  This paper treats a distributed optimal control problem for a tumor growth
model of viscous Cahn--Hilliard type. The evolution of the tumor fraction is
governed by a thermodynamic force induced by a double-well potential of
logarithmic type. The cost functional contains a nondifferentiable term like
the $L^1$-norm in order to enhance the occurrence of sparsity effects in the
optimal control, i.e., of subdomains of the space-time cylinder where the
controls vanish. In the context of cancer therapies, sparsity is very important
in order that the patient is not exposed to unnecessary intensive medical
treatment. In this work, we focus on the derivation of second-order sufficient
optimality conditions for the optimal control problem. While in previous works
on the system under investigation such conditions have been established for the
case without sparsity, the case with sparsity has not been treated before.
