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Research article

State feedback stabilization problem of stochastic high-order and low-order nonlinear systems with time-delay

  • Received: 17 September 2022 Revised: 18 October 2022 Accepted: 26 October 2022 Published: 16 November 2022
  • MSC : 93D12, 93B52, 93E15

  • This manuscript considers the state feedback stabilization problem for a class of stochastic high-order and low-order nonlinear systems with time-delay. Compared with the previous results, a distinctive feature to be studied is that the considered systems involve high-order, low-order, intricate stochastic diffusion terms and time-delay simultaneously. First, the homogeneous domination approach and suitable coordinate transformations are introduced to obtain the updating laws. Then, a state feedback controller is devised to make the closed-loop systems globally asymptotically stable in probability. Finally, a simulation example is shown to prove the proposed approach powerfully.

    Citation: Yanghe Cao, Junsheng Zhao, Zongyao Sun. State feedback stabilization problem of stochastic high-order and low-order nonlinear systems with time-delay[J]. AIMS Mathematics, 2023, 8(2): 3185-3203. doi: 10.3934/math.2023163

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  • This manuscript considers the state feedback stabilization problem for a class of stochastic high-order and low-order nonlinear systems with time-delay. Compared with the previous results, a distinctive feature to be studied is that the considered systems involve high-order, low-order, intricate stochastic diffusion terms and time-delay simultaneously. First, the homogeneous domination approach and suitable coordinate transformations are introduced to obtain the updating laws. Then, a state feedback controller is devised to make the closed-loop systems globally asymptotically stable in probability. Finally, a simulation example is shown to prove the proposed approach powerfully.



    Recently, the theory of nonlinear systems with time-delay has been a hot topic, due to its wide application in practical problems, such as physical engineering, biological systems and economic processes. Among these, the Lyapunov-Krasovskii methodology plays a crucial role in dealing with time-delay systems. Based on the above method, Pepe [1] addressed the input state stability of nonlinear systems with time-delay. Zhang [2] designed a stabilized controller for time-delay feed-forward nonlinear systems to achieve system stability. In order to address the stabilization problem of high-order nonlinear systems with time-delay, some researchers try to find new ways to design corresponding controllers. Yang and Sun [3] investigated the state feedback stabilization problem of controlled systems with high-order or/and time-delay via the homogeneous domination idea. With the help of the saturation function technique, homogeneous domination idea and Lyapunov approach, Song [4] studied the stabilization problem of high order feed-forward time-delay nonlinear systems. In addition to the above works, many results in [5,6,7,8,9,10] have established and improved the concept framework of nonlinear systems with time-delay.

    Ever since the stochastic stability theory was founded and enriched by Deng and Zhu [11,12], great progress has been made on the global stabilization of stochastic nonlinear systems [13,14,15,16]. Subsequently, Florchinger [17] extended the theory of control with the Lyapunov-Krasovskii functional. With the stochastic stability theory in mind, it is still important and meaningful to address high-order stochastic nonlinear systems with time-delay. Zha [18] investigated the issue of output feedback stabilization. Liu [19] studied the output feedback stabilization problem for time-delay stochastic feed-forward systems. By using a power integrator approach, the work in [20,21,22,23] also considered the state-feedback stabilization problems. However, the state feedback stabilization problem for stochastic high-order and low-order nonlinear systems with time-delay has not been well addressed, which leads us to take the interesting problem into account.

    How to deal with the state feedback stabilization problem for high-order and low-order nonlinear systems with time-delay? By using a power integrator approach, Liu & Sun [24] constructed a time-delay independent controller for the aforementioned systems to relax the growth condition and the power order limitations. However, to the best of our knowledge, research on the corresponding stochastic version is limited with scarcely a few convincing results. The main difficulties are explained from two aspects. On one hand, the Itˆo formula brings the gradient terms and the Hessian terms in the Lyapunov analysis. On the other hand, the particularity of its structure has made many traditional methods inapplicable. Therefore, we need to give a new way to consider stochastic nonlinear systems. Inspired by a large number of results in [25,26,27,28,29], stochastic high-order and low-order nonlinear systems with time-delay will be considered as follows:

    {dxi(t)=xpii+1(t)dt+fi(ˉxi(t),ˉxi(tτ),t)dt+gTi(ˉxi(t),ˉxi(tτ),t)dω(t),dxn(t)=upn(t)dt+fn(x(t),x(tτ),t)dt+gTn(x(t),x(tτ),t)dω(t), (1.1)

    where x(t)=[x1(t),,xn(t)]TRn is state, and u(t)R is input; the nonnegative real number τ is the time-delay of the states. ω(t)=[ω1(t),,ωr(t)]T. The high-order can be revealed by piR>1odd=:{pq|pq>0 andp,qareoddintegers}. The drift terms fi:Ri×Ri×R+R and the diffusion terms gi:Ri×Ri×R+Rr,i=1,,n are considered as locally Lipschitz with fi(0,0,t)=0 and gi(0,0,t)=0.

    The contributions are highlighted in the following:

    (i) Systems considered are more general. Systems in [24] only solve the control issues for deterministic cases. It is more complex to consider the stochastic disturbance. By using the homogeneous domination idea, one can give a novel perspective to generalize the control strategy for deterministic systems to the corresponding stochastic cases.

    (ii) The result extends the works [30,31,32] by relaxing the growth condition and the power order limitations. The low order of the nonlinear terms is successfully relaxed to the high-order and low-order of the nonlinear terms. Based on the above situations, we use a proper Lyapunov-Krasovskii functional to handle the stabilization problem under the weaker assumptions.

    Notations: R+{x|x0,xR},Rn{xn|x0}. For a given vector/matrix D, DT denotes its transpose, Tr{D} is the trace when D is square, and the Euclidean norm of a vector |D|. Ci is composed of continuous and ith partial derivable functions. K is composed of continuous functions and strictly increasing; K is composed of functions with K. One sometimes denotes X(t) by X to simplify the procedure.

    Now, the time-delay stochastic nonlinear systems are addressed as follows:

    dx(t)=f(ˉxi(t),ˉxi(tτ),t)dt+gT(ˉxi(t),ˉxi(tτ),t)dω(t). (2.1)

    {x(s):ds0}=zCbF0([d,0];Rn) is an initial data, and ω(t) denotes a Brownian motion with dimension r defined on a complete probability space (Ω,F,{Ft}t0,P).

    The following assumptions are needed:

    Assumption 1. For i=1,,n, there exist two constants a1>0 and a2>0 such that

    |fi(ˉxi(t),ˉxi(tτ),t)|a1ij=1(|xj(t)|ri+θrj+|xj(tτ)|ri+θrj)+a1i1j=1(|xj(t)|1pjpi1+|xj(tτ)|1pjpi1)+a1(|xi(t)|+|xi(tτ)|),gi(ˉxi(t),ˉxi(tτ),t)a2ij=1(|xj(t)|2ri+θ2rj+|xj(tτ)|2ri+θ2rj)+a2i1j=1(|xj(t)|12pjpi1+|xj(tτ)|12pjpi1)+a2(|xi(t)|+|xi(tτ)|), (2.2)

    in which θ=mn0, n is an odd integer, m is an even integer, and ris have the following definitions:

    r1=1,ri=ri1+θpi1,i=2,3,,n+1. (2.3)

    Remark 1. Assumption 1 encompasses and extends high-order and/or low-order results. We discuss this point from two cases.

    Case I: Condition (2.2), when τ=0 it reduces to high-order growth condition with θ0,

    |fi(ˉxi(t),t)|a1ij=1(|xj(t)|ri+θrj+|xj(t)|1pjpi1+a1|xi(t)|,gi(ˉxi(t),t)a2ij=1(|xj(t)|2ri+θ2rj+|xj(t)|12pjpi1+a2(|xi(t)|,

    and low-order growth condition with θ=0,

    |fi(ˉxi(t),t)|a1ij=1|xj(t)|1pjpi1+a1|xi(t)|,gi(ˉxi(t),t)a2ij=1|xj(t)|12pjpi1+a2(|xi(t)|.

    We further discuss its significance from value ranges of both low-order and high-order. From θ(1pjpi1,0], it is easy to see that 0<ri+θrj1pjpi1, which implies that both low-order and high-order in Assumption 1 can take any value in (0,1pjpi1],[1pjpi1,+), respectively.

    Case II: When τ0, several new results [18,19,20,21,22] have been achieved on feedback stabilization of high-order nonlinear time-delay systems. The nonlinearities in [18,19,20,21,22] only have high-order terms. The nonlinearities in [24] include linear and nonlinear parts, and their nonlinear parts only allow low-order 1pjpi1 and high-order ri+θrj with θ0.

    While in this paper, (2.2) not only includes time-delays but relaxes the intervals of low-order and high-order.

    Remark 2. When pi=1,i=1,2,,n1, and τ=0, equation (1) reduces to the well-known form, for which the feedback control problem has been well developed in recent years[16,24,26].

    Proposition 1. For r1,,rn and σ=p1pnrn+1 having the following properties:

    rkR1odd,σrkR1odd,σR1odd,σrkpk1p1R1odd.

    σmax1kn{rk+θ}.

    There hold

    441p1pk1+rk+1pkrkpk1p1,4σrk+1pkrkpk1p1+1p1pk14σrkpk1p1;
    441p1pk1+rk+1pk2rkpk1p1,4σrk+1pk2rkpk1p1+12p1pk14σrkpk1p1.

    For i=1,,k1, one has

    44rk+1pkp1ripi1p1,4σripi1p14σripi1p1.

    Remark 3. It is not difficult to see that system (1.1) is a class of high-order and low-order stochastic nonlinear systems with time-delay satisfying Assumption 1. Compared with [30], it is significant to point out that system (1.1) addressed here is more general. The systems can be composed by time-delay and the coupling of the high-order and low-order terms. Moreover, if g=0, Assumption 1 will generate the same assumption as in [24]. When pi>3, the state feedback stabilization problem under constraint pi=p can give similar results as [19]. Under Assumption 1 with τ=0, we can obtain the same results with [30], if there are no low-order nonlinearities.

    Remark 4. For the case of τ=0 in system (1.1), with the help of adding a power integrator, fruitful results have been achieved over the past years. However, for the case of τ0, some essential difficulties will inevitably be encountered in constructing the desired controller. For instance, the time-delay effect will make the common assumption on the high-order system nonlinearities infeasible, and what conditions should be placed to the nonlinearities remains unanswered. Second, due to the higher power, time-delay and assumptions on the nonlinearities, it is more complicated to find a Lyapunov-Krasovskii functional which can be behaved well in theoretical analysis.

    For ease of the controller design, some helpful definitions are presented.

    Definition 1. [19] Consider the stochastic system dx(t)=f(x,t)dt+g(x,t)dω. For any given C2 function V(x,t), the differential operator L is defined as follows:

    LV=Vt+Vxf(x,t)+12Tr{gT2V2tg},

    where 12Tr{gT2V2tg} is called the Hessian term of L.

    Definition 2. [25] There exists coordinate (x1,,xn)Rn,hi>0,i=1,,n, for arbitrarily ε>0.

    The dilation Δε(x)=(εh1x1,,εhnxn), and hi is referred to as the weights. And one defines dilation weight as =(h1,,hn).

    A function UC(Rn,R) is considered as homogeneous of degree μ, if μR, then U(Δε(x))=εμU(x1,,xn), for arbitrarily xRn{0}.

    A vector field fiC(Rn,R) is considered as homogeneous of degree μ, if μR, then fi(Δε(x))=εμ+hifi(x), for arbitrarily xRn{0}, i=1,,n.

    A homogeneous γ-norm is considered as x,γ=(ni=1|xi|γhi)1γ, for any xRn, where γ1. We use x or x,2 to a exhibit 2-norm.

    With the above definitions, we give some lemmas which will be crucial for controller design.

    Lemma 1. [13] For mR1odd, aRandbR, there hold

    (|a|+|b|)1m|a|1m+|b|1m2m1m(|a|+|b|)1m,
    |ab|m2m1|ambm|.

    Lemma 2. [13] For given a,b0 and a given positive function f(x,y), there exists a positive function g(x,y), such that

    |f(x,y)xayb|g(x,y)|x|a+b+ba+b(a(a+b)g(x,y))ab|f(x,y)|a+bb|y|a+b,x,yR.

    Lemma 3. [13] For a continuous function g, if it is monotone, and g(s)=0, then

    |tsg(x)dx||g(t)||ts|.

    Lemma 4. [19] Given τiR,i=1,,n satisfying 0τ1τn and for given nonnegative functions ai(x,y),i=1,,n, there holds

    a1(x,y)|x|τ1+an(x,y)|x|τnnj=1aj(x,y)|x|τj(|x|τ1+|x|τn)nj=1aj(x,y),x,yR.

    Consider the stochastic high-order and low-order nonlinear systems with time-delay as follows:

    {dxi=(xpii+1+fi)dt+gidω(t),i=1,,n1,dxn=upndt. (3.1)

    Step 0: To begin with, introducing the complete form of the controller,

    {zi(t)=xp1pi1i(t)αp1pi1i1(t),i=1,,n,αi(t)=ϱ1p1pii(zi(t)+zri+1pirii(t))1p1pi,i=1,,n,u(t)=αn(t). (3.2)

    The purpose of this work is to construct a state controller to render system (1.1) globally asymptotically stable in probability. To achieve this goal, propositions are presented as follows.

    Proposition 2. For c1>0, c2>0, i=1,,n, there hold

    |fi(t,ˉxi(t),ˉxi(tτ))|c1ij=1(|zj(t)|1p1pi1+|zj(t)|ri+θrjp1pi1)+c1ij=1(|zj(tτ)|1p1pi1+|zj(tτ)|ri+θrjp1pi1)gi(t,ˉxi(t),ˉxi(tτ))c2ij=1(|zj(t)|12p1pi1+|zj(t)|2ri+θ2rjp1pi1)+c2ij=1(|zj(tτ)|12p1pi1+|zj(tτ)|2ri+θ2rjp1pi1). (3.3)

    Step 1. First, we will construct a Lyapunov-candidate-function V1=x10s3ds+x10s4σr2p1r1ds+nttτ(z41(l)+z4σr11(l))dl. Along the solution of (3.1), one has

    LV1=x31(xp12+f1)+x4σr2p1r11(xp12+f1)+n(z41(t)+z4σr11(t))n(z41(tτ)+z4σr11(tτ))+Ψ1,

    where Ψ1=12Tr{gT12V1x21g1}, which leads to

    LV1=(z31+z4σr2p1r11)(xp12αp11)+(z31+z4σr2p1r11)αp11+(z31+z4σr2p1r11)f1+n(z41(t)+z4σr11(t))n(z41(tτ)+z4σr11(tτ))+Ψ1. (3.4)

    With Proposition 2, Lemma 1 and Lemma 2 in mind, one has

    (z31+z4σr2p1r11)f1c1(|z1|3+|z1|4σr2p1r1)(|z1|+|z1|r2p1r1+|z1(tτ)|+|z1(tτ)|r2p1r1)c1(|z1|4+|z1|4σr2p1r1z1+|z1|r2p1r1z31+|z1|4σr1)+c1(|z1|3|z1(tτ)|+|z1|3|z1(tτ)|r2p1r1+|z1|4σr2p1r1z1|z1(tτ)|+|z1|4σr2p1r1|z1(tτ)|r2p1r1); (3.5)

    with the help of Lemma 4, we can see it satisfies |z1|r2p1r1z31|z1|r2+θ+3r1r1|z1|4+θz41+z4σr11 when 44+θ2r2p1σ+34σr1. Similarly, one can obtain

    (z31+z4σr2p1r11)f1β1(z41+z4σr11)+(z1(tτ)4+z1(tτ)4σr1), (3.6)

    where β1=4c1+2c21+4σr2p1σ(2r2p1σ)r2p14σr2p1c2σ4σr2p11. Now, one designs the virtual controller α1 as

    αp11(x1)=(2n+β1)(z1+zr2p1r11)=ϱ1(z1+zr2p1r11), (3.7)

    where ϱ1>1 is a positive constant. Noticing that

    ϱ1z1+4σr2p1r110,ϱ1z1+r2p1r110,

    and using (4.1) and (3.7) with (3.4) after complex calculations, one finally obtains

    LV1n(z41+z4σr11)+(z31+z4σr2p1r11)(xp12αp11)(n1)(z41(tτ)+z4σr11(tτ))+Ψ1. (3.8)

    To complete the induction, at the kth step, we now define

    WLk=xkαk1(sp1pk1αp1pi1k1)41p1pi1dsWHk=xkαk1(sp1pk1αp1pi1k1)4σrk+1pkrkp1pi1dsWDk=(nk+1)ttτ(z4k(l)+z4σrkp1pi1k(l))dl.

    Lyapunov function Vk=Vk1+WLk+WHk+WDk is C2, proper and positive definite. Moreover, for i=1,,k1, WLk(),WHk(),WDk() satisfy

    WLkxk=z41p1pi1k,WHkxk=z4σrk+1pkrkp1pi1k,frac2WLkx2k=(41p1pi1)z31p1pi1k(p1pk1)xp1pk11 (3.9)
    2WHkx2k=(4σrk+1pkrkp1pi1)z4σrk+1pkrkp1pi11k(p1pk1)xp1pk11WLkxi=(41p1pi1)xkαk1(sp1pk1αp1pi1k1)31p1pi1pi1dsαp1pi1k1xi2WLkxkxi=(41p1pi1)z31p1pi1kαp1pi1k1xi,2WHkxkxi=(4σrk+1pkrkp1pi1)z4σrk+1pkrkp1pi11kαp1pi1k1xi2WLkx2i=xkαk1(41p1pi1)(31p1pi1)(sp1pk1αp1pi1k1)21p1pi1pi1ds(αp1pi1k1xi)2+xkαk1(41p1pi1)(sp1pk1αp1pi1k1)31p1pi1pi1ds(2αp1pi1k1x2i)WHkxi=(4σrk+1pkrkp1pi1)xkαk1(sp1pk1αp1pi1k1)4σrk+1pkrkp1pi11dsαp1pi1k1xi2WHkx2i=xkαk1(4σrk+1pkrkp1pi1)(4σrk+1pkrkp1pi11)(sp1pk1αp1pi1k1)4σrk+1pkrkp1pi12ds(αp1pi1k1xi)2+xkαk1(4σrk+1pkrkp1pi1)(sp1pk1αp1pi1k1)4σrk+1pkrkp1pi11ds(2αp1pi1k1x2i). (3.10)

    Step k (k = 2, 3, …, n): As in step k-1, there exists Lyapunov-candidate-function Vk1, implying

    LVk1(nk+2)k1i=1(z4i+z4σripi1p1i)(nk+1)k1i=1(z4i(tτ)+z4σripi1p1i(tτ))+(z41p1pk2k1+z4σrkpk1rk1pk2p1k1)(xpk1kαpk1k1)+Ψk1, (3.11)

    where Ψk1= 12Tr{ˉψTk1 2Vk1ˉx2k1 ˉψk1}, ˉψk1 = (g1, , gk1). Hence, one will consider Vk=Vk1+WLk+WHk+WDk and define an appropriate virtual controller αk. Similar to step 1, one can obtain

    LVk(nk+2)k1i=1(z4i+z4σripi1p1i)(nk+1)k1i=1(z4i(tτ)+z4σripi1p1i(tτ))+(nk+1)(z4k+z4σrkpk1p1k)+(z41p1pk1k+z4σrkpk1rkpk1p1k)(xpk1k+1αpkk)+(z41p1pk1k+z4σrkpk1rkpk1p1k)αpkk+(z41p1pk1k+z4σrkpk1rkpk1p1k)fk+(z41p1pk2k1+z4σrkpk1rk1pk2p1k1)(xpk1kαpk1k1)+k1i=1(WLkxi+WHkxi)(xpii+1+fi)+Ψk, (3.12)

    where Ψk=12Tr{ˉψTk2Vkˉx2kˉψk}, ˉψk=(g1,,gk). Obviously, the virtual controller αk is used to eliminate the last three terms of (3.12). In light of (3.2) and Lemma 1, it yields that

    xpk1kαpk1k1|(xp1pk1k)1p1pk2(αp1pk1k1)1p1pk2|231p1pk2|zk|1p1pk2.

    In the case of 4σθ4σ, by Lemma 2, one obtains that

    (z41p1pk2k1+z4σrkpk1rk1pk2p1k1)(xpk1kαpk1k1)211p1pk2|zk|1p1pk2(|zk1|41p1pk2+|zk1|4σrkpk1rk1pk2p1)βk1(z4k+z4σrkpk1p1k)+13(z4k1+z4σrk1pk2p1k1), (3.13)

    where βk1 denotes a positive constant. On the basis of Proposition 2 and Lemma 3, one has

    (z41p1pk2k1+z4σrkpk1rk1pk2p1k1)fk12k2i=1(z4i+z4σripi1p1i)+13(z4k1+z4σrk1pk2p1k1)+12k1i=1(z4i(tτ)+zi(tτ)4σripi1p1)+z4k(tτ)+zk(tτ)4σrkpk1p1+βk2(z4k+z4σrkpk1p1k), (3.14)

    where βk2 denotes a positive constant. In the sequel, one estimates the last term. With the help (3.2), Lemmas 2 and 4, it is not hard to achieve

    xkαk1(sp1pk1αp1pk1k1)31p1pk1ds|zk|31p1pk1|xkαk1|231p1pk1|zk|. (3.15)

    Similarly, one can obtain

    xkαk1(sp1pk1αp1pk1k1)4σrk+1pkrkpk1p11ds231p1pk1|zk|4σθrkpk1p11. (3.16)

    On the basis of the previous inequality, one has

    (WLkxi+WHkxi)(xpii+1+fi)λk(|zk|+|zk|4σθrkpk1p11)|αp1pk1k1xi|(xpii+1+fi)dki(z4k+z4σrkpk1pk1k)+12(k1)(k2j=1(z4j+z4σrjp1pj1j)+13(k1)(z4k1+z4σrk1p1pk2k1)+12(k1)k1j=1(z4j(tτ)+z4σrjp1pj1j(tτ)), (3.17)

    in which dki denotes a positive constant. Define βk=βk1+βk2+βk3 with βk3=k1i=1dki and choose the virtual controller αk as

    αp1pkk(Xk)=ϱk(zk+zrk+1pkrkk). (3.18)

    By Lemma 2, one can arrive at

    (z41p1pk1k+z4σrk+1pkrkpk1p1k)αpkk(2(nk+1)+βk)(z4k+z4σrkpk1p1k). (3.19)

    Substituting (3.13)–(3.18) into (3.12) yields

    LVk(nk+1)ki=1(z4i+z4σripi1p1i)(nk)ki=1(z4i(tτ)+z4σripi1p1i(tτ))+(z41p1pk1k+z4σrk+1pkrkpk1p1k)(xpkk+1αpkk)+Ψk.

    It is shown that the above formula holds for k=n with virtual controllers (3.18). Similarly, we choose Vn(x)=ni=1(WLi()+WHi()+WDi()). There is an actual control law

    u(x)=ϱ1p1pnn(zn+zrn+1pnrnn)1p1pn, (3.20)

    such that

    LVkni=1(z4i+z4σripi1p1i). (3.21)

    Until now, the recursive design has been completed. Under the new coordinates

    ξ1=x1,ξi=xiLki,νp=upLkn+1, (3.22)

    where k1=0, ki=ki1+1pi1, i=2,,n and L>1 is a constructed constant, system (1.1) can be rewritten in the form

    dξi=Lξpii+1dt+fi()Lkidt+gi()Lkidω(t),dξn=Lνpnndt+fn()Lkndt+gn()Lkndω(t). (3.23)

    By (3.18) and (3.20), the system (3.1) can be integrated into the complex format

    dξ=LR(ξ)dt+T(t,ξ,ξ(tτ))dt+ψT(t,ξ,ξ(tτ))dω(t), (3.24)

    where ξ=(ξ1,,ξn)T, R(ξ)=(ξp2, ,ξpn, νp)T, T(t,ξ, ξ(tτ))=(f1, f2Lk2,, fnLkn)T, ψ(t,ξ,ξ(tτ))=(g1, g2Lk2,, gnLkn). Introducing the dilation weight =(r1,r2,,rn), one gets

    Vn(ε(ξ))=ni=1εrixiεriαi1(sp1pi1εripi1p1αp1pi1i1)41p1pi1ds+ni=1εrixiεriαi1(sp1pi1εripi1p1αp1pi1i1)4σrk+1pkrkpk1p1ds+(nk+1)ttτ(z4k(s)+z4σrkpk1p1k(s))ds=ni=1xiαi1(εripi1p1(ζp1pi1αp1pi1i1))41p1pi1εripi1p1dζ+ni=1εrixiαi1(sp1pi1εripi1p1αp1pi1i1)4σrk+1pkrkpk1p1εripi1p1dζ=ε4σθVn(ξ), (3.25)

    where s is defined as s=riζ. With the help of the above formula and Definition 2, it can be concluded that Vn(ξ) is homogeneous of degree 4σθ.

    The main result of this manuscript will be stated as follows.

    Theorem 1. Suppose Assumptions 1 apply to stochastic system (1), under the state feedback controller up=Lkn+1νp and (3.20), then:

    (i) There exists a unique solution on [d,);

    (ii) The equilibrium at the origin is globally asymptotically stable in probability.

    Proof. Four steps are used to verify Theorem 1.

    Step 1: By the definition of ϱ>0, we know that p1pj11>1, which implies that 41p1pk11>2, 4σrk+1pkrkpk1p1>2. Therefore, αp1pii(t)xj(t) is continuous, and upn=Lkn+1νpn is C. As is known to all, the function is C. The closed-loop system satisfies the locally Lipschitz condition based on fi and gi being locally Lipschitz.

    Step 2: Consider the Lyapunov-candidate-function:

    V(ξ)=Vn(ξ)+ni=1h1+h21γttτξ4σdη, (3.26)

    where h1 and h2 are positive parameters. It is straightforward to prove that V(ξ) is C2 on ξ. Since Vn(ξ) is continuous, positive definite and radially unbounded, from Lemma 1, one can have

    α20(|ξ|)V(ξ)α21(|ξ|), (3.27)

    where α20 and α21 are K functions. With the help of the homogeneous theory, one finally has

    ˉc0ξi4σΔU(ξ)c_0ξi4σΔ, (3.28)

    in which ˉc0>0, c_0>0, and U(ξ) denotes a positive definite function of the 4σ homogeneous degree. Hence, one has the formula

    α20(|ξ|)U(ξ)α21(|ξ|). (3.29)

    (3.29) leads to

    h1+h21γttτξ4σdη˜cttτˉα22(|ξi|)dη˜c0τα21(|ξi(t+s)|)d(t+s)csupτs0ˉα22(|ξi(s+t)|)α22(supτs0|ξi(s+t)|), (3.30)

    where η=s+t, ˜c>0, c>0 and α22 is a class K function. Since

    |ξ|(supτs0|ξ(s+t)|),α21|ξ|α21(supτs0|ξ(s+t)|).

    Defining β2=α21+α22, by (3.26)-(3.30), one gets

    β1(|ξ|)V(|ξ|)β2(supτs0|ξ(s+t)|).

    Step 3: With the help of Lemma 1 and (3.20), c01 is a positive constant, and one has

    Vn(ξ)ξLR(ξ)c01Lξ4σΔ. (3.31)

    By Proposition 2 and L>1, one can have

    |fi(t,ˉξ(t),ˉξ(tτ))Lki|ˉδ1L1γi1(ij=1|ξ(t)|ri+θrj+ij=1|ξ(tτ)|ri+θrj)δ1L1γi1(ξ(t)ri+θ+ξ(tτ)ri+θ), (3.32)

    in which ˉδ1,δ1>0. With the help of Lemmas 1, 2 and (3.32), one can obtain

    |Vnξ(t)T(t,ξ(t),ξ(tτ))|˜c02L1¯γ0(ni=1ξ(t)4σriθξ(t)ri+θ+ij=1ξ(tτ)4σriθξ(tτ)ri+θ)L1¯γ0(ˉc02ξ(t)4σ+ˉc02ξ(t)4σξ(tτ)4σ), (3.33)

    where c02, ˉc02, ˜c02 and ¯γ0=min1inγi1 are positive constants. Similar to (3.32), we use δ2 and γi2<1/2 to show that

    |gi(t,ˉξ(t),ˉξ(tτ))Lki|1Lkic2ij=1(|zj(t)|12p1pi1+|zj(t)|2ri+θ2rjp1pi1)+1Lkic2c2ij=1(|zj(tτ)|12p1pi1+|zj(tτ)|2ri+θ2rjp1pi1)L12γi2(ξ(t)+ξ(tτ))ri+θ2δ2L12γi2(ξ(t)ri+θ2+ξ(tτ)ri+θ2).

    Using Lemma 1, Lemma 3, Lemma 4 and (3.34), one obtains

    12Tr{ψ(t,ξ(t),ξ(tτ))2Vnξ2ψT(t,ξ(t),ξ(tτ))}12rrni,j=1|2Vnξ2||ψT(t,ξ(t),ξ(tτ))||ψ(t,ξ(t),ξ(tτ))|˜c03L1¯γ0ni,j=1ξ(t)4σrirjθ×(ξ(tτ)ri+θ2+ξ(t)ri+θ2)×(ξ(t)rj+θ2+ξ(tτ)rj+θ2)˜c03L1¯γ0(c03ξ(t)4σ+˜c03ˉc03ξ(t)4σξ(tτ)4σ)L1¯γ0(c03ξ(t)4σ+ˉc03ξ(t)4σξ(tτ)4σ), (3.34)

    in which ˜γ0=min1i,jn{γi2+γj2}>0, c03>0, ˉc03>0 and ˜c03>0 are constants. Based on L>1, we have

    V(ξ)Vn(ξ)+h1+h21γL1γ0ttτξ4σΔdη.

    By Definition 1, (3.26), (3.31), (3.33) and (3.34), one has

    LVVnξLR(ξ)+Vn(ξ)ξT(t,ξ(t),ξ(tτ)+12Tr{ψT(t,ξ(t),ξ(tτ))2Vnξ2ψ(t,ξ(t),ξ(tτ))}+(h1+h2)L1γ0(11γξ(t)4σξ(tτ)4σ)L(c01(c02+c03+h1+h21γ)Lγ0)ξ4σ, (3.35)

    which satisfies γ0=min{ˉγ0, ˜γ0}<1. Because c01 is a constant independent of c02,c03, we choose L>L=max{(c02+c03+h1+h21γc1)1γ0,1}, and there exists a constant B=c01(c02+c03+h1+h21γ)Lγ0>0, such that

    LVLBξ4σ=c0|ξ4σ.

    With the help of the above formula and (3.28), one obtains

    LV(ξ(t))(c0˜c)α_22|(|ξ(t)|).

    Briefly, following Steps 13, the system has a unique solution on [d,], and ξ(t)=0 is globally asymptotically stable in probability.

    Step 4: Because (3.20) is an equivalent transformation, the system composed by (1) and up=Lkn+1vp is similar to the systems (3.20) and (3.22).

    Remark 5. Compared with [24], we construct a state-feedback controller independent of time delays for the stochastic nonlinear system. Compared with [30], we use the methods of adding a power integrator to relax the nonlinear growth condition to cover both high-order and low-order nonlinearities. Not only does it not need to know anything information about the unknown function, but also it can reduce burdensome computations.

    Remark 6. The homogeneous domination method is used for the first time to solve the stabilization problem of stochastic high-order and low-order nonlinear system (1.1) with time-delay.

    Remark 7. In this paper, it is hard to adopt a Lyapunov-Krasovskii functional. In order to solve the above the problem, a suitable Lyapunov-candidate-function is designed to guarantee good system performance, and stabilization analysis is proposed to save better resources

    Remark 8. The construction of the controller effectively keeps away from the zero-division problem of 2ξμ/riiξ2j. It need be noted that the non-zero-division problem and the locally Lipschitz condition (see Step 1 in the proof of Theorem 1) should to be guaranteed simultaneously, which will increase more difficulties.

    Consider the following stochastic high-order and low-order nonlinear systems with time-delay:

    {dx1(t)=[x32(t)+x22(t)x1(t1)]dt+14x1(t)sinx1(t1)dω(t),dx2(t)=[u3(t)+x2(t)cosx2(t1)]dt. (4.1)

    One can see that Assumption 1 is satisfied with p1=p2=3,τ=1,C=1,r1=1,θ=25. One can easily get

    |f1|(|z1|+|z1|75+|z1(t1)|+|z1(t1)|75)/5,
    |g1|(|z1|+|z1|165+|z1(t1)|+|z1(t1)|165)/8,
    |f2|(|z1|13+|z1|1345+|z2|13+|z2|136+|z1(t1)|13+|z1(t1)|1345+|z2(t1)|13+|z2(t1)|136)/5. (4.2)

    In this simulation, we choose V1(z1)=15z51+110z101+2tt1(z41+z4σr11)dl. Several calculations lead to

    LV12(z41+z5251)(z41(t1)+z1(t1)525)+(z1+z91)(x32α31), (4.3)

    where αp1=(2n+β1)(z31+z751). By choosing V2(¯η2)=V1(η1)+WL2+WH2+WD2, a direct calculation leads to

    LV2(z41+z5251)(z42+z26072). (4.4)

    From the previous manipulations, one obtains the following actual controller

    u(t)=ϱ19(z2+z13452)19=306.771(z2+z13452)19. (4.5)

    The initial condition can be given as ξ0(θ)[1,1]T. Figure 1 illustrates that the globally asymptotically stable in probability has been achieved and the responses of (4.7) is given in Figure 2.

    Figure 1.  The trajectories of x1(t) and x2(t).
    Figure 2.  The trajectories of u.

    In this technical note, we investigate the state feedback stabilization problem of stochastic high-order and low-order nonlinear systems with time-delay successfully. According to the homogeneous domination method and the design of integral Lyapunov functions, the control strategy is achieved with the controller design. The above results indicate that the closed-loop system is globally asymptotically stable in probability. There still remain problems to be investigated, such as how to take into account output feedback control and how to extend our results under weaker conditions.

    This work is supported by National Natural Science Foundation of China under Grant 62173208, Taishan Scholar Project of Shandong Province of China under Grant tsqn202103061, Shandong Qingchuang Science and Technology Program of Universities under Grant 2019KJN036.

    The authors declare no conflicts of interest.



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