  We consider the conditional control problem introduced by P.L. Lions in his
lectures at the Coll\`ege de France in November 2016. In his lectures, Lions
emphasized some of the major differences with the analysis of classical
stochastic optimal control problems, and in so doing, raised the question of
the possible differences between the value functions resulting from
optimization over the class of Markovian controls as opposed to the general
family of open loop controls. The goal of the paper is to elucidate this
quandary and provide elements of response to Lions' original conjecture. First,
we justify the mathematical formulation of the conditional control problem by
the description of practical model from evolutionary biology. Next, we relax
the original formulation by the introduction of \emph{soft} as opposed to hard
killing, and using a \emph{mimicking} argument, we reduce the open loop
optimization problem to an optimization over a specific class of feedback
controls. After proving existence of optimal feedback control functions, we
prove a superposition principle allowing us to recast the original stochastic
control problems as deterministic control problems for dynamical systems of
probability Gibbs measures. Next, we characterize the solutions by
forward-backward systems of coupled non-linear Partial Differential Equations
(PDEs) very much in the spirit of the Mean Field Game (MFG) systems. From
there, we identify a common optimizer, proving the conjecture of equality of
the value functions. Finally we illustrate the results by convincing numerical
experiments.
