SONG Xuewei(宋雪薇),LIAN Chunbo(廉春波),GE Bin(葛斌)

(College of Mathematical Sciences,Harbin Engineering University,Harbin 150001,China)

Abstract: In this paper,controllability and Mayer-style optimal control of a class of the incomplete Boolean control networks with delay is studied.The system is converted into a classical Boolean control networks via semi-tensor product.We propose the top-up conditions for controllability and the necessary conditions for optimal control of the Mayer-style.The main contribution is that the system is transformed into Boolean control network algebraic by using matrix semi tensor product method,which overcomes the restriction of the dimension of matrix product.

Key words: Boolean control networks;Delay;Controllability;Mayer-style optimal control;Semi-tensor

1.Introduction

In order to simulate the complex nonlinear system,a Boolean networks is first proposed in [6].The Boolean network is a discrete dynamic system about Boolean variables based on directed graph.D.Z Cheng took the lead in putting forward the semi-tensor product(STP)of matrix when exploring the Morgan problem of control system.The logical dynamic system of Boolean networks is transformed into discrete time linear system.

Following[17],many standard control problems of Boolean control networks have been investigated,for instance,the controllability and observability[3,10],stability and stabilization[1,8],structure and disturbance decoupling[2,5,7],optimization[4].In fact,because of the instantaneous change of some external environment,the state of biological system often changes suddenly at some time in the process of evolution.From a biological point of view,mRNAs and proteins can be synthesized at different positions(nuclear and cytoplasmic,respectively).the distance between the two positions makes the transport or diffusion of mRNAs and proteins considerably delayed.Usually we use high-order networks to describe this type of delay phenomenon[9,11].Laschov and Margaliot[10]studied the optimal control of the Mayer type of the Boolean network based on the principle of Pontryagin extreme value of optimal control of discrete systems,and obtained the design method of the optimal controller.CHEN[12],analyzed the minimum time control problem with pulse affecting Boolean network,and gave the necessary conditions for minimum time control in the form of the principle of great value.ZHAO[13]and others discussed the optimal control problem on boolean control network infinite time domain,and gave the feedback form solution of optimal control by calculating the limit ring of Boolean control network.The paper [14]summarizes the optimal control problem of Boolean control network in the poor time domain and infinity time domain,and puts forward the sufficient necessary conditions for the optimal control problem to be solved.

In order to solve two classic intellectual problems: the wolf-sheep-cabbage puzzle and the missionaries-cannibals problem,ZHANG[15]proposed a new logic control system,which called incomplete logic control system,some of which can only be applied to certain states.In addition,FU et al.[16]established a network evolution game with the risk of bankruptcy as a multi-value logical network,some of which control-state should be avoided.These cases show that the study of incomplete Boolean network is of great theoretical value and important practical significance.Main research models in each chapter of this paper belong to the incomplete Boolean Control Networks with delay(briefly,IDBCN).

The paper is organized as follows: Section 2 provides a discrete time algebraic form for IDBCN.The top-up conditions for controllability,the necessary conditions for mayer-style optimal control are considered in Sections 3,4,respectively.

Before ending this introduction we give a list for notations,which are used in the sequel:

?R: the set of reals;Mm×n: the set ofm×nreal matrices;0m×n: them×nzero matrix;

2.Problem Formulation

This subsection gives a brief review on STP of matrices.STP is the main tool used in this paper.We refer to [4]for details.

Lemma 2.1LetA ∈Mm×n,B ∈Mp×qbe two real matrices,and the least common multiple ofnandpist=lcm{n,p}.The STP ofAandB,denoted byAB,is defined:

whereIkis thek×kidentity matrix,and?is the Kronecker product.

STP has some commutative properties,which will be used in the sequel:

Lemma 2.2GivenA ∈Mm× n,letZ ∈Rn×1be a column vector.Then

Define.Then we have the following result:

Lemma 2.3Givenx ∈?pandy ∈?q,thenDfxy=x,Dsxy=y.

Lemma 2.4Given two column vectorsX ∈Rm,Y ∈RnthenW[m,n]XY=Y X,whereW[m,n]=δmn[1,m+1,···,(n?1)m+1;···;m,···,nm],the main function of a swap matrix is to exchange the order of two vector factors.

Lemma 2.5we construct a matrixΦn,n ≥1,called the group power-reducing matrix,forzj=p1p2···pjwherepi ∈?,i=1,2,···,j,then

Lemma 2.6Letxi ∈D,i=1,2,···,n.A mappingf:Dμn+m →Dis called a mix-valued logical function.A 2-valued logical function is called a Boolean function.Let

The delay boolean control system

wherefi:Dμn+m →D,then there exists a unique matrixM1i ∈L2×2μn+m,such that

Definition 2.1Denote byx(t)=.Then (2.6) can be expressed as

whereL1=M11?M12?...?M1n,andM1i?M1j=[Col1(M1i)?Col1(M1j);···;Colμn+m(M1i)?Colμn+m(M1j)],i,j=1,2,···,n.

The equation (2.7) is called the algebraic state space expression of the mixed-valued logical dynamic system (2.6).

Similar argument shows the following:

Definition 2.2Given the equations

can be expressed as

whereG=M21(I2m ?M22)Φm(I2m ?M23)Φm···(I2m ?M2m)Φm.

Without loss of generality,we can divideL1into 2m+nblocksL1=[L1,1L1,2··· L1,2m+n],whereL1,i ∈L2n×2(μ?1)×nand convert (2.7) into the following form:

Recall thatSandWare equivalent.Then the state trajectories of incomplete systems and systems with control state avoidance sets are equivalent.

3.Main Result

Define a control-transition matrix of (2.15) as

ProofWe first prove the conclusion (i).The dynamic evolution of incomplete system(2.3) can be described as

Then,by using inductive method,we deduce thatz(s)=QG(s)u(0)x(0) and

ProofThe proof method is similar to (2.11).

Consider IDBCN in algebraic state-space representation (2.15).From here we consider the case of a single control(i.e.,m=1)and fix some(arbitrary)initial conditionz(0)=δ2qμn.

Later,we refer toC(k,j;u) as the transition matrix from timejtokcorresponding to the controlu,for anyk ≥l ≥j,we haveC(k,j;u)=C(k,l;u)C(l,j;u).Consider the system withm=1 andz(0)=z0.Suppose thatu?={u?(0),u?(1),···,u?(N)} ?Uis an optimal control sequence for problem (3.2),and letx?denote the corresponding solution.Thenz?(p)=x?(p)x?(p ?1)···x?(p ?μ+1),wherex?(p)=Df[2n,2(μ?1)n]z?(p).

Define the adjoint functionλ:{1,2,···,N}→R2μnas the solution of

at the same timesλ(N)=Df[2n,2(μ?1)n]Tr.And the switching functionm:{0,1,···,N ?1}→R by

We investigate the Mayer-style optimical control of (2.15) using the Pontryagin max principle.Refer to [10]for specific proof method.

Remark 3.1Obviouslym(·) is scalar function.

Remark 3.2Ifm(·) for somek,then there exists anu?optimal control satisfyingu?(k)=and there exists an optimal controlw?satisfyingw?=

4.Example

To demonstrate the application of the results,we consider a simple example.

Example 4.1Consider the BCN

Consider the vector form of the logical variablesx(t)=A(t)B(t) and the algebraic statespace form isx(t+1)=L1u(t)x(t)x(t ?1)x(t ?2) with

This indicates that for the initial statez(0)=,the final statex(5)=is not accessible att=5.Therefore,by the definition of an incomplete system,the system(4.1)is uncontrollable in the initial statez(0)=.

5.Conclusion

In this paper some fundamental control problems,including controllability,Mayer-type optimal control problem of IDBCN were investigated.Firstly,this paper presents a method to convert a IDBCN into a standard Boolean Control system.Then the controllability is also studied,sufficient and necessary condition is proposed.Finally,we considered a Mayer-type optimal control optimal control problem for single-input IDBCN deried a necessary condition for optimality in the form of maximum principle.Though we have solved the problems,there are some control nodes that are necessary and add to computation.Choosing pinning nodes is interesting and fundamental.We leave them for further study.