Pollution management models of Shallow Lake type identify a well-known class of infinite horizon optimal control problems, characterized by the absence of concavity in the state equation and of compactness in the control space. The seminal paper by Mäler et al. (2003) generated a consistent stream of literature over the years, even though existence of solutions to the optimization problem had for long remained an open question. In a recent paper by the author, an optimal policy has been proven to exist, for models describing an endogenous pollution dynamics that decreases globally with the total amount of pollution. The present paper is concerned with the complementary situation, the one in which, mathematically speaking, the velocity field changes its monotonicity with respect to the space variable. For Shallow Lake models, such property corresponds to an hysteresis phenomenon. We prove the existence of an optimum in the same class as in the last-mentioned article, under the assumption that the discount exponent in the objective functional is sufficiently big compared to the spatial derivative of the velocity field. From the methodological viewpoint, this goal requires a significant improvement of the technique introduced to solve the monotonic problem.

浅湖型污染管理模型确定了一类众所周知的无限水平最优控制问题，其特征是状态方程中没有凹度，控制空间中没有紧凑性。Mäler 等人的开创性论文。(2003) 多年来产生了一致的文献流，尽管优化问题的解决方案的存在长期以来一直是一个悬而未决的问题。在作者最近的一篇论文中，已证明存在最优策略，用于描述随污染总量在全球范围内减少的内源性污染动态的模型。本文关注的是互补情况，即从数学上讲，速度场相对于空间变量改变其单调性。对于浅湖模型，这种特性对应于滞后现象。在假设目标函数中的折扣指数与速度场的空间导数相比足够大的情况下，我们证明了与上一篇文章在同一类中存在最优解。从方法论的角度来看，这个目标需要对解决单调问题所引入的技术进行重大改进。