This manuscript deals with analytical studies on detectability of Boolean networks with pinning and injection mode (BNPCIM).
For a Boolean control network, observability is to study whether the system can uniquely judge its initial states by the known information. As a generalization of observability, detectability of Boolean control networks means that the current states can be uniquely determined by input-output data. In this paper, the authors propose three types of detectability for BNPCIM by Cheng product of matrices. They define BNPCIM weakly detectable, detectable and strongly detectable, depending of the existence of \(s \in \mathbb{N}\) and injection modes such that for some or any control and some or any output sequences, state \(x(s)\) can be uniquely determined.
The main contributions of this manuscript can be summarized as follows:

(a)

Three types of detectability are presented to better describe the characters of the system.

(b)

According to free and networked patterns of injection mode and control, the authors design a sequence of matrices to reflect the corresponding output and state information under different injection modes and controls.

(c)

By using these matrices, the authors obtain several necessary and sufficient conditions for different detectability under free and networked input conditions, respectively.

(d)

The number of maximum steps to verify whether the system can satisfy multiple detectability is bounded.

Finally, two examples are presented to show the feasibility of the proposed methods.