the optimal growth model in which there exist infinitely many solutions to the Hamilton–Jacobi–Bellman equation but the value function does not satisfy this equation. We consider the cause of this phenomenon, and find that the lack of a solution to the original problem is crucial. We show that under several conditions, there exists a solution to the original problem if and only if the value function solves the Hamilton–Jacobi–Bellman equation. Moreover, in this case, the value function is the unique nondecreasing concave solution to the Hamilton–Jacobi–Bellman equation. We also show that without our conditions, this uniqueness result does not hold.