Approximate and Approximate Null-Controllability of a Class of Piecewise Linear Markov Switch Systems

TL;DR

This paper introduces an explicit algebraic criterion for approximate null-controllability of piecewise linear Markov switch systems with multiplicative noise, addressing an open problem and distinguishing it from approximate controllability.

Contribution

It provides a new, easily computable criterion for approximate null-controllability and clarifies the difference between null-controllability and controllability in such systems.

Findings

The criterion is explicit and algebraic.

Approximate controllability is strictly stronger than null-controllability.

Illustrated with a biological system model.

Abstract

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 [13]. The proof relies on recent results in [4] 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 [19] and reduced in [5]).