Existence of the Optimum in Shallow Lake Type Models with Hysteresis Effect

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.