Approximate and approximate null-controllability of a class of piecewise linear Markov switch systems

We propose an explicit, easily-computable algebraic criterion for approximate null-controllability of a class of general piecewise linear switch systems with multiplicative noise. This gives an answer to the general problem left open in Goreac and Martinez (2015). The proof relies on recent results in Confortola et al. (2015) allowing to reduce the dual stochastic backward system to a family of ordinary differential equations. Second, we prove by examples that the notion of approximate controllability is strictly stronger than approximate null-controllability. A sufficient criterion for this stronger notion is also provided. The results are illustrated on a model derived from repressed bacterium operon (given in Krishna et al. (2005) and reduced in Crudu et al. (2009)).