#### Abstract

This paper proposes a matrix-based approach to investigate the controllability, reachability, and stabilizability of probabilistic finite automata (PFA). Firstly, the state transition probabilistic structure matrix is constructed for PFA, based on which a kind of controllability matrix is defined for PFA. Secondly, some necessary and sufficient conditions are presented for the controllability, reachability, and stabilizability of PFA with positive probability by using the controllability matrix. Finally, an illustrate example is given to validate the obtained new results.

#### 1. Introduction

Finite automaton is a kind of finite-state machine which plays a key role in theoretical computer science. There are many kinds of finite automata, such as deterministic finite automata (DFA) [1, 2], nondeterministic finite automata (NFA) [3], probabilistic finite automata (PFA) [4, 5], and so on. In the past few decades, finite automata has attracted lots of scholars’ research interest in many disciplines such as computer science, applied mathematics, and engineering, and some fundamental results have been presented [6, 7].

Especially, finite automaton is an important model of discrete event systems (DESs). Another effective model of DESs is Petri net. As we all know, controllability, reachability, and observability are fundamental concepts in DESs [8–10]. The concepts of controllability and observability are extensively studied in Petri nets [11–14], while the concept of reachability is intrinsic to both finite automata [1, 2] and Petri nets [15]. In finite automata, the concept of reachability comes from the classic control theory, which depends on the occurrence events of every state [1, 2]. However, in bounded Petri nets, the concept of reachability depends on transitions and tokens of every marking, which is quite different from that of finite automata [15].

Recently, PFA has been extensively studied because it generalizes the concept of NFA by adding transition probability to the transition function and has wider applications [16–18]. The state estimation and detectability of PFA were considered in [4, 19], and some necessary and sufficient conditions were presented. The supervisory control of PFA was investigated in [20], and several supervisor synthesis problems were proposed.

In some recent works, a semitensor product (STP) approach [21] has been introduced to the reachability and stabilizability of finite automata [1, 2, 22]. For other applications of STP in Boolean networks [23–25] and game theory [26–28], please refer to [29–35]. Xu et al. [1] developed an STP-based method for the reachability of DFA. Yan et al. [2] presented some new criteria for the controllability, reachability and stabilizability of DFA by using STP. Zhang et al. [18] established the algebraic form of PFA via STP and studied the reachability of PFA with positive probability and with probability one, respectively. However, to our best knowledge, there are fewer results on the analysis of controllability and stabilizability of PFA.

In this paper, we propose a matrix-based approach to investigate the controllability, reachability, and stabilizability of PFA with positive probability. The main contributions of this paper are twofold. On one hand, we construct the controllability matrix for PFA, which is effective for the analysis of PFA. On the other hand, some necessary and sufficient conditions are obtained for the controllability, reachability, and stabilizability of PFA with positive probability. These conditions are based on the controllability matrix, and easily verified via MATLAB. Compared with the existing results [18], this paper introduces the concept of controllability and stabilizability for PFA with positive probability.

The rest of this paper is organized as follows. Section 2 introduces some necessary preliminaries on PFA. Section 3 presents some necessary and sufficient conditions for the controllability, reachability, and stabilizability of PFA with positive probability. In Section 4, an example is given to illustrate the new results, which is followed by a brief conclusion in Section 5.

* Notations. * is the -th element of matrix . . represents the -th column of the identity matrix . . The matrix product of this paper is the semitensor product of matrices [21]. We often omit the symbol “” if no confusion arises.

#### 2. Preliminaries on PFA

A probabilistic finite automata is a five-tuple , where is the finite set of states, is the initial state, is the finite set of events called alphabet, is the finite input string set on , and is the partial transition function. Given a state and a finite input string , we define . In this paper, we assume that, for any state , there exists an event such that . is the state transition probabilistic function, that is, means the probability from state to state with the occurrence of event . We notice that holds for any .

Now, we present the concepts of controllability, reachability, and stabilizability for PFA.

*Definition 1. * The state is said to be controllable to the state with positive probability, if there exists a finite input sequence , , such that . Here, means the length of finite input sequence .

The state is said to be controllable with positive probability, if the state is controllable to any state with positive probability.

*Definition 2. * The state is said to be reachable from the state with positive probability, if there exists a finite input sequence , , such that .

The state is said to be reachable with positive probability, if the state is reachable from any state with positive probability.

Given two nonempty sets and , we assume that Then, we give the concepts of controllability, reachability and stabilizability of nonempty sets of states.

*Definition 3. *The nonempty set of states is said to be controllable with positive probability, if for any state , there exists a state such that is controllable to with positive probability.

*Definition 4. *The nonempty set of states is said to be reachable with positive probability, if for any state , there exists a state such that is reachable from with positive probability.

*Definition 5. *The nonempty set of states is said to be one-step returnable with positive probability, if for any state , there exist an event and a state such that .

*Definition 6. *The nonempty set of states is said to be stabilizable with positive probability, if is reachable with positive probability and one-step returnable with positive probability.

*Example 7. *Consider a PFA , where , , , and and are shown in Figure 1.

From Figure 1, we can see that . Therefore, by Definitions 1 and 2, , , , and are controllable and reachable with positive probability.

Assume that and . Since and , by Definition 3, is controllable with positive probability. Similarly, is also controllable with positive probability. Since and , by Definition 4, we can obtain that and are reachable with positive probability. Since and , by Definition 5, is one-step returnable with positive probability. Hence, by Definition 6, is stabilizable with positive probability.

#### 3. Main Results

In this section, we give a controllability matrix to investigate the probability finite automata (PFA) about its controllability, reachability, and stabilizability.

Given a PFA , we identify , , , , and .

We construct the state transition probabilistic structure matrix (STPSM) of as where is defined as follows:

Based on the construction of , the considered PFA has the following algebraic form: where is the mathematic expectation of states at time , is the vector of events at time , and is the STPSM.

For any positive integer , letThen, the -th row of matrix , denoted by , represents the choice of in .

Define where . Then, denotes the set of -step transition probability matrices associated with all possible inputs with length .

Given a finite input sequence with , by (4) and (6), we have

Set Comparing (6) with , one can find that

Theorem 8. *(1) The state is controllable to the state with positive probability, if and only if there exists a positive integer such that , where **(2) The state is controllable with positive probability, if and only if there exists a positive integer such that *

*Proof. *We firstly prove conclusion (1).

(Necessity) Suppose that the state is controllable to the state with positive probability. By Definition 1, there exists a finite input sequence such that . We assume that . By (7), one can see that , which shows that Therefore, there exists a positive integer such that holds.

(Sufficiency) Suppose that there exists a positive integer such that . Then, there exists a positive integer such that . Since , there exists a positive integer such that . By (7), one can find a finite input sequence with , such that . By Definition 1, is controllable to with positive probability.

In the following, we prove conclusion .

(Necessity) Suppose that the state is controllable with positive probability. By Definition 1, for any state , is controllable to with positive probability. Then, by conclusion (1) of Theorem 8, there exists a positive integer such that Let . Then, one can obtain that (Sufficiency) Suppose that there exists a positive integer such that (11) holds. Then for any , we have . By conclusion (1) of Theorem 8, the state is controllable to the state with positive probability. Since is arbitrary, the state is controllable to any state with positive probability. By Definition 1, the state is controllable with positive probability.

Theorem 9. *(1) The state is reachable from the state with positive probability, if and only if there exists a positive integer such that .**(2) The state is reachable with positive probability, if and only if there exists a positive integer such that *

*Proof. *The proof of conclusion (1) is very similar to that of Theorem 8. We just prove conclusion (2).

(Necessity) Suppose that the state is reachable with positive probability. By Definition 2, for any state , is reachable from with positive probability. From conclusion (1) of Theorem 9, there exists a positive integer such that Set . It is easy to see that (15) holds.

(Sufficiency) Suppose that there exists a positive integer such that (15) holds. Then for any , we have . By conclusion (1) of Theorem 9, the state is reachable from the state with positive probability. Since is arbitrary, one can obtain that the state is reachable from any state with positive probability. By Definition 2, the state is reachable with positive probability.

Given the two nonempty sets of states it is obvious that and .

According to (9) and (10), we define the following matrices:

Theorem 10. * is controllable with positive probability, if and only if .*

*Proof. *(Necessity) Suppose that is controllable with positive probability. By Definition 3, for any state , , there exist a state and a control sequence such that . Based on conclusion (1) of Theorem 8, one can obtain , . Therefore, which shows that .

(Sufficiency) Assume that . Then for any , we have . Thus, for any state , , there exist and a control sequence such that . By Definition 3, is controllable with positive probability.

Theorem 11. * is reachable with positive probability, if and only if .*

*Proof. *(Necessity) Suppose that is reachable with positive probability. By Definition 4, for any state , , there exist a state and a control sequence such that , which together with conclusion (1) of Theorem 9 implies that , . Hence, From the arbitrariness of , one can see that .

(Sufficiency) Assume that . Then for any , we have . Hence, for any state , there exist and a control sequence such that . By Definition 4, is reachable with positive probability.

Theorem 12. * is one-step returnable with positive probability, if and only if .*

*Proof. *(Necessity) Suppose that is one-step returnable with positive probability. By Definition 5, for any state , there exist a state and an event such that . Therefore, which implies that . Thus, .

(Sufficiency) Assume that . Then for any , we have . Hence, for any state , there exist and a control sequence such that . By Definition 5, is one-step returnable with positive probability.

Based on Definition 6, Theorems 11 and 12, we have the following result on the stabilizability of PFA.

Theorem 13. * is stabilizable with positive probability, if and only if and .*

#### 4. An Illustrate Example

In this section, we present an example to illustrate the main results.

*Example 14. *Consider a PFA shown in Figure 2, where , and . Assume that and .

Identify and . The state transition probabilistic structure matrix of PFA shown in Figure 2 is

Set . ThenBy Theorem 8, the state is controllable to the any state except the state with positive probability.

A simple calculation shows that , , , , , and . By Theorem 10, is controllable with positive probability and is not controllable with positive probability. By Theorems 11, 12, and 13, both and are reachable, one-step returnable, and stabilizable with positive probability, respectively.

#### 5. Conclusion

In this paper, we have proposed a controllability matrix approach for the investigation of PFA via STP. Based on the controllability matrix, we have presented some necessary and sufficient conditions for the controllability, reachability, and stabilizability of PFA with positive probability. The obtained conditions are easily verified via MATLAB.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The research was supported by the National Natural Science Foundation of China under Grants 61873150 and 61503225 and the Natural Science Fund for Distinguished Young Scholars of Shandong Province under Grant JQ201613.