Existence of the optimum for Shallow Lake type models

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 $L_{loc}^{1}\left(0,+\infty\right)$. Any direct proof seems to be missing in the literature. Dealing with admissible controls that can be unbounded (even locally) is necessary in order to represent properly the concrete optimization problem; on the other hand, the non-compactness of the control space together with the infinite horizon setting prevents from having good \emph{a priori} estimates - and this makes the existence problem considerably harder. We present 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) two uniform localization lemmas providing, given $T\geq1$ and a maximizing sequence of controls, another sequence of controls which is bounded in $L^{\infty}\left(\left[0,T\right]\right)$ and still maximizing. ii) A special diagonal procedure dealing with sequences which are not extracted one from the other. iii) A 'standard' diagonal procedure. The optimum results to be locally bounded by construction. Keywords: Optimal control, infinite horizon, non compact control space, uniform localization, convex-concave dynamics, logarithmic utility.