  This paper deals with some reachability issues for piecewise linear switched
systems with time-dependent coefficients and multiplicative noise. Namely, it
aims at characterizing data that are almost reachable at some fixed time T > 0
(belong to the closure of the reachable set in a suitable L 2-sense). From a
mathematical point of view, this provides the missing link between approximate
controllability towards 0 and approximate controllability towards given
targets. The methods rely on linear-quadratic control and Riccati equations.
The main novelty is that we consider an LQ problem with controlled backward
stochastic dynamics and, since the coefficients are not deterministic (unlike
some of the cited references), neither is the backward stochastic Riccati
equation. Existence and uniqueness of the solution of such equations rely on
structure arguments (inspired by [7]). Besides solvability, Riccati
representation of the resulting control problem is provided as is the synthesis
of optimal (non-Markovian) control. Several examples are discussed.
