  In this work, we investigate a distributed optimal control problem for an
extended phase field system of Cahn--Hilliard type which physical context is
that of tumor growth dynamics. In a previous contribution, the author has
already studied the corresponding problem for the logarithmic potential. Here,
we try to extend the analysis by taking into account a non-smooth singular
nonlinearity, namely the double obstacle potential. Due to its non-smoothness
behavior, the standard procedure to characterize the necessary conditions for
the optimality cannot be performed. Therefore, we follow a different strategy
which in the literature is known as the "deep quench" approach in order to
obtain some optimality conditions that have to be interpreted in a more general
framework. We establish the existence of optimal controls and some first-order
optimality conditions for the system are derived by employing suitable
approximation schemes.
