  In this paper we study the exact null-controllability property for a class of
controlled PDMP of switch type with switch-dependent, piecewise linear dynamics
and multiplicative jumps. First, we show that exact null-controllability
induces a con-trollability metric. This metric is linked to a class of backward
stochastic Riccati equations. Using arguments similar to the euclidian-valued
BSDE in [4], the equation is shown to be equivalent to an iterative family of
deterministic Riccati equations that are solvable. Second, we give an example
showing that, for switch-dependent coefficients, exact null-controllability is
strictly stronger than approximate null-controllability. Finally, we show by
convenient examples that no hierarchy holds between approximate (full)
controllability and exact null-controllability. The paper is intended as a
complement to [15] and [14].
