  We consider the optimal control problem for a linear conditional
McKean-Vlasov equation with quadratic cost functional. The coefficients of the
system and the weigh-ting matrices in the cost functional are allowed to be
adapted processes with respect to the common noise filtration. Semi closed-loop
strategies are introduced, and following the dynamic programming approach in
[32], we solve the problem and characterize time-consistent optimal control by
means of a system of decoupled backward stochastic Riccati differential
equations. We present several financial applications with explicit solutions,
and revisit in particular optimal tracking problems with price impact, and the
conditional mean-variance portfolio selection in incomplete market model.
