Existence of Solutions to Shallow Lake Type Optimal Control Problems

We consider the optimal control problem associated with a general version of the well known Shallow Lake model, and we prove the existence of an optimum in the class of all positive, locally integrable functions with finite discounted integral. Any direct proof seems to be missing in the literature. In order to represent properly the concrete optimization problem, locally unbounded controls must be admitted. In an infinite horizon setting, the non-compactness of the control space can make the existence problem quite hard to treat, because of the lack of good a priori estimates. To face the technical difficulties, we develop an original method, which is in a way opposite to the classical control theoretic approach used to solve finite horizon Mayer or Bolza problems. Synthetically, our method is based on the following scheme: (i) a couple of uniform localization lemmas providing, for any given finite time interval, a maximizing sequence of controls, which is uniformly essentially bounded in that interval; (ii) a special diagonal procedure dealing with sequences not extracted one from the other; (iii) a “standard” diagonal procedure. Further, this approach does not require any concavity assumption on the state equation. The optimum turns out to be locally bounded by construction.