  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.
