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Research article

On definition of solution of initial value problem for fractional differential equation of variable order

  • Received: 27 December 2020 Accepted: 13 April 2021 Published: 22 April 2021
  • MSC : 26A33

  • We propose a new definition of continuous approximate solution to initial value problem for differential equations involving variable order Caputo fractional derivative based on the classical definition of solution of integer order (or constant fractional order) differential equation. Some examples are presented to illustrate these theoretical results.

    Citation: Shuqin Zhang, Jie Wang, Lei Hu. On definition of solution of initial value problem for fractional differential equation of variable order[J]. AIMS Mathematics, 2021, 6(7): 6845-6867. doi: 10.3934/math.2021401

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  • We propose a new definition of continuous approximate solution to initial value problem for differential equations involving variable order Caputo fractional derivative based on the classical definition of solution of integer order (or constant fractional order) differential equation. Some examples are presented to illustrate these theoretical results.



    The optimal control problems (OCPs) of partial differential equations have been extensively studied in numerous fields of science and engineering applications, including fluid mechanics, earth science, petroleum engineering, telecommunication, etc., (see [1,2,3]). In the past few decades, scholars have conducted extensive research on OCPs (see [4,5,6]). From the perspective of control types, there is distributed control and boundary control (see [7,8,9]). With respect to the types of state equations, there are elliptic equations, parabolic equations, and second-order hyperbolic equations (SOHEs) [10]. Among the common numerical discretization schemes, there are finite element methods (FEM) [11], mixed element methods (MFVM) [12], and finite volume methods (FVM)[13]. In terms of handling optimization problems, there are two approaches: optimize-then-discretize and discretize-then-optimize [14].

    The OCPs constrained by SOHEs is an active area of research, attracting significant attention from numerous scholars. Gugat et al. in [15] proposed a valid method based on Lavrentiev regularization and obtained a result similar to the penalty function method for second-order hyperbolic optimal control problems (SOHOCPs) with state constraint. Kröner in [16] used the space-time FEM to discretize SOHOCPs and derived a posteriori error estimates, which separately considers the influence of space, time, and control. In [17], Kröner et al. analyzed the convergence of three types of controls constrained by wave equations utilizing the semismooth Newton method, and numerically implemented the solution in conjunction with the space-time FEM. Lu et al. derived a priori error estimates using the mixed finite element method for a general SOHOCPs in [18]. Luo et al. in [19,20] studied linear SOHOCPs using the FVM and obtained priori error estimates for the Euler and Crank-Nicolson schemes. Lu et al. in [21] used the FVM to study nonlinear SOHOCPs and obtained optimal error estimates for a semi-discrete system. Li et al. in [22] used the FEM and variational discretization approach to investigate linear SOHOCPs by introducing an intermediate variable and obtained optimal priori error estimates.

    Inspired by [21,22], we consider the following nonlinear SOHOCPs:

    minuUadJ(y,u)=12T0Ω|yyd|2dxdt+α2T0Ω|u|2dxdt, (1.1)

    such that

    {yttdiv(Ay)+ϕ(y)=f+Bu,xΩ,t(0,T),y(x,t)=0,xΩ,t(0,T),y(x,0)=y0(x),yt(x,0)=g(x),xΩ, (1.2)

    where ΩR2 is a bounded convex polygon domain with boundary Ω. α is a positive number. T>0 is a constant. f,ydL2(0,T;L2(Ω)) are given functions. ϕ(y) is a nonlinear function that satisfies ϕ()C2. For any R>0 and yH1(Ω), the function ϕ()W2,(R,R), ϕ()L2(Ω), and ϕ()0. A=(aij(x))2×2(W1,(ˉΩ))2×2 is symmetric and uniformly positive definite, i.e., for any XR2, there exist two positive constants C1,C2 such that

    0<C1XTXXTA(x)XC2XTX<+,xΩ.

    B:L2(0,T;L2(Ω))L2(0,T;L2(Ω)) is a bounded linear operator. The space Uad is defined by

    Uad={uL2(0,T;L2(Ω)):uau(x,t)ub,a.e.inΩ×(0,T]}.

    In this work, by introducing a new variable, we transform the hyperbolic equation into two parabolic equations. For the SOHOCPs (1.1) and (1.2), we obtain the continuous first-order necessary condition (FNC) and the second-order sufficient optimality condition (SSC). Using the discretize-then-optimize procedure, we derive the discrete optimality condition for the fully discrete scheme. Based on these, we obtain some optimal error estimates.

    The paper is organized as follows. We give some notations in Section 2. In Section 3, we derive the first and second-order optimality conditions. The Crank-Nicolson finite element approximation and a priori error estimates are presented in Section 4. In Section 5, a numerical experiment is presented to confirm the validity of the proposed numerical scheme.

    After this, C represents different positive constants in different places, each of which is independent of h and Δt.

    Here, we first introduce some notations. Wm,p(Ω) is a standard Sobolev space with norm ||||Wm,p(Ω). W1,2(Ω)=H1(Ω) with norm ||||1, and H10(Ω)={υH1(Ω):υ|Ω=0}. Lk(Ω) denotes a k-squared integrable function space in region Ω with norm ||||. The norm of Lk(Ω) is denoted by ||||0,U. We denote by Lk(0,T;Wm,p(Ω)) the Banach space of all Lk integrable functions from (0,T) into Wm,p(Ω) with norm ||υ||Lk(Wm,p)=||υ||Lk(0,T;Wm,p(Ω))=(T0||υ||kWm,p(Ω))1k for k[1,) and the standard modification for k=.

    For a positive integer N, define time step size Δt by Δt=TN. For n=0,1,...,N1, tn=nΔt, In+1=[tn,tn+1]. write

    dtζn+1=ζn+1ζnΔt,ˉdtζn+1=ζnζn+1Δt,

    and Q=Ω×(0,T]. For any given sequence {ζn}Mn=0, ζn=ζ(x,tn) for the function ζ(x,t) defined in Q. For 1q, a discrete time-dependent norm is given by

    ||υ||lq(Wm,p):=||υ||lq(0,T;Wm,p)=(Nn=1Δt||υn||qWm,p)1q,

    and the standard modification for q=, where

    lq(Wm,p):={υ:||υ||lq(0,T;Wm,p))<}.

    The inner product is noted by

    (υ1,υ2)=Ωυ1υ2dx,υ1,υ2L2(Ω).

    For convenience, we take A to be the identity matrix and write

    a(υ1,υ2)=ΩAυ1υ2dx=(υ1,υ2),υ1,υ2H10(Ω).

    It is obvious that

    a(υ,υ)c1||υ||21,|a(υ1,υ2)|c2||υ1||1||υ2||1,υ,υ1,υ2H10(Ω).

    Let Th be a quasi-uniform triangulation of Ω, hK denotes the diameter of element K, and h=maxKTh{hK}. The space Vh associated with Th is defined by

    Vh:={νh|νhC(ˉΩ),νh|KP1(K),KTh},

    where P1(K) denotes the polynomials space with the degree being no more than one on KTh.

    We consider the following piecewise constant finite element space

    Uh:={uhL2(0,T;Ω):uh|Kisconstant,KTh},

    which is a finite dimensional subspace of Uad.

    For any μU, we define the orthogonal projection operator Πh:UUh such that

    (wh,μΠhμ)=0,whUh. (2.1)

    By the definition of Eq (2.1), we have

    Πhμ|K=1|K|Kμ,KTh, (2.2)

    where |K| is the measure of K. For the operator Πh defined in Eq (2.1), we have

    |μΠhμ|0,p,KCh|μ|1,p,K, (2.3)

    for all μW1,p(Ω) and 1p (see [23]).

    For t(0,T], define L2 projection Rhυ(t)Vh for υ(t)V by

    (υ(t)Rhυ(t),νh)=0,νhVh. (2.4)

    As in [24], for 1r2, the Rhυ(t) satisfies

    ||υRhυ(t)||l2(L2)+h||υRhυ(t)||l2(H1)Chr||υ||l2(Hr), (2.5)
    ||(υRhυ(t))t||L2(L2)+h||(υRhυ(t))t||L2(H1)Chr||υ||H1(Hr). (2.6)

    In this section, we first give the weak form of the state equation (1.2) as: We seek y(,t)H10(Ω) such that

    {(ytt,v)+(y,v)+(ϕ(y),v)=(f+Bu,v),vH10(Ω),t(0,T],y(x,0)=y0(x),ω(x,0)=g(x),xΩ. (3.1)

    Next, we introduce a new variable ω=yt, then the problem (1.1) and (1.2) can be written as

    minuUadJ(y,u)=12T0Ω|yyd|2dxdt+α2T0Ω|u|2dxdt, (3.2)

    subject to

    {(ω,v)=(yt,v),vH10(Ω),t(0,T],(ωt,v)+(y,v)+(ϕ(y),v)=(f+Bu,v),vH10(Ω),t(0,T],y(x,0)=y0(x),ω(x,0)=g(x),xΩ. (3.3)

    The problem (3.2) and (3.3) is formulated in standard reduced functional form as

    {minJ(u)uUad. (3.4)

    Since the problem (3.4) is non-convex, we cannot guarantee the global unique solutions of (3.4). Therefore, we consider the local optimal solutions (see [25,26,27]).

    Definition 3.1. (Local solution [28]). The control ˉuUad is called a local solution of the problem (3.4) if for each fixed t[0,T], there exists a constant ι>0, such that for all uUad with ||uˉu||<ι, it satisfies

    J(u)J(ˉu). (3.5)

    For the problem (3.4), the existence of the local solution can be guaranteed in [10]. Next, we can derive the following FNC for the local solution ˉu.

    Theorem 3.1. If ˉu is a local optimal control for the problem (3.4), then there exists a set of functions (ω(t),y(t),ˉu(t),q(t),p(t))(H1(L2)L2(H1))×(H2(L2)L2(H1))×Uad×(H1(L2)L2(H1))×(H2(L2)L2(H1)) such that

    {(ω,v1)=(yt,v1),v1H10(Ω),t(0,T],(ωt,v2)+(y,v2)+(ϕ(y),v2)=(f+Bˉu,v2),v2H10(Ω),t(0,T],y(x,0)=y0(x),ω(x,0)=g(x),xΩ, (3.6)
    {(q,v1)=(pt,v1),v1H10(Ω),t(0,T],(qt,v2)+(q,v2)+(ϕ(y)p,v2)=(yyd,v2),v2H10(Ω),t(0,T],p(x,T)=0,q(x,T)=0,xΩ, (3.7)
    T0(αˉu+Bp,vˉu)dt0,vUad, (3.8)

    where B is the adjoint operator of B.

    Proof. Applying the variational rule, the optimal condition reads

    J(ˉu)(vˉu)=T0(yyd,Dˉuy(vˉu))dt+αT0(ˉu,vˉu)dt0, (3.9)

    where

    Dˉuy(vˉu)=limt0y(ˉu+t(vˉu))y(ˉu)t.

    Next, differentiating the state equation (3.6) at ˉu in the direction ϱ, we have

    (ωDˉuy(ϱ),v1)=(Dˉuy(ϱ)t,v1), (3.10)
    (ωtDˉuy(ϱ),v2)+(Dˉuy(ϱ)),v2)+ϕ(y)Dˉuy(ϱ),v2)=(Bϱ,v2), (3.11)
    Dˉuy(ϱ)(t=0)=0,  ωDˉuy(ϱ)(t=0)=0. (3.12)

    Defining the co-state (p,q) satisfying Eq (3.7), and letting v1=ωDˉuy(ϱ),v2=Dˉuy(ϱ) in Eq (3.7), we can obtain

    (q,ωDˉuy(ϱ))=(pt,ωDˉuy(ϱ)), (3.13)
    (qt,Dˉuy(ϱ))+(p,Dˉuy(ϱ))+(ϕ(y)p,Dˉuy(ϱ))=(yyd,Dˉuy(ϱ)), (3.14)

    Meanwhile, letting v1=q in Eq (3.10) and v2=p in Eq (3.11), we get

    (ωDˉuy(ϱ),q)=(Dˉuy(ϱ)t,q), (3.15)
    (ωtDˉuy(ϱ),p)+(Dˉuy(ϱ),p)+(ϕ(y)Dˉuy(ϱ),p)=(Bϱ,p). (3.16)

    Since ωDˉuy(ϱ)(t=0)=0,Dˉuy(ϱ)(t=0)=0,q(T)=0,p(T)=0, integrating by parts and a direct calculation, we can yield

    T0(Dˉuy(ϱ)t,q)dt=Dˉuy(ϱ)q|T0T0(Dˉuy(ϱ),qt)dt=T0(Dˉuy(ϱ),qt)dt, (3.17)
    T0(ωtDˉuy(ϱ),p)dt=ωDˉuy(ϱ)p|T0T0(ωDˉuy(ϱ),pt)dt=T0(ωDˉuy(ϱ),pt)dt. (3.18)

    Substituting Eqs (3.17) and (3.18) into Eqs (3.15) and (3.16), we derive

    (ωDˉuy(ϱ),q)=(Dˉuy(ϱ),qt), (3.19)
    (ωDˉuy(ϱ),pt)+(Dˉuy(ϱ),p)+(ϕ(y)Dˉuy(ϱ),p)=(Bϱ,p), (3.20)

    Combining Eqs (3.19) and (3.20) with Eqs (3.13) and (3.14) yields

    (ϱ,Bp)=(Bϱ,p)=(yyd,Dˉuy(ϱ)). (3.21)

    Substituting Eq (3.21) into Eq (3.9) implies

    J(ˉu)(vˉu)=T0(αˉu+Bp,vˉu)dt0,vUad. (3.22)

    This completes the proof of Theorem 3.1.

    According to [26] and [29], we can know that for the non-convex problem (3.4), the FNC is not sufficient. So, we need to consider the SSC:

    (SSC) There exist constants κ>0 and λ>0 such that

    J(ˉu)(v,v)κ||v||2L2(L2(Ω), (3.23)

    for all vL2(0,T;L2(Ω)) satisfying

    v{=0,if|αu+Bp|λ>0,0,ifˉu=ua,0,ifˉu=ub, (3.24)

    where

    J(ˉu)(v,v)=T0(Dˉuy(v),Dˉuy(v))dt+T0(y(ˉu)yd,D2ˉuy(v,v)dt+αT0(v,v)dt, (3.25)

    and the notation D2ˉuy(v,v) is defined as follows: Let ˜y=Dˉuy(v). Then, its directional derivative in the direction v, denoted as ˜y(v), is given by

    D2ˉuy(v,v)=Dˉu˜y(v)=limt0˜y(ˉu+tv)˜y(ˉu)t.

    More details about the notation D2ˉuy(v,v) can be found in [2].

    Referring to [30], we give the coercive property of the problem (3.4) in a neighborhood of the local solution by the following lemma.

    Lemma 3.1. Assume that ˉu is a local solution of the problem (3.4) and satisfies the SSC (3.23). There exists a sufficient small constant ι>0 such that

    J(ˉu)(v,v)κ2||v||2L2(L2(Ω), (3.26)

    for all uUad with ||uˉu||<ι.

    In this section, we develop a fully discrete Crank-Nicolson scheme for the optimality system (3.2)–(3.3) as follows:

    Jδ(yn+12h,un+12h)=ΔtN1n=0[12||yn+12hyn+12d||20,Ω+α2||un+12h||20,Ω], (4.1)
    (ωn+12h,v)=1Δt(yn+1hynh,v),vVh (4.2)
    1Δt(ωn+1hωnh,v)+(yn+12h,v)+(ϕ(yn+12h),v)=(fn+12+Bun+12h,v),vVh (4.3)
    y0h=Rhy0(x),ω0h=Rhg(x),xΩ, (4.4)

    where δ=δ(h,Δt) denotes the fully discrete in space and time. We can reformulate the problem (4.1)–(4.4) as

    {minJδ(un+12h)un+12hUh. (4.5)

    We also give the definition of the local solution of the discrete control problem (4.1)–(4.4).

    Definition 4.1. The control ˉun+12hUh is called a fully discrete local solution of the problem (4.1)–(4.4) if for each fixed tn+12, there exists a constant ι>0, such that for un+12hUh with ||un+12hˉun+12h||<ι, it satisfies

    Jδ(un+12h)Jδ(ˉun+12h). (4.6)

    We present the first-order necessary optimality condition for problem (4.1)–(4.4) at the local solution ˉun+12h by the following theorem.

    Theorem 4.1. Assume that ˉun+12h,n=0,1,...,N1 is a local solution of discrete control problem (4.1)–(4.4) , then there are state (ωn+12h,yn+12h)Vh×Vh and co-state (qn+12h,pn+12h)Vh×Vh, n=0,1,...,N1, such that the following optimality conditions hold:

    (ωn+12h,v)=1Δt(yn+1hynh,v),vVh, (4.7)
    1Δt(ωn+1hωnh,v)+(yn+12h,v)+(ϕ(yn+12h),v)=(fn+12+Bˉun+12h,v),vVh, (4.8)
    y0h=Rhy0(x),ω0h=Rhg(x),xΩ, (4.9)
    (qn+12h,v)=1Δt(pn+1hpnh,v),vVh, (4.10)
    1Δt(qn+1hqnh,v)+(pn+12h,v)+(ϕ(yn+12h)pn+12h,v)=(yn+12hyn+12d,v),vVh, (4.11)
    pNh(x)=0,qNh(x)=0,xΩ. (4.12)
    Jδ(ˉun+12h)(vhˉun+12h)=ΔtN1n=0(αˉun+12h+Bpn+12h,vhˉun+12h)0,vhUh. (4.13)

    Proof. Differentiate the Eq (4.5) at ˉun+12h in the direction vhˉun+12h, and the discrete optimal condition reads

    Jδ(ˉun+12h)(vhˉun+12h)=limt0Jδ(ˉuδ+t(vhˉuδ))Jδ(ˉuδ)t=ΔtN1n=0(yn+12hyn+12d,Dyn+12h(vhˉun+12h))+αΔtN1n=0(ˉun+12h,vhˉun+12h)0. (4.14)

    Similarly, differentiating the Eqs (4.7) and (4.8) at ˉun+12h in the direction ν, we have

    (ω(yn+12h)Dyn+12h(ν),v)=1Δt(Dyn+1h(ν)Dynh(ν),v), (4.15)
    1Δt(ω(yn+1h)Dyn+1h(ν)ω(ynh)Dynh(ν),v)+(Dyn+12h(ν),v)+(ϕ(yn+12h)Dyn+12h(ν),v)=(Bν,v), (4.16)
    Dy0h(ν)=0,ω(y0h)Dy0h(ν)=0. (4.17)

    where

    Dyn+12h(ν)=limt0yn+12h(ˉun+12h+tv)yn+12h(ˉun+12h)t.

    Choosing the discrete co-state (pδ,qδ) satisfying Eqs (4.10)–(4.12), and selecting v=ω(yn+12h)Dyn+12h(ν) in Eq (4.10) and v=Dyn+12h(ν) in Eq (4.11), we get

    (qn+12h,ω(yn+12h)Dyn+12h(ν))=1Δt(pn+1hpnh,ω(yn+12h)Dyn+12h(ν)), (4.18)
    1Δt(qn+1hqnh,Dyn+12h(ν))+(pn+12h,Dyn+12h(ν))+(ϕ(yn+12h)pn+12h,Dyn+1h(ν))=(yn+12hyn+12d,Dyn+12h(ν)). (4.19)

    Taking v=qn+12h in Eq (4.15) and v=pn+12h in Eq (4.16), we have

    (ω(yn+12h)Dyn+12h(ν),qn+12h)=1Δt(Dyn+1h(ν)Dynh(ν),qn+12h), (4.20)
    1Δt(ω(yn+1h)Dyn+1h(ν)ω(ynh)Dynh(ν),pn+12h)+(Dyn+12h(ν),pn+12h)+(ϕ(yn+12h)Dyn+12h(ν),pn+12h)=(Bν,pn+12h), (4.21)

    Since Dy0h(ν)=0,ω(y0h)Dy0h(ν)=0,pNh=0,qNh=0, we know that

    N1n=0(ω(yn+1h)Dyn+1h(ν)ω(ynh)Dynh(ν),pn+12h)=N1n=0(ω(yn+1h)Dyn+1h(ν)ω(ynh)Dynh(ν),pnh+pn+1h2)=12N1n=0[(ω(yn+1h)Dyn+1h(ν),pnh+pn+1h)(ω(ynh)Dynh(ν),pnh+pn+1h)]=12N1n=0[(ω(yn+1h)Dyn+1h(ν),pnhpn+1h)+(ω(yn+1h)Dyn+1h(ν),pn+1h)(ω(ynh)Dynh(ν),pn+1hpnh)(ω(ynh)Dynh(ν),pnh)]=N1n=0(ω(yn+12h)Dyn+12h(ν),pnhpn+1h), (4.22)

    and

    N1n=0(Dyn+1h(ν)Dynh(ν),qn+12h)=N1n=0(qn+1hqnh,Dyn+12h(ν)). (4.23)

    Substituting Eqs (4.22) and (4.23) into Eqs (4.20) and (4.21), we obtain

    (ω(yn+12h)Dyn+12h(ν),qn+12h)=1Δt(qn+1hqnh,Dyn+12h(ν)), (4.24)
    1Δt(ω(yn+1h)Dyn+1h(ν),pn+1hpnh)+(Dyn+12h(ν),pn+12h)+(ϕ(yn+12h)Dyn+12h(ν),pn+12h)=(Bν,pn+12h). (4.25)

    Combining Eqs (4.24) and (4.25) with Eqs (4.18) and (4.19), and summing from 0 to N1 leads to

    N1n=0Δt(Bpn+12h,ν)=N1n=0Δt(Bν,pn+12h)=N1n=0Δt(yn+12hyn+12d,Dyn+12h(ν)). (4.26)

    Therefore, substituting Eq (4.26) into Eq (4.14), we get

    Jδ(ˉun+12h)(vhˉun+12h)=ΔtN1n=0(αun+12h+Bpn+12h,vhun+12h)0,vhUh. (4.27)

    This completes the proof of Theorem 4.1.

    Similarly, we also provide the discrete SSC for the local solution ˉun+12h as

    Jδ(ˉun+12h)(vh,vh)κ||vh||2l2(L2(Ω),vhUh. (4.28)

    where κ>0, and ˉun+12h satisfies Eq (4.13). From Eq (4.1), we can get

    Jδ(ˉun+12h)(vh,vh)=ΔtN1n=0(Dyn+12h(vh),Dyn+12h(vh))+ΔtN1n=0(yn+12hyn+12d,D2yn+12h(vh,vh))+αΔtN1n=0(vh,vh). (4.29)

    The following lemma shows the coercive property of the second derivative of the discrete objective function Jδ in a neighborhood of a local solution ˉu.

    Lemma 4.1. Let ˉu be a local solution of the problem (3.4) and the SSC (3.23) is valid. There exists sufficiently small constant ι>0, and h, for all uUad with ||uˉu||<ι and vUad,

    Jδ(u)(v,v)κ4||v||2l2(L2(Ω) (4.30)

    holds.

    So as to get a priori estimates, it is needed to introduce an auxiliary problem: find (ωn+12h(u),yn+12h(u)) Vh×Vh, n=0,1,,N1, such that for all vVh,

    (ωn+12h(u),v)=1Δt(yn+1h(u)ynh(u),v), (4.31)
    1Δt(ωn+1h(u)ωnh(u),v)+(yn+12h(u),v)+(ϕ(yn+12h(u)),v)=(fn+12+Bun+12,v), (4.32)
    y0h(u)=Rhy0(x),ω0h(u)=Rhg(x). (4.33)

    Another auxiliary problem: find (qn+12h(u),pn+12h(u))Vh×Vh, n=0,1,,N1, such that for all vVh,

    (qn+12h(u),v)=1Δt(pn+1h(u)pnh(u),v), (4.34)
    1Δt(qn+1h(u)qnh(u),v)+(pn+12h(u),v)+(ϕ(yn+12h(u))pn+12h(u),v)=(yn+12h(u)yn+12d,v), (4.35)
    qNh(u)=0,pNh(u)=0. (4.36)

    For convenience, we denote

    θnω=ωnhωnh(u),θny=ynhynh(u),n=0,1,,N,θnq=qnhqnh(u),θnp=pnhpnh(u),n=N,,1,0.

    It is clear that

    θ0ω=0,θ0y=0,θNq=0,θNp=0. (4.37)

    Next, we describe the error caused by the control discretization using the following lemma.

    Lemma 4.2. Let (ωδ,yδ,qδ,pδ) and (ωδ(u),yδ(u),qδ(u),pδ(u)) be the solutions of Eqs (4.7)–(4.12) and Eqs (4.31)–(4.36), respectively. Then,

    ||ωδωδ(u)||l(L2)+||yδyδ(u)||l(H1)C||uδu||l2(L2(Ω)), (4.38)
    ||qδqδ(u)||l(L2)+||pδpδ(u)||l(H1)C||uδu||l2(L2(Ω)). (4.39)

    Proof. To begin, we develop the inequality for θω and θy. Subtracting Eqs (4.7) and (4.8) from Eqs (4.31) and (4.32), we derive

    (θn+12ω,v)=1Δt(θn+1yθny,v), (4.40)
    1Δt(θn+1ωθnω,v)+(θn+12y,v)+(ϕ(yn+12h)ϕ(yn+12h(u)),v)=(B(un+12hun+12),v). (4.41)

    Choosing v=θn+12y, v=θn+12ω as the test function in Eqs (4.40) and (4.41), respectively, we have

    1Δt(θn+1yθny,θn+12y)=(θn+12ω,θn+12y), (4.42)
    1Δt(θn+1ωθnω,θn+12ω)+(θn+12y,θn+12ω)=(ϕ(yn+12h)(u)ϕ(yn+12h),θn+12ω)+(B(un+1hun+1),θn+12ω). (4.43)

    Substituting Eq (4.42) into Eq (4.43), we get

    12Δt[||θn+1ω||2||θnω||2+||θn+1y||2||θny||2]=(ϕ(yn+12h)(u)ϕ(yn+12h),θn+12ω)+(B(un+1hun+1),θn+12ω). (4.44)

    Summing up from n=0 up to N1, it yields

    ||θNω||2+||θNy||2=2ΔtN1n=0(B(un+12hun+12),θn+12ω)+2ΔtN1n=0(ϕ(yn+12h(u))ϕ(yn+12h),θn+12ω). (4.45)

    For the first term, we derive

    2ΔtN1n=0(B(un+12hun+12),θn+12ω)CΔtN1n=0[||un+12hun+12||20,Ω+||θn+12ω||2]CΔtN1n=0[||un+12hun+12||20,Ω+(||θn+1ω||2+||θnω||2)]. (4.46)

    For the second term, we get

    2ΔtN1n=0(ϕ(yn+12h(u))ϕ(yn+12h),θn+12ω)CΔtN1n=0[||θn+12y||2+||θn+12ω||2]CΔtN1n=0[||θn+1ω||2+||θnω||2+||θn+1y||2+||θny||2]. (4.47)

    Now, combining Eqs (4.46) and (4.47) with Eq (4.45), we arrive at

    ||θNω||2+||θNy||2CΔtN1n=0||un+12hun+12||2+ΔtN1n=0(||θn+1ω||2+||θn+1y||2). (4.48)

    The discrete Gronwall's inequality leads to

    ||ωδωδ(u)||l(L2)+||yδyδ(u)||l(H1)C||uδu||l2(L2(Ω)). (4.49)

    Next, we develop the inequality for θq and θp. Subtracting Eqs (4.10) and (4.11) from Eqs (4.34) and (4.35), we get

    (θn+12q,v)=1Δt(θn+1pθnp,v), (4.50)
    1Δt(θn+1qθnq,v)+(θn+12p,v)+(ϕ(yn+12h)pn+12hϕ(yn+12h(u))pn+12h(u),v)=(θn+12y,v). (4.51)

    Choosing v=θn+12q as the test function in Eq (4.51), we have

    12Δt||θnq||212Δt||θn+1q||2(θn+12p,θn+12q)=(ϕ(yn+12h)pn+12hϕ(yn+12h(u))pn+12h(u),θn+12q)(θn+12y,θn+12q). (4.52)

    Substituting Eq (4.50) into Eq (4.52), we know

    12Δt[||θnq||2||θn+1q||2+||θnp||2||θn+1p||2]=(ϕ(yn+12h)pn+12hϕ(yn+12h(u))pn+12h(u),θn+12q)(θn+12y,θn+12q). (4.53)

    By calculation, the following formula holds:

    (ϕ(yn+12h)pn+12hϕ(yn+12h(u))pn+12h(u),θn+12q)=(ϕ(yn+12h)pn+12hϕ(yn+12h(u))pn+12h,θn+12q)+(ϕ(yn+12h(u))pn+12hϕ(yn+12h(u))pn+12h(u),θn+12q). (4.54)

    Multiplying both sides of Eq (4.53) by 2Δt, then summing it over n from M to N1, it leads to

    ||θMq||2+||θMp||2=2ΔtN1n=M[(ϕ(yn+12h)pn+12hϕ(yn+12h(u))pn+12h,θn+12q)+(ϕ(yn+12h(u))pn+12hϕ(yn+12h(u))pn+12h(u),θn+12q)(θn+12y,θn+12q)] (4.55)

    Here, the estimates of \Pi_{1} , \Pi_{2} , \Pi_{3} are similar to those of \Upsilon_{1} , \Upsilon_{2} , \Upsilon_{3} , and we have

    \begin{align*} \Pi_{1}&\leq \Delta t\sum\limits_{n = M}^{N-1}||\phi'(y_{h} ^{n+\frac{1}{2}})-\phi'(y_{h} ^{n+\frac{1}{2}}(u))||_{0, 4}||p_{h} ^{n+\frac{1}{2}}||_{0, 4}||||\theta_{q} ^{n+\frac{1}{2}}||\\ &\leq C\Delta t\sum\limits_{n = M}^{N-1}(||\nabla\theta_{y}^{n+1}||^{2}+||\nabla\theta_{y}^{n}||^{2}+||\theta_{q} ^{n}||^{2}), \end{align*}
    \begin{align*} \Pi_{2}&\leq 2\Delta t\sum\limits_{n = M}^{N-1}||\phi'(y_{h} ^{n+\frac{1}{2}}(u))||_{0, \infty}||p_{h} ^{n+\frac{1}{2}}(u)-p_{h} ^{n+\frac{1}{2}}||\; ||\theta_{q} ^{n+\frac{1}{2}}||\\ &\leq C \Delta t\sum\limits_{n = M}^{N-1}(||\theta_{p} ^{n}||^{2}+||\theta_{q} ^{n}||^{2}), \\ \Pi_{3}&\leq \Delta t\sum\limits_{n = M}^{N-1}(||\nabla\theta_{y} ^{n}||+||\theta_{q} ^{n}||^{2}). \end{align*}

    Finally, inserting the above estimates of \Pi_{1} \Pi_{3} into Eq (4.55), we get

    \begin{align} ||\theta_{q} ^{M}||^{2}+||\nabla\theta_{p} ^{M}||^{2}\leq C\Delta t\sum\limits_{n = M}^{N-1}||\nabla\theta_{y} ^{n}||+C\Delta t\sum\limits_{n = M}^{N-1}(||\theta_{q} ^{n}||^{2}+||\theta_{p} ^{n}||^{2}). \end{align} (4.56)

    Thus, combine the Poincaré inequality and the discrete Gronwall's inequality, such that

    \begin{align} &||q_{\delta}-q_{\delta}(u)||_{l^{\infty}(L^{2})}+||p_{\delta}-p_{\delta}(u)||_{l^{\infty}(H^{1})}\leq C||y_{\delta}-y_{\delta}(u)||_{l^{\infty}(H^{1})}. \end{align} (4.57)

    Eqs (4.38) and (4.57) follows Eq (4.39), which also completes the proof of Lemma 4.2.

    Next, so as to estimate the error of the control discretization, we choose a local solution \bar{u} of the continuous problem (3.6)–(3.8), and an associated approximate solution u_{\delta} of the discrete problem (4.1)–(4.4). We introduce the following auxiliary problem:

    \begin{align} \min\limits_{u_{h} ^{n+\frac{1}{2}}\in U_{h}^{\iota}}\mathcal{J}_{\delta}(u_{h} ^{n+\frac{1}{2}}), \end{align} (4.58)

    where U_{h}^{\iota}(\bar{u}) = \{u_{h} ^{n+\frac{1}{2}}\in U_{h}:||u_{\delta}-\bar{u }|| < \iota\} . In addition, since \mathcal{J}''_{\delta} satisfies Lemma 4.1 for \bar{u}\in U_{h}^{\iota}(\bar{u}), \; v\in U_{ad} , the existence and uniqueness of the problem (4.58) are guaranteed; see [28].

    From the definitions of the local solution (4.6) and U_{h}^{\iota} , we can get

    \begin{align} \mathcal{J}_{\delta}(\bar{u}_{h} ^{n+\frac{1}{2}}) = \min\limits_{u_{h} ^{n+\frac{1}{2}}\in U_{h}^{\iota}(\bar{u})}\mathcal{J}_{\delta}(u_{h} ^{n+\frac{1}{2}}), \quad\quad\text{for}\quad ||u_{h} ^{n+\frac{1}{2}}-\bar{u}_{h} ^{n+\frac{1}{2}}|| < \iota. \end{align} (4.59)

    Utilizing (4.58) and (4.59), we can deduce that \bar{u}_{h} ^{n+\frac{1}{2}}\in U_{h}^{\iota} is the unique solution of the problem (4.58).

    Lemma 4.3. Let \bar{u} be a local solution of the problem (3.4) and the SSC (3.23) is valid. Then, the discrete problem (4.58) has a unique solution \bar{u}_{h} ^{n+\frac{1}{2}} , and the following estimate holds:

    \begin{align} ||\bar{u}-\bar{u}_{\delta}||_{l^{2}(L^{2}(\Omega ))} &\leq C||p-p_{\delta}(\bar{u})||^{2}_{l^{2}(L^{2})}+Ch^{2} (||\bar{u}||^{2}_{ l^{2}(H^{1}(\Omega ))}+||p||^{2}_{ l^{2}(H^{1})}). \end{align} (4.60)

    for \iota > 0, h > 0 sufficiently small.

    Proof. From Lemma 4.1, it is clear to infer that

    \begin{align*} \mathcal{ J }''_{\delta}(u_{\delta})(v, v)\geq \frac{\kappa}{4}||v||^{2}_{l^{2}(L^{2}(\Omega ))}, \; \; \text{for}\; \; \forall u_{\delta}\in U_{h}^{\iota}(\bar{u}) \; \; \text{and}\; \; v\in U_{ad}. \end{align*}

    By the FNC (4.13), we can get

    \begin{align*} &\mathcal{ J }'_{\delta}(\bar{u}^{n+\frac{1}{2}})(\bar{u}^{n+\frac{1}{2}}-\bar{u}_{h}^{n+\frac{1}{2}})\leq 0, \\ &\mathcal{ J }'_{\delta}(\bar{u}_{h}^{n+\frac{1}{2}})(\bar{u}_{h}^{n+\frac{1}{2}}-\Pi_{h}\bar{u}^{n+\frac{1}{2}})\leq 0. \end{align*}

    for h sufficiently small. Utilizing \upsilon^{n+\frac{1}{2}} = \theta \bar{u}^{n+\frac{1}{2}}+(1-\theta)\bar{u}_{h}^{n+\frac{1}{2}} with \theta\in [0, 1] , we get

    \begin{align} &\frac{\kappa}{4}|| \bar{u}-\bar{u}_{\delta}||^{2}_{l^{2}(L^{2}(\Omega ))}\\ \leq& \mathcal{ J }''_{\delta}(\upsilon^{n+\frac{1}{2}})(\bar{u}^{n+\frac{1}{2}}-\bar{u}_{h}^{n+\frac{1}{2}}, \bar{u}^{n+\frac{1}{2}}-\bar{u}_{h}^{n+\frac{1}{2}})\\ = &\mathcal{ J }'_{\delta}(\bar{u}^{n+\frac{1}{2}})(\bar{u}^{n+\frac{1}{2}}-\bar{u}_{h}^{n+\frac{1}{2}})-\mathcal{ J }'_{\delta}(\bar{u}_{h}^{n+\frac{1}{2}})(\bar{u}^{n+\frac{1}{2}}-\bar{u}_{h}^{n+\frac{1}{2}})\\ = &\Delta t\sum^{N-1}_{n = 0}\Big[(\alpha \bar{u}^{n+\frac{1}{2}}+B^{*}p_{h}^{n+\frac{1}{2}}(\bar{u}), \bar{u}^{n+\frac{1}{2}}-\bar{u}_{h}^{n+\frac{1}{2}}) -(\alpha \bar{u}_{h}^{n+\frac{1}{2}}+B^{*}p_{h}^{n+\frac{1}{2}}, \bar{u}^{n+\frac{1}{2}}-\bar{u}_{h}^{n+\frac{1}{2}} ) \Big]\\ \leq&\Delta t\sum^{N-1}_{n = 0} \bigg[(B^{*}(p_{h}^{n+\frac{1}{2}}(\bar{u})-p^{n+\frac{1}{2}}), \bar{u}^{n+\frac{1}{2}}-\bar{u}_{h}^{n+\frac{1}{2}}) +(\alpha \bar{u}_{h}^{n+\frac{1}{2}}, \Pi_{h}\bar{u}^{n+\frac{1}{2}}-\bar{u}^{n+\frac{1}{2}} ) \\ &\quad\quad\quad+ (B^{*}p^{n+\frac{1}{2}}, \Pi_{h}\bar{u}^{n+\frac{1}{2}}-\bar{u}^{n+\frac{1}{2}} ) + (B^{*}(p^{n+\frac{1}{2}}-p_{h}^{n+\frac{1}{2}}(\bar{u})), \bar{u}^{n+\frac{1}{2}}-\Pi_{h}\bar{u}^{n+\frac{1}{2}}) \\ &\quad\quad\quad+ (B^{*}(p_{h}^{n+\frac{1}{2}}(\bar{u})-p_{h}^{n+\frac{1}{2}}), \bar{u}^{n+\frac{1}{2}}-\Pi_{h}\bar{u}^{n+\frac{1}{2}}) \bigg]\\ \triangleq&\sum\limits_{i = 1}^{5} \mathcal{S}_{i}. \end{align} (4.61)

    To start, using the Cauchy-Schwarz inequality, we obtain

    \begin{align*} \mathcal{S}_{1}\leq C||p-p_{\delta}(\bar{u})||^{2}_{l^{2}(L^{2})}+\epsilon|| \bar{u}-\bar{u}_{\delta}||^{2} _{l^{2}(L^{2}(\Omega ))}. \end{align*}

    For the \mathcal{S}_{2} and \mathcal{S}_{3} , by the definition of the operator \Pi_{h} , we have

    \begin{align*} \mathcal{S}_{2} = 0, \end{align*}

    and

    \begin{align*} \mathcal{S}_{3} = & \sum^{N-1}_{n = 0}\Delta t(B^{*}(p^{n+\frac{1}{2}}-\Pi_{h}p^{n+\frac{1}{2}}, \Pi_{h}\bar{u}^{n+\frac{1}{2}}-\bar{u}^{n+\frac{1}{2}} )\\ \leq&Ch^{2} (||p||^{2}_{ l^{2}(0, T;H^{I}(\Omega))}+||\bar{ u}||^{2}_{ l^{2}(0, T;H^{1}(\Omega ))}). \end{align*}

    Then, for the \mathcal{S}_{4} and \mathcal{S}_{5} , from Lemma 4.2 and the Cauchy inequality, it yields

    \begin{align*} & \mathcal{S}_{4}\leq C||p-p_{\delta}(\bar{u})||^{2}_{l^{2}(L^{2})}+Ch^{2} ||\bar{u}||^{2}_{ l^{2}(0, T;H^{I}(\Omega ))}, \end{align*}
    \begin{align*} \mathcal{S}_{5}\leq &\epsilon|| \bar{u}-\bar{u}_{\delta}||^{2} _{l^{2}(L^{2}(\Omega ))}+Ch^{2} ||\bar{u}||^{2}_{ l^{2}(0, T;H^{I}(\Omega ))}. \end{align*}

    Substituting \mathcal{S}_{1} \mathcal{S}_{5} into Eq (4.61), we obtain

    \begin{align*} &|| \bar{u}-\bar{u}_{\delta}||^{2}_{ l^{2}(0, T;L^{2}(\Omega ))}\\ \leq &C||p-p_{\delta}(\bar{u})||^{2}_{l^{2}(L^{2})}+Ch^{2} (||\bar{u}||^{2}_{l^{2}(0, T;H^{I}(\Omega ))}+||p||^{2}_{ l^{2}(0, T;H^{I}(\Omega))}). \end{align*}

    This completes the proof of Lemma 4.3.

    In this subsection, we will establish the error caused by the discretization of the Crank-Nicolson FEM scheme. To do this, we define the error function as

    \begin{align*} e^{n}_{\omega} = \omega(t_{n})-\omega^{n}_{h}(u), \; e^{n}_{y} = y(t_{n})-y^{n}_{h}(u), \; e^{n}_{q} = q(t_{n})-q^{n}_{h}(u), \; e_{p}^{n} = p(t_{n})-p^{n}_{h}(u), \end{align*}

    and introduce the following truncation errors:

    \begin{align*} &\mathcal{T}_{\omega} = \omega(t_{n+1})-\omega(t_{n})-\Delta t \omega_{t}(t_{n+\frac{1}{2}}), \; \; \eta_{\omega} = \frac{\omega(t_{n+1})+\omega(t_{n})}{2}-\omega(t_{n+\frac{1}{2}}), \\ &\mathcal{T}_{y} = y(t_{n+1})-y(t_{n})-\Delta t y_{t}(t_{n+\frac{1}{2}}), \; \; \eta _{y} = \frac{y(t_{n+1})+y(t_{n})}{2}-y(t_{n+\frac{1}{2}}), \\ &\mathcal{T}_{q} = q(t_{n+1})-q(t_{n})-\Delta t q_{t}(t_{n+\frac{1}{2}}), \; \; \eta_{q} = \frac{q(t_{n+1})+q(t_{n})}{2}-q(t_{n+\frac{1}{2}}), \\ &\mathcal{T}_{p} = p(t_{n+1})-p(t_{n})-\Delta t p_{t}(t_{n+\frac{1}{2}}), \; \; \eta _{p} = \frac{p(t_{n+1})+p(t_{n})}{2}-p(t_{n+\frac{1}{2}}). \end{align*}

    For these definitions, we first present the following estimates of the truncation error.

    Lemma 4.4. The following estimates hold:

    \begin{align*} &||\mathcal{T}_{\omega}||^{2}\leq C(\Delta t)^{5}\int_{t_{n}}^{t_{n+1}}||\omega_{ttt} ||^{2}ds, \; \; ||\eta_{\omega}||^{2}\leq C(\Delta t)^{3}\int_{t_{n}}^{t_{n+1}}||\omega_{tt} ||^{2}ds, \\ &||\mathcal{T}_{y}||^{2}\leq C(\Delta t)^{5}\int_{t_{n}}^{t_{n+1}}||y_{ttt} ||^{2}ds, \; \; ||\eta_{y}||^{2}\leq C(\Delta t)^{3}\int_{t_{n}}^{t_{n+1}}||y_{tt} ||^{2}ds, \\ &||\mathcal{T}_{q}||^{2}\leq C(\Delta t)^{5}\int_{t_{n}}^{t_{n+1}}||q_{ttt} ||^{2}ds, \; \; ||\eta_{q}||^{2}\leq C(\Delta t)^{3}\int_{t_{n}}^{t_{n+1}}||q_{tt} ||^{2}ds, \\ &||\mathcal{T}_{p}||^{2}\leq C(\Delta t)^{5}\int_{t_{n}}^{t_{n+1}}||p_{ttt} ||^{2}ds, \; \; ||\eta_{p}||^{2}\leq C(\Delta t)^{3}\int_{t_{n}}^{t_{n+1}}||p_{tt} ||^{2}ds. \end{align*}

    Lemma 4.5. Let (\omega, y, u, q, p) and (\omega_{\delta}(u), y_{\delta}(u), q_{\delta}(u), p_{\delta}(u)) be the solutions of Eqs (3.6)–(3.7) and Eqs (4.31)–(4.36), respectively. Assume that \omega, \; q\in L^{2}(H^{2})\cap H^{1}(H^{2}), \; \omega_{tt}, \; q_{tt}\in L^{2}(H^{1}), y_{tt}\in L^{2}(L^{2})\cap L^{2}(H^{1})\cap L^{2}(H^{2}), p_{tt}\in L^{2}(L^{2})\cap L^{2}(H^{2}), \omega_{ttt}, \; q_{ttt}\in L^{2}(L^{2}), \; y_{ttt}, \; p_{ttt}\in L^{2}(H^{1}), y_{0}(x), \; g(x)\in H^{2}(\Omega) . Then, we have

    \begin{align} &||\omega_{\delta}(u)-\omega||_{l^{\infty}(L^{2})}+||y_{\delta}(u)-y||_{l^{\infty}(H^{1})}\leq C((\Delta t)^{2}+h), \end{align} (4.62)
    \begin{align} &||q_{\delta}(u)-q||_{l^{\infty}(L^{2})}+|| p_{\delta}(u)-p||_{l^{\infty}(H^{1})}\leq C((\Delta t)^{2}+h). \end{align} (4.63)

    Proof. From Eq (3.7), the exact solution (\omega, y) satisfies

    \begin{align} &\frac{1}{2}(\omega(t_{n+1})+\omega(t_{n}), v_{h}) = (\frac{y(t_{n+1})-y(t_{n})}{\Delta t}, v_h)-\frac{1}{\Delta t}(T_{y}, v_{h})+(\eta_{\omega}, v_{h}), \end{align} (4.64)
    \begin{align} &(\frac{\omega(t_{n+1})-\omega(t_{n})}{\Delta t}, v_{h})+(\nabla \frac{y(t_{n+1})+y(t_{n})}{2}, \nabla v_{h})+(\phi (y(t_{n+\frac{1}{2}}) ), v_{h})\\ &\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; = (f(t_{n+\frac{1}{2}})+Bu(t_{n+\frac{1}{2}}), v_{h})+\frac{1}{\Delta t}(T_{\omega}, v_{h})+(\nabla \eta_{y}, \nabla v_{h}). \end{align} (4.65)

    From relations Eqs (4.64) and (4.65) and Eqs (4.31) and (4.32), it holds that

    \begin{align} &(e_{\omega}^{n+\frac{1}{2}}, v_{h}) = \frac{1}{\Delta t}(e_{y}^{n+1}-e_{y}^{n}, v_{h})-\frac{1}{\Delta t}(T_{y}, v_{h})+(\eta_{\omega}, v_{h}), \end{align} (4.66)
    \begin{align} &(e_{\omega}^{n+1}-e_{\omega}^{n}, v_{h} )+\Delta t(\nabla e_{y}^{n+\frac{1}{2}}, \nabla v_{h} ) = \Delta t(\phi(y_{h} ^{n+\frac{1}{2}}(u))-\phi(y ^{n+\frac{1}{2}}), v_{h})+(T_{\omega}, v_{h})+\Delta t(\nabla \eta_{y}, \nabla v_{h}). \end{align} (4.67)

    Taking the discrete inner product of Eq (4.67) with v_h = \mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u) = \mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega(t_{n+\frac{1}{2}}) +\omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u) = \mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega(t_{n+\frac{1}{2}})+e^{n+\frac{1}{2}}_{\omega} and rearranging the terms, we have

    \begin{align} &\frac{1}{2}||e_{\omega}^{n+1}||^{2}-\frac{1}{2}||e_{\omega}^{n}||^{2}+(\nabla e_{y}^{n+\frac{1}{2}}, \nabla e_{\omega}^{n+\frac{1}{2}})\Delta t \\ = &(e_{\omega}^{n+1}-e_{\omega}^{n}, \omega(t_{n+\frac{1}{2}})-\mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})) +(\nabla e_{y}^{n+\frac{1}{2}}, \nabla(\omega(t_{n+\frac{1}{2}})-\mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})))\Delta t\\ &+(\phi(y_{h} ^{n+\frac{1}{2}}(u))-\phi(y ^{n+\frac{1}{2}}), \mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u))\Delta t\\ &+(T_{\omega}, \mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u))+(\nabla \eta_{y}, \nabla(\mathcal{R}_{h} \omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u)))\Delta t. \end{align} (4.68)

    Meanwhile, choosing v_h = e_{y}^{n+\frac{1}{2}} in Eq (4.66), we obtain

    \begin{align} \Delta t(e_{\omega}^{n+\frac{1}{2}}, e_{y}^{n+\frac{1}{2}}) = (e_{y}^{n+1}-e_{y}^{n}, e_{y}^{n+\frac{1}{2}})-(T_{y}^{n}, e_{y}^{n+\frac{1}{2}})+\Delta t(\eta_{\omega}, e_{y}^{n+\frac{1}{2}}). \end{align} (4.69)

    Through Eqs (4.68) and (4.69), and summing from n = 0 up to N 1 , we have that

    \begin{align} &\frac{1}{2}\Big[||e_{\omega}^{N}||^{2}+||\nabla e_{y}^{N}||^{2}-||e_{\omega}^{0}||^{2}-||\nabla e_{y}^{0}||^{2}\Big]\\ = &\sum\limits_{n = 0}^{N-1}\bigg[(e_{\omega}^{n+1}-e_{\omega}^{n}, \omega(t_{n+\frac{1}{2}})-\mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})) + (\nabla e_{y}^{n+\frac{1}{2}}, \nabla(\omega(t_{n+\frac{1}{2}})-\mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})))\Delta t\\ &\quad\quad+ (\phi(y_{h} ^{n+\frac{1}{2}}(u))-\phi(y (t_{n+\frac{1}{2}})), \mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u))\Delta t +(T_{\omega}, \mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u))\\ &\quad\quad- \Delta t(\Delta\eta_{y}, \mathcal{R}_{h} \omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u))) + (\nabla T_{y}, \nabla e_{y}^{n+\frac{1}{2}})- \Delta t(\nabla\eta_{\omega}, \nabla e_{y}^{n+\frac{1}{2}})\bigg]\\ \triangleq&\sum\limits_{i = 1}^{7}\Theta_{i}. \end{align} (4.70)

    By utilizing Young's and H \ddot{o} lder inequalities, we have

    \begin{align*} \Theta_{1} = &\Delta t\sum^{N-1}_{n = 0}(d_{t}\omega^{n+1}-\mathcal{R}_{h}d_{t}\omega^{n+1}, \omega(t_{n+\frac{1}{2}})-\mathcal{R}_{h}\omega(t_{n+\frac{1}{2}}))\\ \leq&Ch^{4}(||\omega||^{2}_{L^{2}(H^{2})}+||\omega||^{2}_{H^{1}(H^{2})}), \\ \Theta_{2}\leq&C \sum^{N-1}_{n = 0}\Delta t||\nabla e_{y}^{n+1}||^{2}+Ch^{2}||\omega||^{2}_{L^{2}(H^{2})}, \\ \Theta_{3}\leq&C \sum^{N-1}_{n = 0}\Delta t(||e_{y}^{n+1}||^{2}+||e_{\omega}^{n+1}||^{2})+Ch^{4}||\omega||^{2}_{L^{2}(H^{2})}. \end{align*}

    For \Theta_{4} \Theta_{7} , by definition of the truncation error and Lemma 4.4, it can be obtained that

    \begin{align*} \Theta_{4}\leq& \frac{1}{\Delta t}\sum^{N-1}_{n = 0}||T_{\omega}||^{2}+\Delta t\sum^{N-1}_{n = 0}||\mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u))||^{2}\\ \leq &C(\Delta t)^{4}||\omega_{ttt}||^{2}_{L^{2}(L^{2})}+Ch^{4}||\omega||^{2}_{L^{2}(H^{2})}+C \sum^{N-1}_{n = 0}\Delta t||e_{\omega}^{n+1}||^{2}, \\ \Theta_{5}\leq& \sum^{N-1}_{n = 0}\Delta t||\Delta \eta_{y}|||| \mathcal{R}_{h} \omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u))||\\ \leq&C (\Delta t)^{4}||y_{tt}||^{2}_{L^{2}(H^{2})}+C \sum^{N-1}_{n = 0}\Delta t|| e_{\omega}^{n+1}||^{2}+Ch^{4}||\omega||^{2}_{L^{2}(H^{2})}, \\ \Theta_{6}\leq &C(\Delta t)^{4}||y_{ttt}||^{2}_{L^{2}(H^{1})}+C \sum^{N-1}_{n = 0}\Delta t||\nabla e_{y}^{n+1}||^{2}, \\ \Theta_{7}\leq &C(\Delta t)^{4}||\omega_{tt}||^{2}_{L^{2}(H^{1})}+C \sum^{N-1}_{n = 0}\Delta t||\nabla e_{y}^{n+1}||^{2}. \end{align*}

    Substituting \Theta_{1} \Theta_{7} into Eq (4.70), we get

    \begin{align} \frac{1}{2}||e_{\omega}^{N}||^{2}+\frac{1}{2}||\nabla e_{y}^{N}||^{2} \leq& \frac{1}{2}||e_{\omega}^{0}||^{2}+\frac{1}{2}||\nabla e_{y}^{0}||^{2}+C \sum^{N-1}_{n = 0}\Delta t(||\nabla e_{y}^{n+1}||^{2}+|| e_{\omega}^{n+1}||^{2})\\ &+Ch^{4}(||\omega||^{2}_{L^{2}(H^{2})}+||\omega||^{2}_{H^{1}(H^{2})})+Ch^{2}||\omega||^{2}_{L^{2}(H^{2})}\\ &+C(\Delta t)^{4}(||\omega_{ttt}||^{2}_{L^{2}(L^{2})}+||y_{ttt}||^{2}_{L^{2}(H^{1})}+||\omega_{tt}||^{2}_{L^{2}(H^{1})}+||y_{tt}||^{2}_{L^{2}(H^{2})}). \end{align} (4.71)

    Note that

    \begin{align*} \frac{1}{2}||e_{\omega}^{0}||^{2}+\frac{1}{2}||\nabla e_{y}^{0}||^{2}& = \frac{1}{2}||g(x)-\mathcal{R}_{h}g(x)||^{2}+\frac{1}{2}||\nabla (y_{0}(x)-\mathcal{R}_{h}y_{0}(x)||^{2}\\ &\leq C h^{4}||g(x)||_{H^{2}}^{2}+Ch^{2}||y_{0}(x)||_{H^{2}}^{2} \end{align*}

    Then, the discrete Gronwall's inequality implies that

    \begin{align*} ||\omega_{\delta}(u)-\omega||_{l^{\infty}(L^{2})}+||y_{\delta}(u)-y||_{l^{\infty}(H^{1})}\leq C(h+(\Delta t)^{2}). \end{align*}

    Furthermore, from Eq (3.7), we can find that the exact solution (p, q) satisfies

    \begin{align} &(\frac{q(t_{n+1})+q(t_{n})}{2}, v) = (\frac{p(t_{n+1})-p(t_{n})}{\Delta t}, v)-\frac{1}{\Delta t}(T_{p}, v)+(\eta_{q}, v), \end{align} (4.72)
    \begin{align} &(\frac{q(t_{n+1})-q(t_{n})}{\Delta t}, v)+(\nabla\frac{p(t_{n+1})+p(t_{n})}{2}, \nabla v)+(\phi'(y(t_{n+\frac{1}{2}})p(t_{n+\frac{1}{2}}), v)\\ &\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; = (\frac{y(t_{n+1})+y(t_{n})}{2}-y_{d}(t_{n+\frac{1}{2}}), v)-(\eta_{y}, v)+\frac{1}{\Delta t}(T_{q}, v)-(\Delta\eta_{p}, v). \end{align} (4.73)

    From relations Eqs (4.72) and (4.73) and Eqs (4.34) and (4.35), we have

    \begin{align} &(e_{q}^{n+\frac{1}{2}}, v_{h}) = \frac{1}{\Delta t}(e_{p}^{n+1}-e_{p}^{n}, v_{h})-\frac{1}{\Delta t}(T_{p}, v_{h})+(\eta_{q}, v_{h}), \end{align} (4.74)
    \begin{align} &\frac{1}{\Delta t}(e_{q}^{n+1}-e_{q}^{n}, v_{h})+( \nabla e_{p}^{n+\frac{1}{2}}, \nabla v_{h} ) = (\phi'(y_{h} ^{n+\frac{1}{2}}(u))p_{h} ^{n+\frac{1}{2}}(u)-\phi'(y(t_{n+\frac{1}{2}})p(t_{n+\frac{1}{2}}), v_{h} )\\ &\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; +(e_{y}^{n+\frac{1}{2}}, v_{h})+\frac{1}{\Delta t}(T_{q}, v_{h})-(\Delta\eta_{p}, v_{h})-(\eta_{y}, v_{h}). \end{align} (4.75)

    Choosing v_{h} = -(\mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q_{h}^{n+\frac{1}{2}}(u)) = q(t_{n+\frac{1}{2}})-\mathcal{R}_{h}q(t_{n+\frac{1}{2}})-e_{q}^{n+\frac{1}{2}} in Eq (4.75), we have

    \begin{align} &\frac{1}{2}(||e_{q}^{n}||^{2}-||e_{q}^{n+1}||^{2})-\Delta t( \nabla e_{p}^{n+\frac{1}{2}}, \nabla e_{q}^{n+\frac{1}{2}} )\\ = &(e_{q}^{n}-e_{q}^{n+1}, q(t_{n+\frac{1}{2}})-\mathcal{R}_{h}q(t_{n+\frac{1}{2}}))+\Delta t( \nabla e_{p}^{n+\frac{1}{2}}, \nabla(q(t_{n+\frac{1}{2}})-\mathcal{R}_{h}q(t_{n+\frac{1}{2}})))\\ &+\Delta t(\phi'(y_{h} ^{n+\frac{1}{2}}(u))p_{h} ^{n+\frac{1}{2}}(u)-\phi'(y(t_{n+\frac{1}{2}})p(t_{n+\frac{1}{2}}), q_{h}^{n+\frac{1}{2}}(u)-\mathcal{R}_{h}^{n+\frac{1}{2}})\\ &-(T_{q}, \mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q_{h}^{n+\frac{1}{2}}(u))+\Delta t(\Delta\eta_{p}, \mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q_{h}^{n+\frac{1}{2}}(u)))\\ &+\Delta t(\eta_{y}, \mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q_{h}^{n+\frac{1}{2}}(u))+\Delta t(e_{y}^{n+\frac{1}{2}}, q_{h}^{n+\frac{1}{2}}(u)-\mathcal{R}_{h}q(t_{n+\frac{1}{2}})). \end{align} (4.76)

    Meanwhile, letting v_{h} = -e_{p}^{n+\frac{1}{2}} in Eq (4.74), we get

    \begin{align} -\Delta t(e_{q}^{n+\frac{1}{2}}, e_{p}^{n+\frac{1}{2}}) = -(e_{p}^{n+1}-e_{p}^{n}, e_{p}^{n+\frac{1}{2}})+(T_{p}, e_{p}^{n+\frac{1}{2}})-\Delta t(\eta_{q}, e_{p}^{n+\frac{1}{2}}). \end{align} (4.77)

    Combining Eqs (4.76) and (4.77), then summing from n = M to N-1 , it can conclude that

    \begin{align} &\frac{1}{2}||e_{q}^{M}||^{2}+\frac{1}{2}|| \nabla e_{p}^{M}||^{2}\\ = &\sum^{N-1}_{n = M}\bigg[(e_{q}^{n}-e_{q}^{n+1}, q(t_{n+\frac{1}{2}})-\mathcal{R}_{h}q(t_{n+\frac{1}{2}}))+\Delta t( \nabla e_{p}^{n+\frac{1}{2}}, \nabla(q(t_{n+\frac{1}{2}})-\mathcal{R}_{h}q(t_{n+\frac{1}{2}})))\\ &\; \; \; \; \; \; \; + \Delta t(\phi'(y_{h} ^{n+\frac{1}{2}}(u))p_{h} ^{n+\frac{1}{2}}(u)-\phi'(y(t_{n+\frac{1}{2}})p(t_{n+\frac{1}{2}}), q_{h}^{n+\frac{1}{2}}(u)-\mathcal{R}_{h}q(t_{n+\frac{1}{2}}))\\ &\; \; \; \; \; \; \; - (T_{q}, \mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q_{h}^{n+\frac{1}{2}}(u))+ \Delta t(\Delta\eta_{p}, \mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q_{h}^{n+\frac{1}{2}}(u)))\\ &\; \; \; \; \; \; \; - (\nabla T_{p}, \nabla e_{p}^{n+\frac{1}{2}})+ \Delta t(\nabla\eta_{q}, \nabla e_{p}^{n+\frac{1}{2}}) +\Delta t(\eta_{y}, \mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q_{h}^{n+\frac{1}{2}}(u))\\ &\; \; \; \; \; \; \; +\Delta t(e_{y}^{n+\frac{1}{2}}, q_{h}^{n+\frac{1}{2}}(u)-\mathcal{R}_{h}q(t_{n+\frac{1}{2}}))\bigg]\\ \triangleq& \sum\limits_{i = 1}^{9}\mathcal{Y}_{i}. \end{align} (4.78)

    By the properties of L^{2} projection, we have

    \begin{align*} \mathcal{Y}_{1} = &\sum^{N-1}_{n = M}\Delta t(\bar{d}_{t}q^{n}-\mathcal{R}_{h}\bar{d}_{t}q^{n}, q(t_{n+\frac{1}{2}})-\mathcal{R}_{h}q(t_{n+\frac{1}{2}}))\\ \leq &Ch^{4}(||q||^{2}_{L^{2}(H^{2})}+||q||^{2}_{H^{1}(H^{2})}), \\ \mathcal{Y}_{2}\leq&C \sum^{N-1}_{n = M}\Delta t||\nabla e_{p}^{n}||^{2}+Ch^{2}||q||^{2}_{L^{2}(H^{2})}. \end{align*}

    A standard algebraic manipulation implies that

    \begin{align*} &\phi'(y_{h} ^{n+\frac{1}{2}}(u))p_{h} ^{n+\frac{1}{2}}(u)-\phi'(y(t_{n+\frac{1}{2}}))p(t_{n+\frac{1}{2}})\\ = &\phi'(y_{h} ^{n+\frac{1}{2}}(u))p_{h} ^{n+\frac{1}{2}}(u)-\phi'(y_{h} ^{n+\frac{1}{2}}(u))\frac{p(t_{n+1})+p(t_{n})}{2}\\ &+\phi'(y_{h} ^{n+\frac{1}{2}}(u))(\frac{p(t_{n+1})+p(t_{n})}{2})-\phi'(\frac{y(t_{n+1})+y(t_{n})}{2}) \frac{p(t_{n+1})+p(t_{n})}{2}\\ &+\phi'(\frac{y(t_{n+1})+y(t_{n})}{2}) \frac{p(t_{n+1})+p(t_{n})}{2}-\phi'(\frac{y(t_{n+1})+y(t_{n})}{2})p(t_{n+\frac{1}{2}})\\ &+\phi'(\frac{y(t_{n+1})+y(t_{n})}{2})p(t_{n+\frac{1}{2}})-\phi'(y(t_{n+\frac{1}{2}}))p(t_{n+\frac{1}{2}}). \end{align*}

    For the bound of \mathcal{Y}_{3} , it holds

    \begin{align*} \mathcal{Y}_{3} \leq& C \sum^{N-1}_{n = M}\Delta t(|| e_{p}^{n}||^{2}+|| e_{q}^{n}||^{2}) +C(\Delta t)^{4}(||y_{tt}||^{2}_{L^{2}(H^{1})}+||p_{tt}||^{2}_{L^{2}(L^{2})})\\ &+Ch^{4}||q||^{2}_{L^{2}(H^{2})}+C||y-y_{\delta}(u)||^{2}_{l^{2}(H^{1})}. \end{align*}

    Then, for \mathcal{Y}_{4} \mathcal{Y}_{9} , noting that \mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q_{h}^{n+\frac{1}{2}}(u) = \mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q(t_{n+\frac{1}{2}})+e_{q}^{n+\frac{1}{2}} and applying Lemma 4.4, we have

    \begin{align*} \mathcal{Y}_{4}\leq& \frac{1}{\Delta t}\sum^{N-1}_{n = M}||T_{q}||^{2}+\Delta t\sum^{N-1}_{n = M}||\mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q^{n+\frac{1}{2}}_{h}(u))||^{2}\\ \leq &C(\Delta t)^{4}||q_{ttt}||^{2}_{L^{2}(L^{2})}+Ch^{4}||q||^{2}_{L^{2}(H^{2})}+C\sum^{N-1}_{n = M}\Delta t||e_{q}^{n}||^{2}, \end{align*}
    \begin{align*} \mathcal{Y}_{5}\leq& \sum^{N-1}_{n = M}\Delta t||\Delta \eta_{p}|||| \mathcal{R}_{h} q(t_{n+\frac{1}{2}})-q^{n+\frac{1}{2}}_{h}(u))||\\ \leq& C(\Delta t)^{4}||p_{tt}||^{2}_{L^{2}(H^{2})}+C\sum^{N-1}_{n = M}\Delta t|| e_{q}^{n}||^{2}+Ch^{4}||q||^{2}_{L^{2}(H^{2})}, \\ \mathcal{Y}_{6}\leq &C(\Delta t)^{4}||p_{ttt}||^{2}_{L^{2}(H^{1})}+C\sum^{N-1}_{n = M}\Delta t||\nabla e_{p}^{n}||^{2}, \\ \mathcal{Y}_{7}\leq &C(\Delta t)^{4}||q_{tt}||^{2}_{L^{2}(H^{1})}+C\sum^{N-1}_{n = M}\Delta t||\nabla e_{p}^{n}||^{2}, \\ \mathcal{Y}_{8}\leq & C(\Delta t)^{4}||y_{tt}||^{2}_{L^{2}(L^{2})}+C\sum^{N-1}_{n = M}\Delta t|| e_{q}^{n}||^{2}+Ch^{4}||q||^{2}_{L^{2}(H^{2})}, \\ \mathcal{Y}_{9}\leq & C||y-y_{\delta}(u)||^{2}_{l^{2}(H^{1})}+C\sum^{N-1}_{n = M}\Delta t|| e_{q}^{n}||^{2}+Ch^{4}||q||^{2}_{L^{2}(H^{2})}. \end{align*}

    Collecting the above bounds and using the discrete Gronwall's inequality, we deduce that

    \begin{align} &||q_{\delta}(u)-q||^{2}_{l^{\infty}(L^{2})}+|| p_{\delta}(u)-p||^{2}_{l^{\infty}(H^{1})}\\ \leq& C(\Delta t)^{4}(||p_{tt}||^{2}_{L^{2}(L^{2})}+||p_{tt}||^{2}_{L^{2}(H^{2})}+||y_{tt}||^{2}_{L^{2}(H^{1})}+||y_{tt}||^{2}_{L^{2}(L^{2})}+||q_{tt}||^{2}_{L^{2}(H^{1})})\\ +& C(\Delta t)^{4}(||q_{ttt}||^{2}_{L^{2}(L^{2})}+||p_{ttt}||^{2}_{L^{2}(H^{1})})+Ch^{4}(||q||^{2}_{L^{2}(H^{2})}+||q||^{2}_{H^{1}(H^{2})})\\ +&Ch^{2}||q||^{2}_{L^{2}(H^{2})}+C||y-y_{\delta}(u)||^{2}_{l^{2}(H^{1})}. \end{align} (4.79)

    The proof of Eq (4.63) can be completed by combining Eq (4.79) with Eq (4.62).

    Above all, the error of the Crank-Nicolson scheme (4.7)–(4.13) is given by the following theorem.

    Theorem 4.2. Let (\omega, y, \bar{u}, q, p) and (\omega_{\delta}, y_{\delta}, \bar{u}_{\delta}, q_{\delta}, p_{\delta}) be the local solutions of Eqs (3.6)–(3.8) and Eqs (4.7)–(4.13), respectively. Moreover, we assume that all conditions in Lemmas 4.2–4.5 are valid. Then, we have

    \begin{align*} &|| \omega-\omega_{\delta}||_{l^{\infty}(L^{2})}+|| y-y_{\delta}||_{l^{\infty}(H^{1})}\\ &+|| q-q_{\delta}||_{l^{\infty}(L^{2})}+|| p-p_{\delta}||_{l^{\infty}(H^{1})}\\ &+||\bar{u}-\bar{u}_{\delta}||_{l^{2}(L^{2}(\Omega ))}\leq C(h+(\Delta t)^{2} ). \end{align*}

    In this section, we provide a numerical example to verify the theoretical results, which will consider the following nonlinear SOHOCPs:

    \begin{align*} \min\frac{1}{2}&\int_{0}^{1}\int_{\Omega}(y-y_{d})^{2}dx+\int_{\Omega}u^{2}dx, \\ &s.t.\; \; y_{tt}-\Delta y+y^{3} = f+u. \end{align*}

    We adopt the same mesh triangular partition for the state and control. Furthermore, we choose

    \begin{align*} \Omega& = (0, 1)\times(0, 1), \; T = 1, \; u_{a} = -1, \; u_{b} = 5, \\ y_{d}& = (e^{t}+2+2\pi^{2}(t-T)^{2})\sin \pi x_{1}\sin \pi x_{2}+3e^{2t}(t-T)^{2}(\sin \pi x_{1}\sin \pi x_{2})^{3}, \\ f& = (1+2\pi ^{2})e^{t}\sin \pi x_{1}\sin \pi x_{2}+e^{3t}(\sin \pi x_{1}\sin \pi x_{2})^{3}\\ &\quad-\max\{u_{a}, \min \{u_{b}, (t-T)^{2}\sin \pi x_{1}\sin \pi x_{2}\}\}. \end{align*}

    such that the exact (y, u, p) is

    \begin{align*} y& = e^{t}\sin \pi x_{1}\sin \pi x_{2}, \\ u& = \max\{u_{a}, \min \{u_{b}, (t-T)^{2}\sin \pi x_{1}\sin \pi x_{2}\}\}, \\ p& = -(t-T)^{2}\sin \pi x_{1}\sin \pi x_{2}. \end{align*}

    We show the convergence results of the Crank-Nicolson scheme in Table 1. The profile of the exact (y, p, u) is drawn in Figure 1. The simulated results for the second-order scheme is presented in Figure 2.

    Table 1.  Numerical results of the Crank-Nicolson scheme ( h = \Delta t^2 ).
    \Delta t || y-y_{\delta}||_{l^{\infty}(H^{1}(\Omega))} Rate || p-p_{\delta}||_{l^{\infty}(H^{1}(\Omega))} Rate || u-u_{\delta}||_{l^{2}(L^{2}(\Omega))} Rate
    1/2 1.7731332397 0.6737539940 0.0349163048
    1/4 0.4450851439 1.9941 0.1680451731 2.0033 0.0056814014 2.6195
    1/8 0.1111334138 2.0017 0.0414201514 2.0204 0.0013428612 2.0809
    1/16 0.0277721376 2.0005 0.0103092780 2.0063 0.0003323956 2.0143
    1/32 0.0069422975 2.0001 0.0025741990 2.0017 0.0000831732 1.9987

     | Show Table
    DownLoad: CSV
    Figure 1.  The exact (y, p, u) with t = 0.5 .
    Figure 2.  The approximate ( y_{\delta}, p_{\delta}, u_{\delta} ) computed by the Crank-Nicolson scheme at t = 0.5 .

    From Table 1, it implies that the numerical results are consistent with the theoretical results. From Figure 1 and Figure 2, we can see the Crank-Nicolson scheme is efficient.

    This paper presents a second-order fully discrete scheme for nonlinear SOHOCPs and, in conjunction with auxiliary problems, derives a priori error estimates. Furthermore, a numerical experiment is conducted to confirm the convergence order of the theoretical results.

    In [31], Li et al. established a mixed-form discrete scheme for the nonlinear stochastic wave equations (SWEs) with multiplicative noise by defining a new variable. In [32], Sonawane et al. studied the existence of an optimal control problem for the bilinear SWEs. In the future, we plan to consider the method based on the definition of the new variable mentioned in this paper and in [31]. We will apply this method to the optimization system described in [32] and further conduct an error analysis of the resulting discrete scheme.

    Huanhuan Li was responsible for the methodology and writing the original draft. Meiling Ding contributed the software. Xianbing Luo handled the review, editing, and supervision. Shuwen Xiang oversaw the supervision and validation.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the National Natural Science Foundation of China (Granted No. 11961008) and Guizhou University Doctoral Foundation (Granted NO. 15 (2022)).

    The authors declare there is no conflict of interest.



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