  We study a linear quadratic optimal control problem with stochastic
coefficients and a terminal state constraint, which may be in force merely on a
set with positive, but not necessarily full probability. Under such a partial
terminal constraint, the usual approach via a coupled system of a backward
stochastic Riccati equation and a linear backward equation breaks down. As a
remedy, we introduce a family of auxiliary problems parametrized by the
supersolutions to this Riccati equation alone. The target functional of these
problems dominates the original constrained one and allows for an explicit
description of both the optimal control policy and the auxiliary problem's
value in terms of a suitably constructed optimal signal process. This suggests
that, for the minimal supersolution of the Riccati equation, the minimizers of
the auxiliary problem coincide with those of the original problem, a conjecture
that we see confirmed in all examples understood so far.
