  In this paper, we consider a class of optimal control problems for a
one-dimensional time-discrete constrained quasilinear diffusion state-systems
of singular Allen--Cahn types and its regularized approximating problems. We
note that the control parameter for each system is given by physical
temperature. The principal part of this paper is started with the verification
of a Key-Theorem dealing with the decompositions of the subdifferentials of the
governing convex energies of the state-systems. On this basis, we will prove
five Main Theorems, concerned with: the solvability and precise regularity
results of state-systems; the continuous-dependence of the solutions to
state-systems including convergences in spatially $C^1$-topologies; the
existence and parameter-dependence of optimal controls; the necessary
optimality conditions for approximate optimal controls; precise
characterizations of the approximating limit of the optimality conditions.
