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Research article

Enhanced evolutionary approach for solving fractional difference recurrent neural network systems: A comprehensive review and state of the art in view of time-scale analysis

  • Received: 03 August 2023 Revised: 17 October 2023 Accepted: 27 October 2023 Published: 14 November 2023
  • MSC : 46S40, 47H10, 54H25

  • The present research deals with a novel three-dimensional fractional difference neural network model within undamped oscillations. Both the frequency and the amplitude of movements in equilibrium are subsequently estimated mathematically for such structures. According to the stability assessment, the thresholds of the fractional order were determined where bifurcations happen, and an assortment of fluctuations bifurcate within an insignificant equilibrium state. For such discrete fractional-order connections, the parameterized spectrum of undamped resonances is also predicted, and the periodicity and strength of variations are calculated computationally and numerically. Several qualitative techniques, including the Lyapunov exponent, phase depictions, bifurcation illustrations, the 01 analysis and the approximate entropy technique, have been presented with the rigorous analysis. These outcomes indicate that the suggested discrete fractional neural network model has crucial as well as complicated dynamic features that have been affected by the model's variability, both in commensurate and incommensurate cases. Furthermore, the approximation entropy verification and C0 procedure are used to assess variability and confirm the emergence of chaos. Ultimately, irregular controllers for preserving and synchronizing the suggested framework are highlighted.

    Citation: Hanan S. Gafel, Saima Rashid. Enhanced evolutionary approach for solving fractional difference recurrent neural network systems: A comprehensive review and state of the art in view of time-scale analysis[J]. AIMS Mathematics, 2023, 8(12): 30731-30759. doi: 10.3934/math.20231571

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  • The present research deals with a novel three-dimensional fractional difference neural network model within undamped oscillations. Both the frequency and the amplitude of movements in equilibrium are subsequently estimated mathematically for such structures. According to the stability assessment, the thresholds of the fractional order were determined where bifurcations happen, and an assortment of fluctuations bifurcate within an insignificant equilibrium state. For such discrete fractional-order connections, the parameterized spectrum of undamped resonances is also predicted, and the periodicity and strength of variations are calculated computationally and numerically. Several qualitative techniques, including the Lyapunov exponent, phase depictions, bifurcation illustrations, the 01 analysis and the approximate entropy technique, have been presented with the rigorous analysis. These outcomes indicate that the suggested discrete fractional neural network model has crucial as well as complicated dynamic features that have been affected by the model's variability, both in commensurate and incommensurate cases. Furthermore, the approximation entropy verification and C0 procedure are used to assess variability and confirm the emergence of chaos. Ultimately, irregular controllers for preserving and synchronizing the suggested framework are highlighted.



    With the rapid development of science and technology, it is becoming more and more important to solve the optimal control problem by using appropriate numerical methods for satisfying various different actual requirements. Many numerical methods, such as finite volume element method, finite element method, mixed finite element method, and spectral method have been applied to approximate the solutions of optimal control problems (see, e.g., [4,7,8,9,12,16,18,20,21,22,23,25]). The optimal control problem of bilinear type considered in this paper includes a useful model of parameter estimation problems. It plays a very important role in many fields of science and engineering, where prior errors can improve accuracy and promote the development of related practical applications, such as air and water pollution control, oil exploration, and other fields. Although numerical analysis for bilinear optimal control problem was considered in a number of [11,23,28], there were few papers that consider the error estimates of finite volume element method for bilinear parabolic optimal control problem.

    The finite volume element methods lie somewhere between finite difference and finite element methods, they have a flexibility similar to that of finite element methods for handling complicated solution domain geometries and boundary conditions, and they have a simplicity for implementation comparable to finite difference methods with triangulations of a simple structure. The finite volume methods are effective discretization technique for partial differential equations. Bank and Rose obtained some results for elliptic boundary value problems that the finite volume element approximation was comparable with the finite element approximation in H1-norm which can be found in [3]. In [15], the authors presented the optimal L2-error estimates for second-order elliptic boundary value problems under the assumption that fH1, they also obtained the H1-norm and L-norm error estimates for those problems. In [27], Luo and Chen used the finite volume element method to obtain the approximation solution for optimal control problem associate with a parabolic equation by using optimize-then-discretize approach and the variational discretization technique. The authors also derived some error estimates for the semi-discrete approximation. Recently, the first author of this paper investigated L-error estimates of the bilinear elliptic optimal control problem by rectangular Raviart-Thomas mixed finite element methods in [23]. In this paper, we will study a priori error estimates for the finite volume element approximation of bilinear parabolic optimal control problem. By using finite volume element method to discretize the state and adjoint equations. Under some reasonable assumptions, we obtained some optimal order error estimates. Moreover, by employing the backward Euler method for the discretization of time, and using finite volume element method to discretize the state and adjoint equations, we will construct the fully discrete finite volume element approximation scheme for the bilinear optimal control problem. Then we obtain a priori error estimates for the fully discrete finite volume element approximation of bilinear parabolic optimal control problem.

    In this paper, we use the standard notations Wm,p(Ω) for Sobolev spaces and their associated norms ||v||m,p (see, e.g., [1]) in these paper. To simplify the notations, we denote Wm,2(Ω) by Hm(Ω) and drop the index p=2 and Ω whenever possible, i.e., ||u||m,2,Ω=||u||m,2=||u||m, ||u||0=||u||. Set H10(Ω)={vH1:v|Ω=0}. As usual, we use (,) to denote the L2(Ω)-inner product.

    We consider the following bilinear parabolic optimal control problem

    minuUad{12T0(||y(x,t)yd(x,t)||2L2(Ω)+α||u(x,t)||2L2(Ω))dt}, (1.1)
    yt(x,t)(Ay(x,t))+u(x,t)y(x,t)=f(x,t),  tJ, xΩ, (1.2)
    y(x,t)=0,  tJ, xΓ,  y(x,0)=y0,  xΩ, (1.3)

    where α is a positive constant,

    (Ay)=xi(aij(x)yxj),

    ΩR2 is a bounded convex polygonal domain and Γ is the boundary of Ω, f(,t)L2(Ω) or H1(Ω), J=(0,T], A=(ai,j)2×2 is a symmetric, smooth enough and uniformly positive definite matrix in Ω, y0(x)=0,xΓ. It is assumed that the functions y have enough regularity and they satisfy appropriate compatibility conditions so that the boundary value problems (1.1)–(1.3) has a unique solution. Uad is a set defined by

    Uad={u: uL2(J;L2(Ω)),u(x,t)0, a.e. in Ω, tJ, a,bR}.

    The remainder of this paper is organized as follows. In Section 2, we present some notations and the finite volume element approximation for the bilinear parabolic optimal control problem. In Section 3, we analyze the error estimates between the exact solution and the finite volume element solution. In Section 4, a priori error estimates for the fully discrete finite volume element approximation of the bilinear optimal control problem are presented. a numerical example is presented to test the theoretical results in Section 5. Finally, we briefly give conclusions and some possible future works in Section 6.

    For the convex polygonal domain Ω, we consider a quasi-uniform triangulation Th consisting of closed triangle elements K such that ˉΩ=KThK. We use Nh to denote the set of all nodes or vertices of Th. To define the dual partition Th of Th, we divide each KTh into three quadrilaterals by connecting the barycenter CK of K with line segments to the midpoints of edges of K as is shown in Figure 1.

    Figure 1.  The dual partition of a triangularK.

    The control volume Vi consists of the quadrilaterals sharing the same vertex zi as is shown in Figure 2.

    Figure 2.  The control volume Vi sharing the same vertex zi.

    The dual partition Th consists of the union of the control volume Vi. Let h=max{hK}, where hK is the diameter of the triangle K. As is shown in [15], the dual partition Th is also quasi-uniform. Throughout this paper, the constant C denotes different positive constant, which is independent of the mesh size h and the time step k.

    We define the finite dimensional space Vh associated with Th for the trial functions by

    Vh={v: vC(Ω), v|KP1(K),  KTh, v|Γ=0},

    and define the finite dimensional space Qh associated with the dual partition Th for the test functions by

    Qh={q: qL2(Ω), q|VP0(V),  VTh; q|Vz=0, zΓ},

    where Pl(K) or Pl(V) consists of all the polynomials with degree less than or equal to l defined on K or V.

    To connect the trial space and test space, we define a transfer operator Ih:VhQh as follows:

    Ihvh=ziNhvh(zi)χi,  Ihvh|Vi=vh(zi),   ViTh,

    where χi is the characteristic function of Vi. For the operator Ih, it is well known that there exists a positive constant C such that for all vVh, we can get

    ||vIhv||Ch||v||1. (2.1)

    Let a(w,v)=ΩAwvdx. We assume a(v,v) satisfies the coercive conditions, then coercive property of a(,) is that there exists a positive constant c such that for all vVh, we can obtain (see, e.g., [5])

    a(v,v)c||v||21. (2.2)

    As is defined in [6], we define the standard Ritz projection Rh:H10Vh by

    a(Rhu,χ)=a(u,χ),   χVh. (2.3)

    For the projection Rh, it has the property that (see, e.g., [6])

    ||Rhuu||Ch2||u||2.  (2.4)

    Now, we will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the bilinear parabolic optimal control problem.

    As is seen in [25], the necessary and sufficient optimal condition of (1.1)–(1.3) consists of the state equation, a co-state equation and a variational inequality, i.e., find y(,t), p(,t)H10(Ω) and u(,t)Uad such that

    (yt,w)+(Ay,w)+(uy,w)=(f,w), wH10(Ω),  y(x,0)=y0(x), (2.5)
    (pt,q)+(Ap,q)+(up,q)=(yyd,q), qH10(Ω),  p(x,T)=0, (2.6)
    T0(αuyp,vu)dt0, vUad. (2.7)

    If y(,t)H10(Ω)C2(Ω) and p(,t)H10(Ω)C2(Ω), then the optimal control problems (2.5)–(2.7) can be written by

    yt(Ay)+uy=f, tJ, xΩ, (2.8)
    y(x,t)=0, tJ, xΓ,   y(x,0)=y0(x), xΩ, (2.9)
    pt(Ap)+up=yyd, tJ, xΩ, (2.10)
    p(x,t)=0, tJ, xΓ,   p(x,T)=0,  xΩ, (2.11)
    T0(αuyp,vu)dt0, vUad. (2.12)

    Then, we use the finite volume element method to discretize the state and co-state equations directly. Then the continuous optimal control problems (2.8)–(2.12) can be approximated by: find (yh(,t),ph(,t),uh(,t))Vh×Vh×Uad such that

    (yht,Ihwh)+ah(yh,Ihwh)+(uhyh,Ihwh)=(f,Ihwh), whVh, (2.13)
    yh(x,0)=Rhy0(x), xΩ, (2.14)
    (pht,Ihqh)+ah(ph,Ihqh)+(uhph,Ihqh)=(yhyd,Ihqh), qhVh, (2.15)
    ph(x,T)=0, xΩ, (2.16)
    T0(αuhyhph,vuh)dt0, vUad, (2.17)

    where ah(ϕ,Ihψ)=ziNhψ(zi)ViAϕnds.

    Similar to [17], we can find that the variational inequality (2.12) is equivalent to

    u(x,t)=max(y(x,t)p(x,t),0). (2.18)

    And then the variational inequality (2.17) is equivalent to

    uh(x,t)=max(yh(x,t)ph(x,t),0). (2.19)

    In this section, we will analyze the error between the exact solution and the finite volume element solution. Let εa(x,y)=a(x,y)ah(x,Ihy), it is well known (see, e.g., [6,13]) that for all yVh:

    |εa(x,y)|Ch||x||1||y||1, xVh. (3.1)

    To deduce the error estimates, let (yh(u), ph(u)) be the solution of

    (yht(u),Ihwh)+ah(yh(u),Ihwh)+(uyh(u),Ihwh)=(f,Ihwh),  (3.2)
    yh(u)(x,0)=Rhy0(x), xΩ, (3.3)
    (pht(u),Ihqh)+ah(ph(u),Ihqh)+(uph(u),Ihqh)=(yh(u)yd,Ihqh), (3.4)
    ph(u)(x,T)=0, xΩ, (3.5)

    where for all wh,qhVh, note that yh=yh(uh), ph=ph(uh), we have the following lemma for yh(u) and ph(u).

    Let (p(u),y(u)) and (ph(u),yh(u)) be the solutions of (2.13)–(2.15) and (3.2)–(3.4), respectively. Let J():UadR be a G-differential convex functional near the solution u which satisfies the following form:

    J(u)=12||yyd||2L2(Ω)+α2||u||2L2(Ω).

    Then we have a sequence of convex functional Jh:UadR:

    Jh(u)=12||yh(u)yd||2L2(Ω)+α2||u||2L2(Ω),Jh(uh)=12||yh(uh)yd||2L2(Ω)+α2||uh||2L2(Ω).

    It can be shown that

    (J(u),v)=(αuyp,v),(Jh(u),v)=(αuyh(u)ph(u),v),(Jh(uh),v)=(αuhyhph,v).

    In the following we estimate uuhL2(J;L2). We assume that the cost function J is strictly convex near the solution u, i.e., for the solution u there exists a neighborhood of u in L2 such that J is convex in the sense that there is a constant c>0 satisfying:

    T0(J(u)J(v),uv)dtcuv2L2(J;L2). (3.6)

    For all v in this neighborhood of u. The convexity of J() is closely related to the second order sufficient optimality conditions of optimal control problems, which are assumed in many studies on numerical methods of the problem. For instance, in many references, the authors assume the following second order sufficiently optimality condition (see [14,26]): there is c>0 such that J(u)v2cv20.

    From the assumption (3.6), by the proof contained in [2,10], there exists a constant c>0 satisfying

    T0(Jh(v)Jh(u),vu)dtcvu2L2(J;L2), vUad. (3.7)

    Now, we estimate the error of the approximate control in L2-norm.

    Theorem 3.1. Let (y,p,u) and (yh,ph,uh) are the solutions of (2.5)–(2.7) and (2.13)–(2.17), respectively. Then there exists a constant h0>0 such that for all 0<hh0, we have

    ||uuh||L2(J;L2)Ch2. (3.8)

    Proof. Let v=uh in (2.7) and v=u in (2.17), then we have

    T0(αuyp,uhu)dt0, (3.9)
    T0(αuhyhph,uuh)dt0. (3.10)

    From (3.9) and (3.10), it is easy to see that

    T0α(uuh,uuh)dtT0(ypyhph,uuh)dt. (3.11)

    By using (3.7) and (3.11), we obtain

    cuuh2L2(J;L2)T0(Jh(u),uuh)dtT0(Jh(uh),uuh)dt=T0(αuyh(u)ph(u),uuh)dtT0(αuhyhph,uuh)dt=T0α(uuh,uuh)dt+T0(yhphyh(u)ph(u),uuh)dtT0(ypyhph,uuh)dt+T0(yhphyh(u)ph(u),uuh)dt=T0(ypyh(u)ph(u),uuh)dt=T0(ypyh(u)p,uuh)dt+T0(yh(u)pyh(u)ph(u),uuh)dt. (3.12)

    Now, we estimate all terms at the right side of (3.12). By using the Theorem 4.2 in [24] and the δ-Cauchy inequality, we have

    T0(ypyh(u)p,uuh)dtCyyh(u)L2(J;L2)uuhL2(J;L2)Ch2uuhL2(J;L2)Ch4+δuuh2L2(J;L2). (3.13)

    In the same way, we also have

    T0(yh(u)pyh(u)ph(u),uuh)dtCph(u)pL2(J;L2)uuhL2(J;L2)Ch2uuhL2(J;L2)Ch4+δuuh2L2(J;L2). (3.14)

    Putting (3.13) and (3.14) into (3.12) and choosing appropriate value for δ, we can obtain the result (3.8).

    Theorem 3.2. Let (y,p,u) and (yh,ph,uh) are the solutions of (2.5)–(2.7) and (2.13)–(2.17), respectively. Then there exists a constant h0>0 such that for all 0<hh0, we have

    yyhL(J;L2)+pphL(J;L2)Ch2, (3.15)
    yyhL(J;H1)+pphL(J;H1)Ch. (3.16)

    Proof. By employing the triangle inequality, we have

    yyhL(J;L2)yyh(u)L(J;L2)+yh(u)yhL(J;L2),pphL(J;L2)pph(u)L(J;L2)+ph(u)phL(J;L2).

    Similar to Lemma 3.1 in [16], it implies that

    yyhL(J;L2)yyh(u)L(J;L2)+CuuhL2(J;L2), (3.17)
    pphL(J;L2)pph(u)L(J;L2)+CuuhL2(J;L2). (3.18)

    By using Theorem 3.1, (3.17)–(3.18) and Theorem 4.2 of [24], we can easily obtain

    yyhL(J;L2)yyh(u)L(J;L2)+CuuhL2(J;L2)Ch2+CuuhL2(J;L2)Ch2. (3.19)

    In the same way, we also have

    pphL(J;L2)pph(u)L(J;L2)+CuuhL2(J;L2)Ch2+CuuhL2(J;L2)Ch2. (3.20)

    Combining (3.19) with (3.20), we can prove (3.15). Similarly, we can obtain (3.18).

    In this section, we will present a fully discrete scheme and error estimates of the finite volume element approximation.

    Now, we shall construct the fully discrete approximation scheme for semi-discrete scheme (2.13)–(2.17). Let 0=t0<t1<<tM1<tm=T, ti=it, t=TM, for i=1,2,,M. And let ψi=ψ(x,ti), ψi=(ψiψi1)/t. We define a discrete time-dependent norm for 1s< by |||ψ|||Ls(J;Hm(Ω))=(Mi=1t||ψi||sm)1/s (e.g., |||ψ|||L2(J;L2)=(Mi=1t||ψi||2)1/2, |||ψ|||L(J;L2)=max1iM||ψi||). By using the backward Euler method for the discretization of time in (2.13) and (2.15), we can obtain the fully discrete scheme of (2.13)–(2.17) is to find (yih,pi1h,uih)Vh×Vh×Uad such that

    (yih,Ihwh)+ah(yih,Ihwh)+(uihyih,Ihwh)=(fi,Ihwh), whVh, (4.1)
    y0h(x)=Rhy0(x), xΩ, i=1,2,,M; (4.2)
    (pih,Ihqh)+ah(pi1h,Ihqh)+(uihpi1h,Ihwh)=(yihyid,Ihqh), qhVh, (4.3)
    pMh(x)=0, xΩ, i=M,M1,,1; (4.4)
    (αuihyi1hpi1h,vuih)0, vUad, i=1,2,,M. (4.5)

    To derive the fully discrete error analysis, let (yih(u),pi1h(u)) be the solution of

    (yih(u),Ihwh)+ah(yih(u),Ihwh)+(uiyih(u),Ihwh)=(fi,Ihwh), whVh, (4.6)
    y0h(u)(x)=Rhy0, xΩ, i=1,2,,M, (4.7)
    (pih(u),Ihqh)+ah(pih(u),Ihqh)+(uihpi1h(u),Ihwh)=(yih(u)yid,Ihqh), (4.8)
     qh(u)Vh, pMh(u)(x)=0, xΩ, i=M,M1,,1. (4.9)

    Let uh=(u0h,u1h,,uMh), yh=(y0h,y1h,,yMh) and ph=(p0h,p1h,,pMh). For (yih(u),pi1h(u)), we have the following lemma.

    Lemma 4.1. Assume that (yih,pi1h,uih) and (yih(u),pi1h(u)) are the solutions of (4.1)–(4.5) and (4.6)–(4.9), respectively. Then we have the following results:

    ||yih(u)yih||H1(Ω)C|||uuh|||L2(J;L2(Ω)), (4.10)
    ||pih(u)pih||H1(Ω)C|||uuh|||L2(J;L2(Ω)). (4.11)

    Proof. Let ηk=ykh(u)ykh (1ki). Subtracting (4.1) from (4.6), we have

    (ηk,Ihwh)+ah(ηk,Ihwh)+(ukηk,Ihwh)=((ukhuk)ykh,Ihwh), whVh.

    Let wh=ηk, we have

    (ηk,Ihηk)+ah(ηk,Ihηk)+(ukηk,Ihηk)=((ukhuk)ykh,Ihηk).

    Due to εa(x,y)=a(x,y)ah(x,Ihy), we can get

    (ηk,Ihηk)+a(ηk,ηk)=εa(ηk,ηk)(ukηk,Ihηk)+((ukhuk)ykh,Ihηk).

    Note that

    a(ηk,ηk)=12t(a(ηk,ηk)a(ηk1,ηk1)+a(ηkηk1,ηkηk1)).

    Thanks to a(ηkηk1,ηkηk1)0, we obtain

    (ηk,Ihηk)+12t(a(ηk,ηk)a(ηk1,ηk1))εa(ηk,ηk)(ukηk,Ihηk)+((ukhuk)ykh,Ihηk).

    The inverse estimate and (3.1) imply that

    εa(ηk,ηk)Ch||ηk||1||ηk||1C||ηk||1||ηk||C||ηk||21+Cδ ||ηk||2.

    Note that

    (ukηk,Ihηk)C ||ηk||2+Cδ (Ihηk,Ihηk),

    and

    ((ukhuk)ykh,Ihηk)C ||ukukh||2+Cδ (Ihηk,Ihηk).

    Using the equivalent properties of (,), (,Ih()) and (Ih(),Ih()), we derive

    Cδ (ηk,ηk)+a(ηk,ηk)a(ηk1,ηk1)+Ct||ηk||21+Ct||ukukh||2.

    Choosing appropriate value for δ, we have

    a(ηk,ηk)a(ηk1,ηk1)+Ct||ηk||21+Ct||ukukh||2.

    Applying the coercive property of a(,), and summing k from 1 to i and noticing η0=0, we can get

    ||ηi||21ik=1Ct||ηk||21+ik=1Ct||ukukh||2.

    By using the discrete Gronwall's lemma (see, e.g., Lemma 3.3 in [9]), Then we have

    ||ηi||1=||yih(u)yih||1C|||uuh|||L2(J;L2(Ω)).

    Then we prove (4.10). In the same way as (4.10), we can obtain (4.11).

    We can get the error estimate for uh in the discrete L2(J;L2)-norm by using Lemma 4.1.

    Theorem 4.1. Let (y,p,u) and (yih,pi1h,uih) be the solutions of problems (2.5)–(2.7) and (4.1)–(4.5), respectively. Then there exists a constant h0>0 such that for all 0<hh0, we have

    |||uuh|||L2(J;L2)C(h3/2+t). (4.12)

    Proof. Note that yih=yih(uih) and pih=pih(uih), it can be shown that

    (Jh(ui),v)=(αuiyih(u)pih(u),v),(Jh(uih),v)=(αuihyihpih,v).

    From (2.7), (3.7), and (4.5), we have

    α|||uuh|||2L2(J;L2)Mi=1titi1(Jh(ui),uiuih)dtMi=1titi1(Jh(uih),uiuih)dt=Mi=1titi1(αuiyih(u)pih(u),uiuih)dtMi=1titi1(αuihyihpih,uiuih)dt=Mi=1t(αuiyih(u)pih(u),uiuih)Mi=1t(αuihyihpih,uiuih)Mi=1t(yipiyi1hpi1h,uiuih)Mi=1t(yih(u)pih(u)yihpih,uiuih)Mi=1t(yipiyih(u)pih(u),uiuih)+Mi=1t(yihpihyi1hpi1h,uiuih)T1+T2.

    For the first term T1, using the Cauchy inequality and the Theorem 4.1 in [29], we can obtain

    T1=Mi=1t(yipiyih(u)pih(u),uiuih)=Mi=1t(yipiyih(u)pi,uiuih)+Mi=1t(yih(u)piyih(u)pih(u),uiuih)C(Mi=1t||yiyih(u)||||uiuih||+Mi=1t||pipih(u)||||uiuih||)C(h3/2+t)|||uuh|||L2(J;L2(Ω)).

    For the second term T2, we can derive

    T2=Mi=1t(yihpihyi1hpi1h,uiuih)Ct |||(yhph)t|||L2(J;L2(Ω))|||uuh|||L2(J;L2(Ω))Ct|||uuh|||L2(J;L2(Ω)).

    Connecting T1 and T2, we can obtain (4.12) easily for sufficiently small h.

    Then we can obtain the following result from Lemma 4.1 and Theorem 4.1.

    Theorem 4.2. Let (y,p,u) and (yih,pi1h,uih) be the solutions of problems (2.5)–(2.7) and (4.1)–(4.5), respectively. Then there exists a constant h0>0 such that for all 0<hh0, we have

    |||yyh|||L(J;L2)+|||pph|||L(J;L2)C(h3/2+t). (4.13)

    In this section, we give a numerical example to validate the error estimates for the control, state and adjoint state. We consider the bilinear parabolic optimal control problem:

    minu(t)Uad1210(||yyd||2L2(Ω)+||uu0||2L2(Ω))dt,ytΔy+uy=f,  (x,t)Ω×J,y(x,t)=0,  (x,t)Ω×J,y(x,0)=0,  xΩ,

    where Ω=[0,1]×[0,1], J=(0,1], and Uad={u:u0}. The dual equation of the state equation is

    ptΔp+up=yyd.

    Then we assume that

    y(x,t)=sin(πx1)sin(πx2)t,p(x,t)=sin(πx1)sin(πx2)(1t),u0(x,t)=0.5sin(πx1)sin(πx2)t,yd(x,t)=y+pt+Δpup,u(x,t)=max(u0+yp,0),f(x,t)=ytΔy+uy.

    Firstly, we adopt the same mesh partition for the state and the control such that t=h32 in our test. In this case, we investigate the convergence order for the solutions which compute on a series of uniformly triangular meshes. We present the L2(J;L2), L(J;L2) and L(J;L2) errors for u, y and p in Table 1, which means that the convergent rates are O(h32+t). We show the convergence orders in Figure 3, where dofs denotes degree of freedoms. It is easy to see that this is consistent with the results proved in the previous.

    Table 1.  The numerical errors for state and control variables.
    Resolution Errors
    ||uuh||L2(J;L2) yyhL(J;L2) pphL(J;L2)
    16×16 6.24692E-03 3.05468E-02 1.91005E-02
    32×32 2.10214E-03 9.84713E-03 6.35234E-03
    64×64 6.66687E-04 3.30168E-03 2.11847E-03
    128×128 2.39392E-04 1.18101E-03 7.44588E-04

     | Show Table
    DownLoad: CSV
    Figure 3.  Convergence orders of uuh, yyh, and pph in different norms.

    In order to explore more on the rates of convergence separately in time and space, we try to validate the estimates by separating the discretization errors. We consider the behavior of the errors under refinement of the spatial triangulation for fixed t=180. Then, we show the errors for u, y, and p in Table 2. Figure 4 depicts the convergence orders under refinement of the spatial triangulation for fixed t=180. We can observe the order is O(h32).

    Table 2.  The numerical errors for state and control variables.
    h Errors
    ||uuh||L2(J;L2) yyhL(J;L2) pphL(J;L2)
    116 4.98753E-03 2.53465E-02 1.87542E-02
    132 1.66221E-03 8.44288E-03 6.22578E-03
    164 5.54334E-04 2.81459E-03 2.15089E-03
    1128 1.98298E-04 9.89101E-04 7.71995E-04

     | Show Table
    DownLoad: CSV
    Figure 4.  Convergence orders of uuh, yyh, and pph in different norms.

    Finally, we examine the behavior of the errors for a sequence of discretizations with decreasing size of the time steps and a fixed spatial triangulation with h=1256. The L2(J;L2), L(J;L2) and L(J;L2) error norms for control variable state variable and adjoin variable are shown in Table 3. In Figure 5, the convergence orders under refinement of the time steps for h=1256 are shown. Up to the discretization errors it exhibits the proven convergence order O(t). From the numerical results, we observe that convergence of order O(t) which demonstrates our theoretical results.

    Table 3.  The numerical errors for state and control variables.
    t Errors
    ||uuh||L2(J;L2) yyhL(J;L2) pphL(J;L2)
    110 4.68547E-03 2.34568E-02 1.62486E-02
    120 1.56117E-03 7.81571E-03 5.23753E-03
    140 5.22744E-04 2.60798E-03 1.74768E-03
    180 1.91003E-04 9.36223E-04 6.30968E-04

     | Show Table
    DownLoad: CSV
    Figure 5.  Convergence orders of uuh, yyh, and pph in different norms.

    Seen from the numerical results listed in Tables 13 and Figures 35, it is easy to find that the convergent orders match the theories derived in the previous sections.

    In this paper, we established semi-discrete and fully discrete finite volume element approximation scheme of bilinear parabolic optimal control problem. Then we used the finite volume element method to discretize the state and adjoint equations of the system. Under some reasonable assumptions, we obtained some error estimates. To our best knowledge in the context of optimal control problems, the priori error estimates of finite volume element method for bilinear parabolic optimal control problems are new.

    In the future, we shall consider the finite volume element method for bilinear hyperbolic optimal control problems. Furthermore, we shall consider a priori error estimates and superconvergence of the finite volume element solutions for hyperbolic optimal control problems.

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

    This work is supported by National Science Foundation of China (11201510), National Social Science Fund of China (19BGL190), Natural Science Foundation of Chongqing (CSTB2022NSCQ-MSX0286), Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJZD-K202001201), Chongqing Key Laboratory of Water Environment Evolution and Pollution Control in Three Gorges Reservoir Area (WEPKL2018YB04), Research Center for Sustainable Development of Three Gorges Reservoir Area (2022sxxyjd01), Guangdong Basic and Applied Basic Research Foundation of Joint Fund Project (2021A1515111048), and Guangdong Province Characteristic Innovation Project (2021WTSCX120).

    The authors declare that they have no competing interests.



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