# State Feedback Stabilization of Generic Logic Systems via Ledley Antecedence
Solution

Yingzhe Jia  Student Member, IEEE  Daizhan Cheng  Fellow, IEEE and Jun-e Feng
This work is supported partly by the National Natural Science Foundation of
China (NSFC) under Grants 61773371, 61733018 and 61877036.Yingzhe Jia and
Jun-e Feng are with School of Mathematics, Shandong University, Jinan 250100,
P. R. China (e-mail for Feng<EMAIL_ADDRESS>e-mail for Jia:
yingzhe.jia@postgrad.manchester.ac.uk). Daizhan Cheng is with the Key
Laboratory of Systems and Control, Academy of Mathematics and Systems
Sciences, Chinese Academy of Sciences, Beijing 100190, P. R. China (e-mail
dcheng@iss.ac.cn).Corresponding author: Jun-e Feng. Tel.: +86 531 88364652.

###### Abstract

In this paper, the application of Ledley antecedence solutions in designing
state feedback stabilizers of generic logic systems has been proposed. To make
the method feasible, two modifications are made to the original Ledley
antecedence solution theory: (i) the preassigned logical functions have been
extended from being a set of equations to an admissible set; (ii) the domain
of arguments has been extended from the whole state space to a restricted
subset. In the proposed method, state feedback controls are considered as a
set of extended Ledley antecedence solutions for a designed iterative
admissible sets over their corresponding restricted subsets. Based on this, an
algorithm has been proposed to verify the solvability, and simultaneously to
provide all possible state feedback stabilizers when the problem is solvable.
All stabilizers are optimal, which stabilize the logic systems from any
initial state to the destination state/state set in the shortest time. The
method is firstly demonstrated on Boolean control networks to achieve point
stabilization. Then, with some minor modifications, the proposed method is
also proven to be applicable to set stabilization problems. Finally, it is
shown that in $k$-valued and mix-valued logical systems, the proposed method
remains effective.

Boolean control network, Stabilization, Ledley antecedence/consequence
solution, State feedback stabilizer, Semi-tensor product of matrices.

##  1 Introduction

The problem of stability and stabilization is one of the most important issues
in studying dynamic (control) systems. It is also a fundamental topic for
Boolean (control) networks as well as for $k$-valued or mix-valued (control)
networks. For statement ease, in the following text a logical network could be
a Boolean, $k$-valued, or mix-valued network. The systematic investigation of
stability of logical networks may be traced back to F. Robert [27], where it
was called the convergence of discrete iteration. The vector metric was
introduced and used to provide some sufficient conditions for convergence of
discrete iterations.

Boolean network (BN) was introduced firstly by Kauffman [14, 15] to formulate
cellular networks. To manipulate BN, the Boolean control network (BCN) was
merged naturally [13]. Since then, the stability and stabilization become an
interesting and challenging topic to study. Later on, Cheng and his colleagues
proposed a new matrix product called semi-tensor product (STP) of matrices,
and used STP to convert a logical network into an algebraic form called the
algebraic state space representation (ASSR) of logical networks [2, 5].
Motivated by ASSR, the investigation of Boolean (control) networks has been
developed quickly. Many fundamental control problems about logical networks
have been studied, for instance, controllability and observability[1, 8],
state space decomposition and disturbance decoupling [4, 11], optimization and
optimal control [31, 9], with some applications to finite automata [28, 29],
coding [33], just to mention a few. A general outline on recent development of
logical networks via STP/ASSR and its applications can be discovered by some
survey papers [10, 22, 24, 21, 30].

The stability of logical networks have also been studied via ASSR widely.
Three basic approaches have been developed: (i) incidence-matrix-based
stability analysis [3]; (ii) transition-matrix-based stability analysis [2,
7]; (iii) Lyapunov function-based analysis [19]. Of course, stability is
closely related to the stabilization of logical control networks, because
stabilization is to find a suitable controller, which is commonly called the
stabilizer, to make the controlled system being stable.

Various stabilization problems have also been investigated via STP and ASSR.
For instance, set stability and stabilization were firstly proposed and
investigated by [12]; stability and stabilization of BNC with delay have been
discussed by several authors, say [23]; state feedback stabilization with
design technique was proposed by [20]; partial stability and stabilization
were investigated by [6]; robust stability and stabilization of BCN with
disturbances were considered in [32].

This paper investigates the state feedback stabilization of generic logic
systems. The technique proposed in this paper is based on Ledley antecedence
solution. The antecedence/consequence solution for a set of logical equations
was firstly proposed by R.S. Ledley [17, 18]. It has been applied to several
logical problems such as error diagnosis of disease, digital computational
method in symbolic logic, digital circuit design, etc. We also refer to [16]
for a systematic description. It was firstly applied to design of BCN by [26].
Using STP, some useful formulas were developed by [26] to calculate
anticedence/consequence solutions. The method of Ledley
antecedence/consequence solution has also been used to design control of
singular Boolean networks [25].

It is interesting that Ledley antecedence solution may provide a suitable
state feedback stabilizer in a very natural and convenient way. To this end,
two generalizations of Ledley antecedence solution have been done: (i) Ledley
antecedence solution has been originally used with respect to a set of
equalities, in which the preassigned functions equal to a set of constants
respectively. In this paper, it has been developed such that the constants can
be converted to an admissible set. (ii) Originally, the domain of arguments
for Ledley antecedence solution is over the whole state space. In this paper
it has been developed into a restricted set.

The main advantages of this new approach are: (i) The algorithm is necessary
and sufficient. It can verify whether the problem is solvable, and
simultaneously it can also produce a state feedback stabilizer provided the
problem is solvable. (ii) The stabilizer provides an optimal solution, which
means each point can reach its destination point/set in the shortest time.
(iii) All the feasible state feedback stabilizers can be found with one
search. (iv) With a mild revision, the method is applicable to set
stabilization. In addition, the computations involved are easy and
straightforward.

Before ending this section, we give a brief list for notations:

  1. 1.

${\mathcal{M}}_{m\times n}$: the set of $m\times n$ real matrices.

  2. 2.

$\operatorname{Col}(M)$ ($\operatorname{Row}(M)$): the set of columns (rows)
of $M$. $\operatorname{Col}_{i}(M)$ ($\operatorname{Row}_{i}(M)$): the $i$-th
column (row) of $M$.

  3. 3.

${\mathcal{D}}:=\\{0,1\\}$.

  4. 4.

${\mathcal{D}}_{k}:=\left\\{0,\frac{1}{k-1},\cdots,\frac{k-2}{k-1},1\right\\}$,
$k\geq 2$.

  5. 5.

$\delta_{n}^{i}$: the $i$-th column of the identity matrix $I_{n}$.

  6. 6.

$\Delta_{n}:=\left\\{\delta_{n}^{i}|i=1,\cdots,n\right\\}$;
$\Delta:=\Delta_{2}$.

  7. 7.

${\bf 0}_{\ell}:=(\underbrace{0,0,\cdots,0}_{\ell})^{\mathrm{T}}$.

  8. 8.

A matrix $L\in{\mathcal{M}}_{m\times n}$ is called a logical matrix if
$\operatorname{Col}(L)\subset\Delta_{m}$. Denote by ${\mathcal{L}}_{m\times
n}$ the set of $m\times n$ logical matrices.

  9. 9.

If $L\in{\mathcal{L}}_{n\times r}$, by definition it can be expressed as
$L=[\delta_{n}^{i_{1}},\delta_{n}^{i_{2}},\cdots,\delta_{n}^{i_{r}}]$. For the
sake of compactness, it is briefly denoted as
$L=\delta_{n}[i_{1},i_{2},\cdots,i_{r}]$.

  10. 10.

$\delta_{n}\\{a_{1},a_{2},\cdots,a_{n}\\}:=\\{\delta_{n}^{a_{1}},\delta_{n}^{a_{2}},\cdots,\delta_{n}^{a_{n}}\\}$.

  11. 11.

$\ltimes$: Semi-tensor product of matrices.

  12. 12.

Let $A\in{\mathcal{M}}_{p\times n}$ and $B\in{\mathcal{M}}_{q\times n}$. Then
$A*B\in{\mathcal{M}}_{pq\times n}$ is the Khatri-Rao product of matrix,
satisfying [2]

$\operatorname{Col}_{j}(A*B)=\operatorname{Col}_{j}(A)\ltimes\operatorname{Col}_{j}(B),\quad
j=1,\cdots,n.$

  13. 13.

Denote by ${\mathcal{B}}_{m\times n}$ the set of $m\times n$ Boolean matrices.

  14. 14.

$A,B\in{\mathcal{M}}_{m\times n}$, $A\geq B$ means that $A_{i,j}\geq B_{i,j}$,
$\forall i,j$.

  15. 15.

$PR_{k}$: power reducing matrix, which is defined as

$PR_{k}=diag(\delta_{k}^{1},\delta_{k}^{2},\cdots,\delta_{k}^{k})\in{\mathcal{M}}_{k^{2}\times
k}.$

The rest of this paper is organized as follows: Section II provides some
necessary preliminaries, including two parts: (i) STP of matrices and the
matrix express of logical functions; (ii) Ledley antecedence/consequnce
solution to a set of logical functions. Section III considers the state
feedback stabilization of BCN to a pre-assigned point. In Section III the
Ledley antecedence/consequence solution has been generalized in two ways: (i)
the set of equalities is generalized to a set of inclusions; (ii) the
antecedence/consequence solution has been generalized to subset
antecedence/consequence. Section IV uses the generalized antecedence to
construct state feedback stabilizer. Section V discusses the stabilization to
a pre-assigned subset. The algorithm for point stabilization has been extended
to a similar algorithm for set stabilization. Section VI provides an example
to show that the technique developed is also applicable to $k$-valued or mix-
valued logical networks. In section VII, some concluding remarks of the paper
are given.

##  2 Preliminaries

###  2.1 Matrix Expression of Logical Functions

We first recall STP, which is the fundamental tool for algebraic expression of
logical functions.

###### Definition 2.1

[2] Let $A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times q}$, and
the least common multiple $\operatorname{lcm}(n,p)=t$. Then the SPT of $A$ and
$B$ is defined by

$\displaystyle A\ltimes B:=\left(A\otimes I_{t/n}\right)\left(B\otimes
I_{t/p}\right),$ (1)

where $\otimes$ is Kronecker product.

Note that STP is a generalization of classical matrix product. That is, when
$n=p$, STP degenerates to classical matrix product. Throughout this paper the
default matrix product is STP, and in most cases the symbol $\ltimes$ is
omitted.

$X$ is called a logical variable if $X\in{\mathcal{D}}$. The vector expression
of $X$, denoted by $x=\vec{X}$, is

$x=\vec{X}:=\begin{bmatrix}X\\\ 1-X\end{bmatrix}\in\Delta.$

Note that as a convention, we always use capital letters, such as $X_{i}$,
$U_{j}$, for logical variables, and use their lower case, such as $x_{i}$,
$u_{j}$, for their vector forms.

###### Definition 2.2

A mapping $\phi:{\mathcal{D}}\rightarrow{\mathcal{D}}$ is called a unary
operator. A mapping $\phi:{\mathcal{D}}^{2}\rightarrow{\mathcal{D}}$ is called
a binary operator. A mapping $\phi:{\mathcal{D}}^{n}\rightarrow{\mathcal{D}}$
is called a (classical) logical function of $n$-variables. A classical logical
function is also called a Boolean function.

So the unary operators are one-variable Boolean functions, and the binary
operators are two-variable Boolean functions. In vector form, for each
operator there is a logical matrix such that the logical expression can be
converted into an algebraic expression, called the ASSR expression.

Consider unary operator negation ( $\neg$)

$\displaystyle\neg x=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}x=\delta_{2}[2,1]x:=M_{n}x,$ (2)

where $M_{n}\in{\mathcal{L}}_{2\times 2}$ is called the structure matrix of
negation.

Similarly, for a binary operator $\sigma$, we can also find its structure
matrix $M_{\sigma}\in{\mathcal{L}}_{2\times 4}$, such that

$x\sigma y=M_{\sigma}xy.$

For instance,

  * •

conjunction ( $\wedge$):

$\displaystyle M_{c}=\delta_{2}[1,2,2,2].$ (3)

  * •

disjunction ( $\vee$):

$\displaystyle M_{d}=\delta_{2}[1,1,1,2].$ (4)

  * •

conditional ( $\rightarrow$):

$\displaystyle M_{i}=\delta_{2}[1,2,1,1].$ (5)

  * •

biconditional ( $\leftrightarrow$):

$\displaystyle M_{e}=\delta_{2}[1,2,2,1].$ (6)

  * •

exclusive or ( $\bar{\vee}$):

$\displaystyle M_{p}=\delta_{2}[2,1,1,2].$ (7)

###### Proposition 2.3

[2] Let $F:{\mathcal{D}}^{n}\rightarrow{\mathcal{D}}$ be a Boolean function.
Then there exists a unique matrix, $M_{F}\in{\mathcal{L}}_{2\times 2^{n}}$
such that in vector form we have

$\displaystyle f(x_{1},\cdots,x_{n})=M_{F}\ltimes_{i=1}^{n}x_{i},$ (8)

where $f$ is the vector form of $F$.

###### Proposition 2.4

Let $x=\delta_{k}^{s}$. Then

$\displaystyle x^{2}=PR_{k}x,\quad x\in\Delta_{k}.$ (9)

###  2.2 Ledley Antecedence/Consequence Solution

The antecedence and consequence solutions of a given logical function were
proposed and investigated firstly by Ledley [17, 18]. A systematic description
with applications can be found in [16]. The following definition comes from
[26], which is a generalization of the original one given in [16].

###### Definition 2.5

Given a set of Boolean equations

$\displaystyle F_{i}(X_{1},\cdots,X_{n},U_{1},\cdots,U_{m})=C_{i},\quad
i=1,\cdots,s,$ (10)

where $C_{i}\in{\mathcal{D}}$ are constants. A set of logical functions

$\displaystyle U_{j}=G_{j}(X_{1},X_{2},\cdots,X_{n}),\quad j=1,2,\cdots,m$
(11)

  * (i)

is called an antecedence solution of (10), if (11) implies (10);

  * (ii)

is called a consequence solution of (10), if (10) implies (11).

Note that in general for a given set of Boolean equations with
$Z=\\{Z_{1},Z_{2},\cdots,Z_{q}\\}$ as their independent variables, an
arbitrary partition

$Z=Z^{1}\bigcup Z^{2}$

is considered. The antecedence/consequence solution may have the form as

$Z_{s}=G_{s}(Z^{1}),\quad Z_{s}\in Z^{2}.$

For our purpose, (11) is enough for control design.

Using vector form expression $x_{i}=\vec{X}_{i}$, $i=1,2,\cdots,n$,
$u_{j}=\vec{U}_{j}$, $j=1,2,\cdots,m$, and denoting
$x=\ltimes_{i=1}^{n}x_{i}$, $u=\ltimes_{j=1}^{m}u_{j}$, the algebraic forms of
(10) and (11) are as the following (12) and (13) respectively:

$\displaystyle M_{F}ux=c,$ (12) $\displaystyle u=M_{G}x.$ (13)

Based on algebraic form (12), [26] proposed a matrix, called truth matrix, to
describe the corresponding relation between $x$ and $u$. Roughly speaking,
truth matrix shows when $x$ and $u$ are consistent. We use a simple example to
describe this.

###### Example 2.6

Given

$\displaystyle\begin{array}[]{l}F_{1}(X_{1},X_{2},U)=(X_{1}\vee U)\rightarrow
X_{2}=\delta_{2}^{1},\\\ F_{2}(X_{1},X_{2},U)=(X_{1}\wedge
U)=\delta_{2}^{2},\end{array}$ (16)

it is easy to calculate that

$\displaystyle M_{F}=\delta_{4}[1,3,2,4,2,4,2,2],$ (17)

and $c=\delta_{4}^{2}$. Next, we construct a matrix, where the columns
represent different values of $x$ and rows represent different values of $u$.
Then the table shows whether corresponding to each pair $(u,x)$ equation (12)
holds. If it is “true”, the corresponding entry is $1$, if it is “false”, the
corresponding entry is $0$. Then we have Table 1. We also say that the truth
matrix is

$\displaystyle T=\begin{bmatrix}0&0&1&0\\\ 1&0&1&1\end{bmatrix}.$ (18)

Table 1: Truth Matrix  $u\backslash x$ | $\delta_{4}^{1}$ | $\delta_{4}^{2}$ | $\delta_{4}^{3}$ | $\delta_{4}^{4}$
---|---|---|---|---
$\delta_{2}^{1}$ | 0 | 0 | 1 | 0
$\delta_{2}^{2}$ | 1 | 0 | 1 | 1

The following result is essential for verifying antecedence/consequence
solutions.

###### Theorem 2.7

[26] Assume the truth table of equations (10) is $T$, and the set of equations
(11) has its algebraic form as (13). Then

  * (i)

(11) is an antecedence solution of (10), if and only if,

$\displaystyle M_{G}\leq T,$ (19)

  * (ii)

(11) is a consequence solution of (10), if and only if,

$\displaystyle T\leq M_{G}.$ (20)

##  3 Generalized Antecedence/Consequence Solutions

By definition, the antecedence solution generates a set of state feedback
controls

$U_{j}=G_{j}(X_{1},X_{2},\cdots,X_{n}),\quad j=1,2,\cdots,m.$

To make this set of state feedback controls meeting our requirement, we need
to generalize the previous concepts about antecedence solution to design
stabilizer.

First, we replace a single state $C=(C_{1},C_{2},\cdots,C_{s})$ by a set of
states. Assume there exists a set $\Omega\subset{\mathcal{D}}_{s}$, called
admissible set. Replacing a constant set $C$, it is said that (10) is true as
long as $C=(C_{1},C_{2},\cdots,C_{s})\in\Omega$. Then we can construct a truth
matrix $T_{\Omega}$. In Theorem 2.7, replacing $T$ by $T_{\Omega}$, it is easy
to verify that Theorem 2.7 remains available.

Second, we may consider a subset of states, $W\subset{\mathcal{D}}_{n}$,
called restricted set. The concept of antecedence/consequence solutions can be
generalized as follows:

###### Definition 3.1

Given a set of Boolean equations (10), where
$C=(C_{1},C_{2},\cdots,C_{s})\in\Omega\subset{\mathcal{D}}_{s}$. Consider a
set of functions

$\displaystyle U_{j}=G_{j}(X_{1},X_{2},\cdots,X_{n}),\quad j=1,2,\cdots,m.$
(21)

  * (1)

(21) is called a subset antecedence solution of (10) with respect to $W$, if,
as long as $(X_{1},\cdots,X_{n})\in W$, (21) implies (10);

  * (ii)

(21) is called a subset consequence solution of (10) with respect to $W$, if,
as long as $(X_{1},\cdots,X_{n})\in W$, (10) implies (21).

Note that at $T_{\Omega}$ each column corresponds to a distinct state $X$, the
restriction of $T_{\Omega}$ on $W$, denoted by $\left.T_{\Omega}\right|_{W}$,
is obtained by deleting columns which corresponding to $X\not\in W$.

###### Definition 3.2

  * (i)

Assume $f$ is a subset antecedence/consequence solution with respect to $W$,
and $g$ is a subset antecedence/consequence solution with respect to
$W^{\prime}$. $f$ is said to superior to $g$, denoted by $f\succ g$ (or
$g\prec f$), if $W^{\prime}\subset W$ and

$f|_{W^{\prime}}=g|_{W^{\prime}}.$

  * (ii)

Assume $f$ is a $W$-antecedence/consequence solution. $f$ is said to be a
maximum antecedence/consequence solution, if $g$ is a
$W^{\prime}$-antecedence/consequence solution and $g\succ f$, then
$W^{\prime}=W$ and $g|_{W}=f|_{W}$.

Note that if a $W$-subset antecedence/consequence solution $f$ is a maximum
antecedence/consequence solution, then $W$ is completely determined by $T$
(precisely speaking, by the non-zero columns of $T$), and is independent of
$f$. Hence, $W$ is called a maximum (restricted) subset.

Using the same argument as in the proof of Theorem 2.7, it is straightforward
to prove the following result.

###### Theorem 3.3

Assume the truth table of equations (10) with respect to
$\Omega\subset{\mathcal{D}}^{s}$ is $T_{\Omega}$, and state subset
$W\subset{\mathcal{D}}^{n}$ is given. Then

  * (i)

(11) is a subset antecedence solution of (10) with respect to subset $W$, if
and only if,

$\displaystyle\left.M_{G}\right|_{W}\leq\left.T_{\Omega}\right|_{W}.$ (22)

  * (ii)

(11) is a subset consequence solution of (10) with respect to subset $W$, if
and only if,

$\displaystyle\left.T_{\Omega}\right|_{W}\leq\left.M_{G}\right|_{W}.$ (23)

A subset antecedence/consequence solution with respect to subset $W$ is
briefly called $W$-antecedence/consequence solution.

###### Example 3.4

Recall Example 2.6.

  * (i)

Since the truth matrix of (16) is (18). According to Theorem 2.7, it is
obvious that (16) has no antecedence solution.

  * (ii)

Let $W=\\{\delta_{4}^{1},\delta_{4}^{3},\delta_{4}^{4}\\}$, which is a maximum
subset. Set $u=\delta_{4}[2,1,2,2]x$. According to Theorem 3.3, this $u$ is a
$W$-antecedence solution, which is a maximum subset antecedence solution.

  * (iii)

Let $W^{\prime}=\\{\delta_{4}^{1},\delta_{4}^{3}\\}$. Then
$u=\delta_{4}[2,1,2,1]x$ is a $W^{\prime}$-antecedence solution. It is also
clear that this $u$ is not a maximum solution.

##  4 Design of State Feedback Stabilizer

###### Definition 4.1

Consider a BCN

$\displaystyle\begin{cases}X_{1}(t+1)=F_{1}(X_{1}(t),\cdots,X_{n}(t),U_{1}(t),\cdots,U_{m}(t)),\\\
X_{2}(t+1)=F_{2}(X_{1}(t),\cdots,X_{n}(t),U_{1}(t),\cdots,U_{m}(t)),\\\
\vdots\\\
X_{n}(t+1)=F_{n}(X_{1}(t),\cdots,X_{n}(t),U_{1}(t),\cdots,U_{m}(t)),\\\
\end{cases}$ (24)

where $X_{i}(t)\in{\mathcal{D}}$, $i=1,\cdots,n$ are state variables,
$U_{j}(t)\in{\mathcal{D}}$, $j=1,\cdots,m$ are controls,
$F_{i}:{\mathcal{D}}^{m+n}\rightarrow{\mathcal{D}}$, $i=1,\cdots,n$ are
Boolean functions.

Let $X^{d}=(X^{d}_{1},X^{d}_{2},\cdots,X^{d}_{n})\in{\mathcal{D}}^{n}$ be a
given state. The BCN is said to be state feedback stabilizable to $X^{d}$, if
there exists a state feedback control sequence

$\displaystyle\begin{cases}U_{1}(t)=G_{1}(X_{1}(t),\cdots,X_{n}(t)),\\\
U_{2}(t)=G_{2}(X_{1}(t),\cdots,X_{n}(t)),\\\ \vdots\\\
U_{m}(t)=G_{m}(X_{1}(t),\cdots,X_{n}(t)),\\\ \end{cases}$ (25)

such that for the closed-loop network, there exists a $T\in\mathbb{Z}_{\geq
0}$ such that $X(t)=X^{d}$, for $t\geq T$.

###### Definition 4.2

Consider BCN (24). A state $X^{d}\in{\mathcal{D}}^{n}$ is called a control
fixed point, if there exists a control sequence
$U=(U_{1},U_{2},\cdots,U_{m})$, such that

$\displaystyle F_{i}(X^{d},U)=X^{d}_{i},\quad i=1,2,\cdots,n.$ (26)

###### Remark 4.3

It is obvious that (26) is equivalent to: there exists a state feedback
control sequence $U(X)$, such that

$\displaystyle F_{i}(X^{d},U(X^{d}))=X^{d}_{i},\quad i=1,2,\cdots,n.$ (27)

In fact, for a BCN, free control sequence is equivalent to state feedback
control. It is obvious that state feedback control can be replaced by free
control sequence. Conversely, since there are only finite states in a BCN, if
a free control sequence can do something, then at each point
$X\in{\mathcal{D}}^{n}$, a control $U$ is selected, then this control can be
described as $U(X)$.

The following necessary condition is obvious.

###### Proposition 4.4

Assume BCN (24) is stabilizable to $X^{d}$, then $X^{d}$ is a control fixed
point.

Next, we provide an algorithm, which provides a constructive procedure for
stabilizer.

###### Algorithm 4.5

  * •

Step 0 (Initial Step): Set initial object
$\Omega(0)=W_{0}:=\\{(C_{1},C_{2},\cdots C_{n})\\}$, where
$X^{d}=(C_{1},C_{2},\cdots,C_{n})$ is the destination point, to which the BCN
is designed to be stabilized. Construct $T_{\Omega(0)}$, and using it to find
the maximum set $W_{1}$ with respect to $\Omega(0)$, and check if $X^{d}\in
W_{1}$. If $X^{d}\not\in W_{1}$, $X^{d}$ is not a control fixed point, and the
corresponding state feedback stabilization is not solvable. Stop the
algorithm.

  * •

Step $k$ (Repeated Step, $k\geq 1$) : Set
$\Omega(k)=W_{k}\backslash\bigcup\limits_{i=0}^{k-1}W_{i}$.

    * (i)

If $\Omega(k)=\emptyset$, then the problem is not solvable. Stop.

    * (ii)

Else (i.e., $\Omega(k)\neq\emptyset$). Check if

$\displaystyle\bigcup_{i=0}^{k}\Omega_{i}=\Delta_{2^{n}}.$ (28)

If (28) holds, the stabilization problem is solvable, set last $k$ as $k_{*}$
and go to Final Step. Else (i.e., (28) does not hold), construct
$T_{\Omega(k)}$, and use it to find the maximum set $W_{k+1}$ with respect to
$\Omega(k)$. Then go back to Step $k+1$.

  * •

Final Step (Stabilizer Constructing Step): Construct $u(t)=M_{G}x(t)$ by the
following inequalities:

$\displaystyle\left.M_{G}\right|_{\Omega(i)}\begin{cases}\leq\left.T_{\Omega_{0}}\right|_{\Omega(0)},\quad
i=0,\\\ \leq\left.T_{\Omega_{i-1}}\right|_{\Omega(i)},\quad
i=1,\cdots,k^{*}.\end{cases}$ (29)

###### Theorem 4.6

BCN (24) is stabilizable to $X^{d}$, if and only if, the Algorithm 4.5 can go
through and provide a state feedback stabilizer.

######  Proof 4.1.

Starting from first step, one easily sees that $X\in\Omega(0)$ means there
exists a $W_{0}$-antecedence solution, as a state feedback control, which
assures $F_{i}(X,U(X))=C_{i}$, $i=1,2,\cdots,n$. That is, the state feedback
control can drive $X\in\Omega(0)$ to $C$. Similar argument shows that at step
k, there is a state feedback control, when restrict it on $\Omega(k)$, it
drives $X\in\Omega(k)$ to $\Omega(k-1)$. Eventually, if there exists a $k^{*}$
such that (28) holds, then all the points can be driven to $C$ within $k^{*}$
steps. So the successful of the algorithm means the state feedback
stabilization is solvable.

To see that if the algorithm fails, the problem is not solvable, we consider
$k^{*}$ as the last step for $\Omega_{k}\neq\emptyset$ and assume

$\bigcup_{i=0}^{k^{*}}\Omega_{i}\neq\Delta_{2^{n}}.$

Then, there exists a
$X\in\Delta_{2^{n}}\backslash\left\\{\bigcup_{i=0}^{k^{*}}\Omega_{i}\right\\}$.
It is clear that no state feedback can drive it to $C$.

######  Remark 7.

From the proof of Theorem 4.6 one sees easily that if the stabilization
problem is solvable, then the state feedback stabilizers obtained from
Algorithm 4.5 are “optimal” in the sense that each point
$X\in{\mathcal{D}}^{n}$ can reach the destination at the shortest time. If we
do not require “time-optimal”, we may have more state feedback stabilizers. If
in Algorithm 4.5 the $\Omega(i)$ are replaced by $W_{i}$, then in addition to
time-optimal stabilizers, there are some non-time optimal stabilizers,
unfortunately, there are also some state feedback controls, which can not
stabilize the network.

We use an example to depict this algorithm.

###### Example 8

Consider the following BCN:

$\displaystyle\begin{cases}X_{1}(t+1)=X_{2}(t)\vee U_{1}(t),\\\
X_{2}(t+1)=X_{4}(t)\vee\left(U_{2}(t)\wedge X_{1}(t)\right),\\\
X_{3}(t+1)=\left(X_{1}(t)\wedge X_{4}(t)\right)\bar{\vee}\left(\neg
X_{3}(t)\right),\\\ X_{4}(t+1)=\left(\neg X_{1}(t)\right)\leftrightarrow
U_{2}(t).\end{cases}$ (30)

Is it possible to stabilize it to $X^{d}=(1,1,0,1)$?

Consider the following equation:

$\displaystyle\begin{cases}F_{1}(X,U)=X_{2}\vee U_{1},\\\
F_{2}(X,U)=X_{4}\vee\left(U_{2}\wedge X_{1}\right),\\\
F_{3}(X,U)=\left(X_{1}\wedge X_{4}\right)\bar{\vee}\left(\neg X_{3}\right),\\\
F_{4}(X,U)=\left(\neg X_{1}\right)\leftrightarrow U_{2}.\end{cases}$ (31)

Let the structure matrix of $F_{i}$ be $M_{i}$, $i=1,2,3,4$. Then
$F:=F_{1}\ltimes F_{2}\ltimes F_{3}\ltimes F_{4}$ has its structure matrix

$\displaystyle\begin{array}[]{ccl}M_{F}&=&M_{1}*M_{2}*M_{3}*M_{4}\\\
{}\hfil&=&\delta_{16}[2,4,4,2,2,4,4,2,3,7,1,5,3,7,1,5,\\\
{}\hfil&{}\hfil&~{}~{}~{}~{}1,7,3,5,1,7,3,5,4,8,2,6,4,8,2,6,\\\
{}\hfil&{}\hfil&~{}~{}~{}~{}2,4,4,2,10,12,12,10,3,7,1,5,11,15,9,13,\\\
{}\hfil&{}\hfil&~{}~{}~{}~{}1,7,3,5,9,15,11,13,4,8,2,6,12,16,10,14].\end{array}$
(37)

Then

$\Omega(0)=W_{0}=\delta^{3}_{16}\sim\\{C=(1,1,0,1)\\}.$

Then the truth matrix of $\Omega(0)$ can be obtained as

$\displaystyle
T_{\Omega(0)}=\left[\begin{array}[]{cccccccccccccccc}0&0&0&0&0&0&0&0&1&0&0&0&1&0&0&0\\\
0&0&1&0&0&0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \end{array}\right].$ (42)

From (42) it is clear that

$\displaystyle W_{1}=\delta_{16}\\{3,7,9,13\\}.$ (43)

Since $(1,1,0,1)\sim\delta_{16}^{3}\in W_{1}$, $(1,1,0,1)$, the objective
state is a control fixed point.

Next, set

$\displaystyle\Omega(1)=W_{1}\backslash W_{0}=\delta_{16}\\{7,9,13\\}.$ (44)

We consider

$\displaystyle(F_{1}(X,U),F_{2}(X,U),F_{3}(X,U),F_{4}(X,U))\in\Omega(1),$ (45)

The truth matrix can be obtained as

$\displaystyle
T_{\Omega(1)}=\left[\begin{array}[]{cccccccccccccccc}0&0&0&0&0&0&0&0&0&1&0&0&0&1&0&0\\\
0&1&0&0&0&1&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0&0&0&0&1&1\\\
0&1&0&0&1&0&0&1&0&0&0&0&0&0&0&0\\\ \end{array}\right].$ (50)

Then we have

$\displaystyle W_{2}=\delta_{16}\\{2,5,6,8,10,14,15,16\\}.$ (51)
$\displaystyle\begin{array}[]{ccl}\Omega(2)&=&W_{2}\backslash\\{W_{1}\cup
W_{0}\\}\\\ {}\hfil&=&\delta_{16}\\{2,5,6,8,10,14,15,16\\}.\end{array}$ (54)

Third step, we consider

$\displaystyle(F_{1}(X,U),F_{2}(X,U),F_{3}(X,U),F_{4}(X,U))\in\Omega(2),$ (55)
$\displaystyle
T_{\Omega(2)}=\left[\begin{array}[]{cccccccccccccccc}1&0&0&1&1&0&0&1&0&0&0&1&0&0&0&1\\\
0&0&0&1&0&0&0&1&0&1&1&1&0&1&1&1\\\ 1&0&0&1&1&0&0&1&0&0&0&1&0&1&0&0\\\
0&0&0&1&0&1&0&0&0&1&1&1&0&1&1&1\\\ \end{array}\right].$ (60)

Then we have

$\displaystyle W_{3}=\delta_{16}\\{1,4,5,6,8,10,11,12,14,15,16\\}.$ (61)

Hence,

$\displaystyle\Omega(3):=W_{3}\backslash\\{W_{2}\cup W_{1}\cup
W_{0}\\}=\delta_{16}\\{1,4,11,12\\}.$ (62)

Now since

$\Omega(0)\cup\Omega(1)\cup\Omega(2)\cup\Omega(3)=\Delta_{16},$

the BCN (30) is state feedback stabilizable to $(1,1,0,1)$.

Finally, we construct the state feedback control. It should satisfy the
following condition

$\displaystyle\begin{array}[]{lr}U|_{\Omega(1)\cup\Omega(0)}\leq
T_{\Omega(0)}|_{\Omega(1)\cup\Omega(0)}&~{}~{}~{}~{}~{}(a)\\\
U|_{\Omega(2)}\leq T_{\Omega(1)}|_{\Omega(2)}&~{}~{}~{}~{}~{}(b)\\\
U|_{\Omega(3)}\leq T_{\Omega(2)}|_{\Omega(3)}&~{}~{}~{}~{}~{}(c)\\\
\end{array}$ (66)

For (66)(a), one feasible choice is:

$\displaystyle\begin{array}[]{l}u(\delta_{16}^{3})=\delta_{4}^{4}\in\delta_{4}\\{2,4\\},\\\
u(\delta_{16}^{7})=\delta_{4}^{2},\\\
u(\delta_{16}^{9})=\delta_{4}^{3}\in\delta_{4}\\{1,3\\},\\\
u(\delta_{16}^{13})=\delta_{4}^{1}.\\\ \end{array}$ (71)

Note that where $\in\delta_{4}\\{*,*\\}$ means it has multiple choices.

For (66)(b), one feasible choice is:

$\displaystyle\begin{array}[]{l}u(\delta_{16}^{2})=\delta_{4}^{2}\in\delta_{4}\\{2,4\\},\\\
u(\delta_{16}^{5})=\delta_{4}^{4},\\\ u(\delta_{16}^{6})=\delta_{4}^{2},\\\
u(\delta_{16}^{8})=\delta_{4}^{4},\\\
u(\delta_{16}^{10})=\delta_{4}^{3}\in\delta_{4}\\{1,3\\},\\\
u(\delta_{16}^{14})=\delta_{4}^{1},\\\ u(\delta_{16}^{15})=\delta_{4}^{3},\\\
u(\delta_{16}^{16})=\delta_{4}^{3}.\\\ \end{array}$ (80)

For (66)(c), one feasible choice is:

$\displaystyle\begin{array}[]{l}u(\delta_{16}^{1})=\delta_{4}^{1}\in\delta_{4}\\{1,3\\},\\\
u(\delta_{16}^{4})=\delta_{4}^{2}\in\delta_{4}\\{1,2,3,4\\},\\\
u(\delta_{16}^{11})=\delta_{4}^{2}\in\delta_{4}\\{2,4\\},\\\
u(\delta_{16}^{12})=\delta_{4}^{4}\in\delta_{4}\\{1,2,3,4\\}.\\\ \end{array}$
(85)

Putting (71)-(85) together, we have a state feedback stabilizer as

$\displaystyle u(t)=M_{G}x(t),$ (86)

where

$M_{G}=\delta_{4}[1,2,4,2,4,2,2,4,3,3,2,4,1,1,3,3].$

Then

$\begin{array}[]{ccl}M_{G}^{1}&=&(I_{2}\otimes{\bf 1}^{T}_{2})M_{G}\\\
{}\hfil&=&\delta_{2}[1,1,2,1,2,1,1,2,2,2,1,2,1,1,2,2],\end{array}$
$\begin{array}[]{ccl}M_{G}^{2}&=&({\bf 1}^{T}_{2}\otimes I_{2})M_{G}\\\
{}\hfil&=&\delta_{2}[1,2,2,2,2,2,2,2,1,1,2,2,1,1,1,1].\end{array}$

Then it is easy to figure out the state feedback stabilizer as

$\displaystyle\begin{array}[]{ccl}U_{1}(t)&=&\left(X_{1}(t)\wedge
X_{2}(t)\wedge X_{3}(t)\right)\\\ {}\hfil&{}\hfil&\vee\left(X_{1}(t)\wedge
X_{2}(t)\wedge\neg X_{3}(t)\wedge\neg X_{4}(t)\right)\\\
{}\hfil&{}\hfil&\vee\left(X_{1}(t)\wedge\neg X_{2}(t)\wedge X_{3}(t)\wedge\neg
X_{4}(t)\right)\\\ {}\hfil&{}\hfil&\vee\left(X_{1}(t)\wedge\neg
X_{2}(t)\wedge\neg X_{3}(t)\wedge X_{4}(t)\right)\\\
{}\hfil&{}\hfil&\vee\left(\neg X_{1}(t)\wedge X_{2}(t)\wedge\neg
X_{3}(t)\wedge X_{4}(t)\right)\\\ {}\hfil&{}\hfil&\vee\left(\neg
X_{1}(t)\wedge\neg X_{2}(t)\wedge X_{3}(t)\right),\\\
U_{2}(t)&=&\left(X_{1}(t)\wedge X_{2}(t)\wedge X_{3}(t)\wedge
X_{4}(t)\right)\\\ {}\hfil&{}\hfil&\vee\left(\neg X_{1}(t)\wedge
X_{2}(t)\wedge X_{3}(t)\right)\\\ {}\hfil&{}\hfil&\vee\left(\neg
X_{1}(t)\wedge\neg X_{2}(t)\right).\\\ \end{array}$ (96)

To verify whether this stabilizer works we calculate the closed-loop BCN,
which is

$\displaystyle\begin{array}[]{ccl}x(t+1)&=&M_{F}u(t)x(t)=M_{F}M_{G}x(t)x(t)\\\
{}\hfil&=&M_{F}M_{G}PR_{16}x(t):=M_{c}x(t),\end{array}$ (99)

where

$\begin{array}[]{ccl}M_{c}&=&M_{F}M_{G}PR_{16}\\\
{}\hfil&=&\delta_{16}[2,7,3,5,9,7,3,13,3,7,2,6,3,7,9,13].\end{array}$

It is ready to verify that

$M_{c}^{3}=\delta_{16}[3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3].$

That is, after three steps all the trajectories of the closed-loop BCN
converge to $\delta_{16}^{3}\sim(1,1,0,1)$.

The state transition graph of (99) is shown in Fig. 1, where the only
attractor is the point $\delta_{16}^{3}\sim(1,1,0,1)$.

$\delta_{16}^{12}$$\delta_{16}^{13}$$\delta_{16}^{8}$$\delta_{16}^{4}$$\delta_{16}^{6}$$\delta_{16}^{3}$$\delta_{16}^{16}$$\delta_{16}^{5}$$\delta_{16}^{10}$$\delta_{16}^{7}$$\delta_{16}^{9}$$\delta_{16}^{15}$$\delta_{16}^{14}$$\delta_{16}^{11}$$\delta_{16}^{2}$$\delta_{16}^{1}$
Figure 1: State Transition Graph of (99)

Finally, it is worthy noting that from (71)-(85) one sees easily that there
are

$2\times 2\times 2\times 2\times 2\times 4\times 2\times 4=1024$

time optimal feasible state feedback stabilizers.

##  5 Set Stabilization

###### Definition 1

Consider BCN (24). Let $\mathcal{M}\subset{\mathcal{D}}^{n}$ be a given set of
states. The BCN is said to be state feedback stabilizable to $\mathcal{M}$, if
there exists a state feedback control sequence (25) such that for any initial
state $X_{0}$ of the closed-loop network, there exists a $T\in\mathbb{Z}_{\geq
0}$ such that,

$\displaystyle X(t;X_{0},U(X))\in\mathcal{M},\quad\forall t\geq T.$ (100)

###### Definition 2

Consider BCN (24), and assume $\mathcal{M}\subset{\mathcal{D}}^{n}$.
$\mathcal{M}$ is said to be control invariant set, if for each state
$X_{0}\in\mathcal{M}$, there exists a control sequence
$U=(U_{1},U_{2},\cdots,U_{m})$ such that $F(X_{0},U)\in\mathcal{M}$.

To verify whether a state set $\mathcal{M}$ is a control invariant set, we
have the following results.

###### Proposition 3

Consider BCN (24) and assume a set of states
$\mathcal{M}\subset\mathcal{D}^{n}$ is given to be stabilized. Construct the
corresponding truth matrix $T_{\mathcal{M}}$. Let the maximum set for
$T_{\mathcal{M}}$ be $\Theta$. Then $\mathcal{M}$ is a control invariant set,
if,

$\displaystyle\mathcal{M}\cap\Theta=\mathcal{M}$ (101)

In other words, $\mathcal{M}\subset\Theta$.

######  Proof 5.1.

By definition, there exists at least one $\Theta$-antecedence solution $U(X)$
such that

$\displaystyle F(X_{0},U)\in\mathcal{M},\quad\forall X_{0}\in\Theta$ (102)

Since $\mathcal{M}\subset\Theta$, then it is obvious that
$F(X_{0},U)\in\mathcal{M},\quad X_{0}\in\mathcal{M}$.

On the other hand, in more general scenarios where
$\mathcal{M}\not\subset\Theta$, yet $\mathcal{M}\cap\Theta\neq\emptyset$, we
have the following remark.

######  Remark 4.

In the aforementioned scenarios where $\mathcal{M}\not\subset\Theta$, yet
$\mathcal{M}\cap\Theta\neq\emptyset$, the control invariant set $W_{0}$ within
the state set $\mathcal{M}$ can be determined by,

$\displaystyle W_{0}=\mathcal{M}\cap\Theta$ (103)

The proof is the same as above, and for brevity, it is not demonstrated here.
In [12], the concept of the largest control invariant set is given, through
which, the optimal state feedback controls can be designed. Based on the
property of Ledley Antecedence Solution, it is quite obvious that the control
invariant set solved using the maximum set of the truth matrix and (103) is
the largest control invariant set. Similar to point stabilization case, we
have the following algorithm.

######  Algorithm 0.

  * •

Step 0 (determining the largest control invariant set): Set the destination
set $\mathcal{M}$, to which the BCN is designed to be stabilized. Construct
$T_{\mathcal{M}}$, and use it to find the maximum set $\Theta$ with respect to
$\mathcal{M}$. Check if $\mathcal{M}\cap\Theta=\emptyset$. If
$\mathcal{M}\cap\Theta=\emptyset$, the corresponding state feedback
stabilization is not solvable. Stop the algorithm. Else (i.e.,
$W_{0}=\mathcal{M}\cap\Theta$), calculate the corresponding truth matrix
$T_{W_{0}}$ and the maximum set $W_{1}$, set $\Omega(1)=W_{1}\backslash W_{0}$
and go to step $k=1$.

  * •

Step $k$ (Repeated Step, $k\geq 1$) : Same as Step $k$ of Algorithm 4.5.

  * •

Final Step (Stabilizer Constructing Step): Same as Final Step of Algorithm
4.5.

The following result can be proved using similar argument for Theorem 4.6.

######  Theorem 6.

BCN (24) is stabilizable to $W_{0}$, if and only if, the Algorithm 5 can go
through and provide a state feedback stabilizer.

It is also worthy noting that this algorithm provides the time-optimal
stabilizers. In the following an example is used to describe the design
process.

###### Example 7

Recall Example 8, and consider BCN (30), and set
$\mathcal{M}=\\{(1,0,1,0)\sim\delta_{16}^{6},(1,0,0,1)\sim\delta_{16}^{7},(0,1,0,1)\sim\delta_{16}^{12}\\}$.
This example investigates whether BCN (30) can be stabilized to $\mathcal{M}$.

First, construct the truth matrix $T_{\mathcal{M}}$ corresponding to
$\mathcal{M}$,

$\displaystyle
T_{\mathcal{M}}=\left[\begin{array}[]{cccccccccccccccc}0&0&0&0&0&0&0&0&0&1&0&0&0&1&0&0\\\
0&1&0&0&0&1&0&0&0&0&0&1&0&0&0&1\\\ 0&0&0&0&0&1&1&0&0&1&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0&0&0&1&1&0&0&0\end{array}\right].$

The maximum set is

$\Theta=\delta_{16}\left\\{2,6,7,10,12,13,14,16\right\\}.$

Since $\mathcal{M}\cap\Theta=\mathcal{M}$, set $\mathcal{M}$ is control
invariant set. For easier understanding, denote $W_{0}=\mathcal{M}$,
$W_{1}=\Theta$, set $\Omega(1):=W_{1}\backslash W_{0}$. Then it is easy to
calculate that

$\displaystyle
T_{\Omega(1)}=\left[\begin{array}[]{cccccccccccccccc}1&0&0&1&1&0&0&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&1&0&0&0&1&0\\\ 1&0&0&1&1&0&0&1&0&0&0&0&0&0&0&1\\\
0&0&0&0&0&0&0&1&0&0&1&0&0&1&1&1\end{array}\right].$

Then $W_{2}=\delta_{16}\\{1,4,5,8,11,14,15,16\\}$, and set

$\begin{array}[]{ccl}\Omega(2)&:=&W_{2}\backslash(W_{1}\cup W_{0})\\\
{}\hfil&=&\delta_{16}\\{1,4,5,8,11,15\\}.\end{array}$

We calculate that

$\displaystyle
T_{\Omega(2)}=\left[\begin{array}[]{cccccccccccccccc}0&1&1&0&0&1&1&0&0&0&1&1&0&0&1&1\\\
1&0&0&1&1&0&0&1&1&1&0&0&1&1&0&0\\\ 0&1&1&0&0&0&0&0&0&0&1&1&1&1&0&0\\\
1&0&0&1&0&1&1&0&1&1&0&0&0&0&0&0\end{array}\right].$

One sees that

$\begin{array}[]{ccl}W_{3}&=\delta_{16}\\{&1,2,3,4,5,6,7,8,\\\
{}\hfil&{}\hfil&9,10,11,12,13,14,15,16\\},\end{array}$

and

$\begin{array}[]{ccl}\Omega(3)&=W_{3}\backslash(W_{2}\cup W_{1}\cup
W_{0})=\delta_{16}\\{3,9\\}.\end{array}$

Since

$\Omega(0)\cup\Omega(1)\cup\Omega(2)\cup\Omega(2)=\Delta_{16},$

we conclude that the BCN (30) is a state feedback stabilizable to
$\mathcal{M}$.

Afterwards, we construct the state feedback stabilizer.

Since $U(X)$ for $X\in\Omega(0)\cup\Omega(1)$ is determined by
$T_{\Omega(0)}$, we choose one feasible solution as follows:

$\displaystyle\begin{array}[]{l}u(\delta_{16}^{2})=\delta_{4}^{4}\in\delta_{4}\\{2,4\\}\\\
u(\delta_{16}^{6})=\delta_{4}^{3}\in\delta_{4}\\{2,3\\}\\\
u(\delta_{16}^{7})=\delta_{4}^{3}\\\
u(\delta_{16}^{10})=\delta_{4}^{1}\in\delta_{4}\\{1,3\\}\\\
u(\delta_{16}^{12})=\delta_{4}^{4}\in\delta_{4}\\{2,4\\}\\\
u(\delta_{16}^{14})=\delta_{4}^{1}\\\
u(\delta_{16}^{16})=\delta_{4}^{2}.\end{array}$ (111)

Using $T_{\Omega(1)}$, we can determine $U(X)$ for $X\in\Omega(2)$, that is

$\displaystyle\begin{array}[]{l}u(\delta_{16}^{1})=\delta_{4}^{1}\in\delta_{4}\\{1,3\\}\\\
u(\delta_{16}^{4})=\delta_{4}^{1}\in\delta_{4}\\{1,3\\}\\\
u(\delta_{16}^{5})=\delta_{4}^{1}\in\delta_{4}\\{1,3\\}\\\
u(\delta_{16}^{8})=\delta_{4}^{4}\in\delta_{4}\\{1,3,4\\}\\\
u(\delta_{16}^{11})=\delta_{4}^{2}\in\delta_{4}\\{2,4\\}\\\
u(\delta_{16}^{15})=\delta_{4}^{4}\in\delta_{4}\\{2,4\\}.\end{array}$ (118)

Using $T_{\Omega(2)}$, we can determine $U(X)$ for $X\in\Omega(3)$ as

$\displaystyle\begin{array}[]{l}u(\delta_{16}^{3})=\delta_{4}^{3}\in\delta_{4}\\{1,3\\}\\\
u(\delta_{16}^{9})=\delta_{4}^{2}\in\delta_{4}\\{2,4\\}.\end{array}$ (121)

Assume

$u(t)=M_{G}x(t),$

where $M_{G}\in{\mathcal{L}}_{4\times 16}$. Summarizing the choices
(111)-(121), a feasible control is

$\displaystyle M_{G}=\delta_{4}[1,2,3,1,1,3,3,4,2,1,2,4,4,1,4,2].$ (122)

We conclude that (122) provides a state feedback stabilizer for BCN (30).

Finally, we may check the closed-loop system. It is

$\displaystyle\begin{array}[]{ccl}x(t+1)&=&M_{F}u(t)x(t)=M_{F}M_{G}x(t)x(t)\\\
{}\hfil&=&M_{F}M_{G}PR_{16}x(t):=M_{C}x(t).\end{array}$ (125)

It is easy to calculate that

$\displaystyle M_{C}=\delta_{16}[2,7,4,2,2,12,12,13,4,7,2,6,12,7,10,6].$ (126)

The state transition graph of this closed-loop network is described in Fig. 2,
which shows that there are only one attractor
$C=\\{\delta_{12}^{6},~{}\delta_{12}^{12}\\}$. Hence, all trajectories
converge to $C\subset W_{0}$.

$\delta_{16}^{9}$$\delta_{16}^{4}$$\delta_{16}^{3}$$\delta_{16}^{5}$$\delta_{16}^{1}$$\delta_{16}^{11}$$\delta_{16}^{2}$$\delta_{16}^{15}$$\delta_{16}^{10}$$\delta_{16}^{7}$$\delta_{16}^{14}$$\delta_{16}^{8}$$\delta_{16}^{13}$$\delta_{16}^{12}$$\delta_{16}^{6}$$\delta_{16}^{16}$
Figure 2: State Transition Graph of (125)

From (111)-(121) it is easy to see that there are 6144 feasible state feedback
stabilizers.

##  6 Stabilization of General Logical Systems

The method discussed in previous sections are all applicable to $k$-valued or
mix-valued logic. In this section we give an example to demonstrate this.

###### Example 1

Consider a mix-valued logical network

$\displaystyle\begin{cases}X_{1}(t+1)=X_{1}(t)\diamondsuit X_{2}(t),\\\
X_{2}(t+1)=X_{1}(t)\Box U(t),\end{cases}$ (127)

where $X_{1}(t)\in{\mathcal{D}}_{2}$, $X_{2}(t),~{}U(t)\in{\mathcal{D}}_{3}$,
$\diamondsuit:{\mathcal{D}}_{2}\times{\mathcal{D}}_{3}\rightarrow{\mathcal{D}}_{2}$,
$\Box:{\mathcal{D}}_{2}\times{\mathcal{D}}_{3}\rightarrow{\mathcal{D}}_{3}$,
with their structure matrices as

$\displaystyle M_{\diamondsuit}=\delta_{2}[1,1,2,2,2,1];$ (128)

$\displaystyle M_{\Box}=\delta_{3}[1,2,3,2,3,1].$ (129)

  * (i)

Is it possible to stabilize (127) to $(1,1)\sim\delta_{6}^{1}$?

It is easy to calculate that

$\displaystyle x(t+1)=M_{F}u(t)x(t),$ (130)

where

$M_{F}=\delta_{6}[1,1,4,5,5,2,2,2,5,6,6,3,3,3,6,4,4,1].$

Set $\Omega(0)=W_{0}=\\{(1,1)\\}$. Using (130), the truth matrix with respect
to $\Omega(0)$ is

$\displaystyle T_{\Omega(0)}=\begin{bmatrix}1&1&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&1\\\ \end{bmatrix}.$ (131)

Hence,

$\displaystyle W_{1}=\delta_{6}\\{1,2,6\\},$ (132)

and

$\displaystyle\Omega(1)=W_{1}\backslash W_{0}=\delta_{6}\\{2,6\\}.$ (133)

The truth matrix with respect to $\Omega(1)$ is

$\displaystyle T_{\Omega(1)}=\begin{bmatrix}0&0&0&0&0&1\\\ 1&1&0&1&1&0\\\
0&0&1&0&0&0\\\ \end{bmatrix}.$ (134)

Hence,

$\displaystyle W_{2}=\delta_{6}\\{1,2,3,4,5,6\\},$ (135)

and

$\displaystyle\Omega(2)=W_{2}\backslash(W_{0}\cup
W_{1})=\delta_{6}\\{3,4,5\\}.$ (136)

Now

$\Omega(0)\cup\Omega(1)\cup\Omega(2)={\mathcal{D}}_{6}.$

Hence, mix-valued logical system (127) can be stabilized to
$(1,1)\sim\delta_{6}^{1}$.

Next, we construct a state feedback stabilizer. Similarly to Boolean cases, a
feasible state feedback control can be obtained as

$\displaystyle u(t)=M_{G}x(t),$ (137)

where

$M_{G}=\delta_{3}[1,1,3,2,2,3].$

To verify the result we consider the closed-loop network, which is

$\displaystyle x(t+1)=M_{F}M_{G}x^{2}(t)=M_{F}M_{G}PR_{6}x(t):=M_{c}x(t).$
(138)

Then

$M_{c}=M_{F}M_{G}PR_{6}=\delta_{6}[1,1,6,6,6,1].$

Since

$M_{c}^{2}=\delta_{6}[1,1,1,1,1,1],$

one sees easily that after two steps the system stabilized to
$\delta_{6}^{1}$.

  * (ii)

Is it possible to stabilize (127) to $\mathcal{M}=\\{(1,0),(0,0)\\}$? To
verify this, we first calculate the truth matrix corresponding to state set
$\mathcal{M}$,

$\displaystyle T_{\mathcal{M}}=\begin{bmatrix}0&0&0&0&0&0\\\ 0&0&0&1&1&1\\\
1&1&1&0&0&0\\\ \end{bmatrix}.$ (139)

where the maximum set $\Theta=\Delta_{6}$, and
$\mathcal{M}\cap\Theta=\mathcal{M}$, therefore, the state set $\mathcal{M}$ is
a control invariant set. From the truth table 139, one can easily sees that
all states of the logic system can be stabilized to the state set
$\mathcal{M}$ within one step, and the control sequence can be obtained as,

$\displaystyle u(t)=\delta_{3}[3,3,3,2,2,2]x(t)$ (140)

Put this state feedback control into the original system, we have the closed-
loop system as

$\displaystyle x(t+1)=M_{F}M_{G}PR_{6}x(t)=M_{c}x(t),$ (141)

where

$M_{c}=M_{F}M_{G}PR_{6}=\delta_{6}[3,3,6,6,6,3].$

Fig. 3 is the state transition graph of the closed-loop network (141). It
shows the convergence of this system to $\mathcal{M}$.

$\delta_{6}^{2}$$\delta_{6}^{3}$$\delta_{6}^{1}$$\delta_{6}^{5}$$\delta_{6}^{6}$$\delta_{6}^{4}$
Figure 3: State Transition Graph of (141)

##  7 Conclusion

The Ledley antecedence/consequence solution has been applied for the design of
feedback stabilizers in generic logic systems in this paper. To achieve such
goal, the original theory has been extended in two ways. First, logical
functions have been considered as an admissible set of values instead of a set
of equations. Second, the domain of arguments has also been applied on a
restricted subset of state space instead of the whole state space. Based on
such knowledge, this paper has depicted that by properly designing admissible
subsets for the logic functions of the systems, the state feedback controls
can be obtained automatically when solving the subset antecedence solutions
with restricted subsets for arguments. By using this approach, the
stabilization of BCN to a pre-assigned point and to a pre-assigned set can be
achieved respectively. An iterative algorithm has been designed to verify if
the problem is solvable or not, and if the problem is solvable, the algorithm
is also capable to provide all (time-optimal) state feedback stabilizers.
Various examples have been demonstrated to verify the effectiveness of the
proposed technique in Boolean control networks, as well as $k$-valued and mix-
valued logical networks.

The technique introduced in this paper is new and useful. It shows that state
feedback controls can be obtained via properly designed admissible sets and
restricted sets. The method provided in this paper has a universal character,
which is via properly designed sequence of subsets, the state feedback control
may be obtained for various purposes (i.e., state feedback control, output
feedback stabilization, tracking).

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[]Yingzhe Jia (S’18) received the B.E. in electrical engineering from Shandong
University, China, in 2015, and the Ph.D. degree in electrical and electronic
engineering from the University of Manchester, Manchester, U.K. in 2020. He is
currently a research associate with the school of Mathematics, Shandong
University, China. His research interests include stabilization of logic
control networks, application of game theory in power systems, etc.

[]Daizhan Cheng (F’05) received the Bachelor s degree in mechanical
engineering from Tsinghua University, Beijing, China, the M.S. degree in
mathematics from the Graduate School, Chinese Academy of Sciences, Beijing,
China, and the Ph.D. degree in systems science and mathematics from Washington
University, St. Louis, WA, USA, in 1970, 1981, and 1985, respectively.

Since 1990 he has been a Professor both with the Institute of Systems Science,
National University of Singapore, Singapore, and Academy of Mathematics and
Systems Science, Chinese Academy of Sciences, Beijing, China. He has authored
or coauthored 12 books, more than 240 journal papers, and more than 130
conference papers. His current research interests include nonlinear control
systems, switched systems, Hamiltonian systems, Boolean control networks, and
game theory.

Dr. Cheng was the recipient of the Second Grade National Natural Science Award
of China in 2008 and 2014, and the Automatica 2008 to 2010 Best
Theory/Methodology Paper Award, bestowed by IFAC, in 2011. He was the Chairman
of the Technical Committee on Control Theory, Chinese Association of
Automation from 2003 to 2010, and a Member of the IEEE CSS Board of Governors
in 2009. He was an IFAC Fellow in 2008 and an IFAC Council Member from 2011 to
2014.

[]Jun-e Feng received the Ph.D. degree in cybernetics from Shandong
University, Jinan, China, in 2003.

From 2006 to 2007, she was a Visiting Scholar with Massachusetts Institute of
Technology, Cambridge, MA, USA, and a Visiting Scholar with the University of
Hong Kong, Hong Kong, in 2013. She is currently a Professor of School of
Mathematics with Shandong University. Her research interests include singular
systems, multiagent systems, logic control networks, etc