# Adaptive Semidiscrete Finite Element Methods for Semilinear Parabolic Integrodifferential Optimal Control Problem with Control Constraint

Authors: Zuliang Lu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The aim of this work is to study the semidiscrete finite element discretization for a class of semilinear parabolic integrodifferential optimal control problems. We derive a posteriori
error estimates inL2(J;L2(Ω))-norm and L2(J;H1(Ω))-norm for both the control and coupled state approximations. Such estimates can be used to construct reliable adaptive finite element approximation for semilinear parabolic integrodifferential optimal control problem. Furthermore, we introduce an adaptive algorithm to guide the mesh refinement. Finally, a numerical example is given to demonstrate the theoretical results.


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## Body

## 1. Introduction

With the advances of scientific computing, optimal control problems are now widely used in multidisciplinary applications such as physical, biological, medicine, engineering design, finance, fluid mechanics, and socioeconomic systems. As a result, more and more people will benefit greatly by learning to solve the optimal control problems numerically. Realizing such growing needs, books and papers on optimal control put more weight on numerical methods.In modeling a wide range of problems for physical, economic, and social processes, optimal control problems described by integrodifferential equations play an important role. Parabolic integrodifferential optimal control problems are very important model in engineering numerical simulation, for example, biology mechanics, nuclear reaction dynamics, heat conduction in materials with memory, visco-elasticity, and so forth. Finite element approximation of optimal control problems has a very important status in the numerical methods for these problems. The finite element approximation of optimal control problem by piecewise constant functions was well investigated by Falk [1] and Geveci [2]. The finite element methods for semilinear elliptic optimal control problems were discussed by Casas et al. in [3]. In [4], the author studied the finite element discretization for a class of quadratic boundary optimal control problems governed by nonlinear elliptic equations and obtained a posteriori error estimates for the coupled state and control approximation. Many introductions about the numerical computation method for optimal control problems can be found in [5–8].As one of important kinds of optimal control problems, parabolic integrodifferential optimal control problem is widely used in scientific and engineering computing. The literature in this aspect was huge, see; for example, [9]. In [10], Brunner and Yan analyzed finite element Galerkin discretization for a class of optimal control problems governed by integral equations and integrodifferential equations and derived a priori error estimates and a posteriori error estimators for the approximation solutions.Adaptive finite element method is the most important method to boost accuracy of the finite element discretization. It ensures a higher density of nodes in certain area of the given domain, where the solution is discontinuous or more difficult to approximate, using an a posteriori error indicator. A posteriori error estimates are computable quantities in terms of the discrete solution and they measure the actual discrete errors without the knowledge of exact solutions. They are essential in designing algorithms for mesh which equidistribute the computational effort and optimize the computation. The literature in this was huge. In [11], the authors presented an a posteriori error analysis for finite element approximation of distributed convex elliptic optimal control problems. They derived a posteriori error estimates for the coupled state and control approximations. Such estimates can be used to construct reliable adaptive finite element approximation schemes for control problems. In [12], Verfürth gave a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear elliptic equations. He obtained a posteriori error estimates, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution. Some of techniques directly relevant to our work can be found in [11, 12]. Recently, in [13–16], we derived a priori error estimates and a posteriori error estimates for optimal control problems using mixed finite element methods. Although a posteriori error estimates of finite element approximation were widely used in numerical simulations, they have not yet been utilized in nonlinear parabolic integrodifferential optimal control problem.In this paper, we adopt the standard notationWm,p(Ω) for Sobolev spaces on Ω with a norm ∥·∥m,p given by ∥v∥m,pp=∑|α|≤m∥Dαv∥Lp(Ω)p and a seminorm |·|m,p given by |v|m,pp=∑|α|=m∥Dαv∥Lp(Ω)p. We set W0m,p(Ω)={v∈Wm,p(Ω):v|∂Ω=0}. For p=2, we denote Hm(Ω)=Wm,2(Ω), H0m(Ω)=W0m,2(Ω), and ∥·∥m=∥·∥m,2, ∥·∥=∥·∥0,2. We denote by Ls(0,T;Wm,p(Ω)) the Banach space of all Ls integrable functions from J to Wm,p(Ω) with norm ∥v∥Ls(J;Wm,p(Ω))=(∫0T‍∥v∥Wm,p(Ω)sdt)1/sfors∈[1,∞), and the standard modification for s=∞. The details can be found in [9].In this paper, we derive a posteriori error estimates for a class of semilinear parabolic integrodifferential optimal control problems. To the best of our knowledge in the context of semilinear parabolic integrodifferential optimal control problems, these estimates are new. We consider the following semilinear parabolic integrodifferential optimal control problems:(1)minu(t)∈K⊂U{∫0T‍(12∥y-y0∥2+α2∥u∥2)dt}
subject to the state equation
(2)yt-div(A∇y(x,t))-∫0t‍div(ψ(t,τ)∇y(x,τ))dτ+ϕ(y)=f+Bu,x∈Ω,t∈J,y(x,t)=0,x∈∂Ω,t∈J,y(x,0)=y0(x),x∈Ω,
where the bounded open set Ω⊂ℝ2 is2regular convex polygon with boundary ∂Ω, J=(0,T], f∈L2(Ω), ψ=ψ(x,t,τ)=ψi,j(x,t,τ)2×2∈C∞(0,T;L2(Ω-))2×2, y0∈H1(Ω), α is a positive constant, and B is a continuous linear operator from K to L2(Ω). For any R>0, the function ϕ(·)∈W2,∞(-R,R), ϕ′(y)∈L2(Ω) for any y∈L2(J;H01(Ω)), and ϕ′(y)≥0. We assume that the coefficient matrix A(x)=(ai,j(x))2×2∈(W1,∞(Ω))2×2 is a symmetric positive definite matrix and there is a constant c>0 satisfying for any vector X∈ℝ2, XtAX≥c∥X∥ℝ22. Here, K denotes the admissible set of the control variable defined by
(3)K={u(x,t)∈U=L2(J;L2(Ω)):u(x,t)≥0,t∈J}.
Assume that there are constants c and C, such that for all t and τ in [0,T]:
(4)a(z,z)≥c∥z∥1,Ω2,∀z∈V,(5)|a(z,w)|≤C∥z∥1,Ω·∥w∥1,Ω,∀z,w∈V,(6)|ψ(t,τ;z,w)|≤C∥z∥1,Ω·∥w∥1,Ω,∀z,w∈V.The plan of this paper is as follows. In the next section, we present the finite element discretization for semilinear parabolic integrodifferential optimal control problems. A posteriori error estimates are established for the optimal control problems in Section3. In Section 4, we introduce an adaptive algorithm to guide the mesh refinement. In Section 5, a numerical example is given to demonstrate our theoretical results. Finally, we analyze the conclusion and future work in Section 6.


## 2. Finite Elements for Integrodifferential Optimal Control

We will now describe the finite element discretization of semilinear parabolic integrodifferential optimal control problems (1)-(2). Let V=H01(Ω) and W=L2(Ω). Let
(7)a(y,w)=∫Ω‍(A∇y)·∇w,∀y,w∈V,ψ(t,τ;z,w)=(ψ(t,τ)∇z,∇w),∀z,w∈V×V,(u,v)=∫Ω‍uv,∀(u,v)∈W×W,(f1,f2)=∫Ω‍f1f2,∀(f1,f2)∈W×W.
Then, the semilinear parabolic integrodifferential optimal control problems (1)-(2) can be restated as
(8)minu(t)∈K{∫0T‍(12∥y-y0∥2+α2∥u∥2)dt}
subject to
(9)(yt,w)+a(y,w)+∫0t‍ψ(t,τ;y(τ),w)dτ+(ϕ(y),w)=(f+Bu,w),∀w∈V,t∈J,y(x,0)=y0(x),x∈Ω,
where the inner product inL2(Ω)orL2(Ω)2is indicated by (·,·). From Yanik and Fairweather [17], we know that the above weak form has at least one solution in y∈W(0,T)={w∈L2(0,T;H1(Ω)),wt′∈L2(0,T;H-1(Ω))}.It is well known (see, e.g., [11]) that the optimal control problem has a solution (y,u), and if a pair (y,u) is the solution of (8)-(9), then there is a costate p∈V such that triplet (y,p,u) satisfies the following optimality conditions:
(10)(yt,w)+a(y,w)+∫0t‍ψ(t,τ;y(τ),w)dτ+(ϕ(y),w)=(f+Bu,w),∀w∈V,(11)y(x,0)=y0(x),x∈Ω,(12)-(pt,w)+a(q,p)+∫tT‍ψ(τ,t;q,p(τ))dτ+(ϕ′(y)p,q)=(y-y0,q),∀q∈V,(13)p(x,T)=0,x∈Ω,(14)∫0T‍(αu+B*p,v-u)Udt≥0,∀v∈K,
whereB*is the adjoint operator ofB. In the rest of the paper, we will simply write the product as(·,·)whenever no confusion should be caused.Let us consider the finite element approximation of the optimal control problem (8)-(9). Again here we consider onlyn-simplex elements and conforming finite elements.Let𝒯h be regular partition of Ω. Associated with 𝒯h is a finite dimensional subspace Vh of C(Ω-), such that χ|τ are polynomials of m-order (m≥1) for all χ∈Vh and τ∈𝒯h. It is easy to see that Vh⊂V. Let hτ denote the maximum diameter of the element τ in 𝒯h, h=maxτ∈𝒯h{hτ}.Due to the limited regularity of the optimal controluin general, there will be no advantage in considering higher-order finite element spaces for the control. So, we only consider the piecewise constant finite element space for the approximation of the control, though higher-order finite element spaces will be used to approximate the state and the costate. Let P0(τ) denote all the0-order polynomial over τ. Therefore, we take Kh={u∈K:u(x,t)|τ∈P0(τ)}. In addition, C orcdenotes a general positive constant independent of h.By the definition of finite element subspace, the finite element discretization of (8)-(9) is as follows: compute (yh,uh)∈Vh×Kh such that
(15)minuh∈Kh{∫0T‍(12∥yh-y0∥2+α2∥uh∥2)dt}(16)(yht,wh)+a(yh,wh)+∫0t‍ψ(t,τ;yh(τ),wh)dτ+(ϕ(yh),wh)=(f+Buh,wh),(17)yh(x,0)=y0h(x),x∈Ω,
where wh∈Vh, y0h∈Vh is an approximation of y0.Again, it follows that the optimal control problems (15)–(17) have a solution (yh,uh), and if a pair (yh,uh) is the solution of (15)–(17), then there is a costate ph∈Vh such that triplet (yh,ph,uh) satisfies the following optimality conditions:
(18)(yht,wh)+a(yh,wh)+∫0t‍ψ(t,τ;yh(τ),wh)dτ+(ϕ(yh),wh)=(f+Buh,wh),(19)yh(x,0)=y0h(x),x∈Ω,(20)-(pht,wh)+a(qh,ph)+∫tT‍ψ(τ,t;qh,ph(τ))dτ+(ϕ′(yh)ph,qh)=(yh-y0,qh),(21)ph(x,T)=0,x∈Ω,(22)∫0T‍(αuh+B*ph,vh-uh)dt≥0,
where wh,qh∈Vh, and vh∈Kh.In the rest of the paper, we will use some intermediate variables. For any control functionuh∈K, we first define the state solution (y(uh),p(uh)) satisfying
(23)(yt(uh),w)+a(y(uh),w)+∫0t‍ψ(t,τ;y(uh)(τ),w)dτ+(ϕ(y(uh)),w)=(f+Buh,w),(24)∀w∈V,y(uh)(x,0)=y0(x),x∈Ω,(25)-(pt(uh),q)+a(q,p(uh))+∫tT‍ψ(τ,t;q,p(uh)(τ))dτ+(ϕ′(y(uh))p(uh),q)(26)=(y(uh)-y0,q),∀q∈V,p(uh)(x,T)=0,x∈Ω.Now, we restate the following well-known estimates in [9].Lemma 1.
Letπ^h be the Clément-type interpolation operator defined in [9]. Then for any v∈H1(Ω) and all element τ,
(27)∥v-π^hv∥L2(τ)+hτ∥∇(v-π^hv)∥L2(τ)≤Chτ∑τ-′∩τ-≠∅‍|v|L2(τ′),∥v-π^hv∥L2(l)≤Chl1/2∑l⊂τ-′‍|∇v|L2(τ′),
where l is the edge of the element.Forφ∈Wh, we will write
(28)ϕ(φ)-ϕ(ρ)=-ϕ~′(φ)(ρ-φ)=-ϕ′(ρ)(ρ-φ)+ϕ~′′(φ)(ρ-φ)2,
where
(29)ϕ~′(φ)=∫01‍ϕ′(φ+s(ρ-φ))ds,ϕ~′′(φ)=∫01‍(1-s)ϕ′′(ρ+s(φ-ρ))ds
are bounded functions in Ω- [12].


## 3. A Posteriori Error Estimates

In this section, we will obtain a posteriori error estimates for semilinear parabolic integrodifferential optimal control problems. Firstly, we estimate the error∥u-uh∥L2(J;L2(Ω)).For givenu∈K, let M be the inverse operator of the state equation (10), such that y(u)=MBu is the solution of the state equation (10). Similarly, for given uh∈Kh, yh(uh)=MhBuh is the solution of the discrete state equation (16). Let
(30)S(u)=12∥MBu-y0∥2+α2∥u∥2,Sh(uh)=12∥MhBuh-y0∥2+α2∥uh∥2.
It is clear that S and Sh are well defined and continuous on K and Kh. Also the functional Sh can be naturally extended on K. Then (8) and (15) can be represented as
(31)minu∈K{S(u)},(32)minuh∈Kh{Sh(uh)}.
It can be shown that
(33)(S′(u),v)=(αu+B*p,v),(S′(uh),v)=(αuh+B*p(uh),v),(Sh′(uh),v)=(αuh+B*ph,v),
where p(uh) is the solution of (23)–(25).In many applications,S(·) is uniformly convex near the solution u (see, e.g., [18]). The convexity of S(·) is closely related to the second-order sufficient conditions of the control problems, which was assumed in many studies on numerical methods of the problems. If S(·) is uniformly convex, then there is a c>0, such that
(34)∫0T‍(S′(u)-S′(uh),u-uh)dt≥c∥u-uh∥L2(J;L2(Ω))2,
where u and uh are the solutions of (31) and (32), respectively. We will assume the above inequality throughout this paper.Letp(uh) be the solution of (23)–(25); we establish the following error estimate.Theorem 2.
Letu and uh be the solutions of (31) and (32), respectively. Assume that Kh⊂K. In addition, assume that (Sh′(uh))|τ∈Hs(τ), for all τ∈𝒯h, (s=0,1), and there is a vh∈Kh such that
(35)|(Sh′(uh),vh-u)|≤C∑τ∈𝒯h‍hτ∥Sh′(uh)∥Hs(τ)∥u-uh∥L2(τ)s.
Then, one has
(36)∥u-uh∥L2(J;L2(Ω))2≤Cη12+C∥ph-p(uh)∥L2(J;H1(Ω))2,
where
(37)η12=∫0T‍∑τ∈𝒯h‍hτ1+s∥αuh+B*ph∥Hs(τ)1+sdt.Proof.
It follows from (31) and (32) that
(38)∫0T‍(S′(u),u-v)≤0,∀v∈K,(39)∫0T‍(Sh′(uh),uh-vh)≤0,∀vh∈Kh⊂K.
Then it follows from (35), (39), and the Schwartz inequality, that
(40)c∥u-uh∥L2(J;L2(Ω))2≤∫0T‍(S′(u)-S′(uh),u-uh)dt≤-∫0T‍(S′(uh),u-uh)dt=∫0T‍{(Sh′(uh),uh-u)000000000+(Sh′(uh)-S′(uh),u-uh)}dt≤∫0T‍{(Sh′(uh),vh-u)000000000+(Sh′(uh)-S′(uh),u-uh)}dt≤C∫0T‍{∑τ∈𝒯h‍hτ1+s∥Sh′(uh)∥Hs(τ)1+s00000000000+∥Sh′(uh)-S′(uh)∥L2(Ω)2∑τ∈𝒯h‍}dt+c2∥u-uh∥L2(J;L2(Ω))2.
It is not difficult to show that
(41)Sh′(uh)=αuh+B*ph,S′(uh)=αuh+B*p(uh),
where p(uh) is defined in (23)–(26). Thanks to (11), it is easy to derive
(42)∫0T‍∥Sh′(uh)-S′(uh)∥L2(Ω)dt=∫0T‍∥B*(ph-p(uh))∥L2(Ω)dt≤C∥ph-p(uh)∥L2(J;L2(Ω))≤C∥ph-p(uh)∥L2(J;H1(Ω)).
Then, by the estimates (40) and (42), we can prove the requested result (36).Now, we estimate the error∥y(uh)-yh∥L2(J;H1(Ω)).Theorem 3.
Let(y(uh),p(uh)) and (yh,ph) be the solutions of (23)–(26) and (18)–(22), respectively. Then
(43)∥y(uh)-yh∥L2(J;H1(Ω))2≤C∑i=24‍ηi2,
where
(44)η22=∫0T‍∑τ∈𝒯h‍hτ2∫τ‍(∫0t‍f+Buh-yht+div(A∇yh)00000000000000+∫0t‍div(ψ(t,τ)∇yh(τ))dτ00000000000000-ϕ(yh)∫0t‍)2dt,η32=∫0T‍∑τ∈𝒯h‍hl∫∂τ‍[∫0t‍(A∇yh)·n00000000000000+∫0t‍(ψ(t,τ)∇yh(τ))·ndτ]2dldt,η42=∥yh(x,0)-y0(x)∥L2(Ω)2,
where l is a face of an element τ, hl is the size of face l, and [(A∇yh)·n] is the A-normal derivative jump over the interior face l, defined by
(45)[(A∇yh)·n]l=(A∇yh∣τl1-A∇yh∣τl2)·n,
where n is the unit normal vector on l=τ-l1∩τ-l2 outwards τl1. For convenience, one defines [(A∇yh)·n]l=0 when l⊂∂Ω.Proof.
LeteIy be the Clément-type interpolator of ey defined in Lemma 1. It follows from (18) and (23) that we have
(46)((yh-y(uh))t,wh)+a(yh-y(uh),wh)+∫0t‍ψ(t,τ;(yh-y(uh))(τ),wh)dτ+(ϕ(yh)-ϕ(y(uh)),wh)=0,∀wh∈Vh.
Let ey=yh-y(uh); by using (46), then we obtain
(47)((yh-y(uh))t,ey)+a(yh-y(uh),ey)+∫0t‍ψ(t,τ;(yh-y(uh))(τ),ey)dτ+(ϕ(yh)-ϕ(y(uh)),ey)=((yh-y(uh))t,ey-eIy)+a(yh-y(uh),ey-eIy)+∫0t‍ψ(t,τ;(yh-y(uh))(τ),ey-eIy)dτ+(ϕ(yh)-ϕ(y(uh)),ey-eIy)=(yht,ey-eIy)+a(yh,ey-eIy)+∫0t‍ψ(t,τ;yh(τ),ey-eIy)dτ+(ϕ(yh),ey-eIy)-(f+Buh,ey-eIy)=∑τ‍∫τ‍(∫0t‍yht-div(A∇yh)0000000000-∫0t‍div(ψ(t,τ)∇yh)dτ0000000000+ϕ(yh)-f-Buh,ey-eIy∫0t‍)+∑τ‍∫∂τ‍[∫0t‍(A∇yh)·n00000000000+∫0t‍(ψ(t,τ)∇yh(τ))·ndτ](ey-eIy)].
Then, we have
(48)12ddt∥yh-y(uh)∥0,Ω2+c∥yh-y(uh)∥1,Ω2≤((yh-y(uh))t,ey)+a(yh-y(uh),ey)+(ϕ(yh)-ϕ(y(uh)),ey)≤∑τ‍∫τ‍(yht-div(A∇yh)-∫0t‍div(ψ(t,τ)∇yh)dτ0000000000+ϕ(yh)-f-Buh,ey-eIy∫0t‍)+∑τ‍∫∂τ‍[(A∇yh)·n+∫0t‍(ψ(t,τ)∇yh(τ))·ndτ]00000000×(ey-eIy)-∫0t‍ψ(t,τ;(yh-y(uh))(τ),ey)dτ.
By integrating time from0totin the above inequality, combining (6) and the Schwartz inequality, we have
(49)12∥yh-y(uh)∥0,Ω2+c∫0t‍∥yh-y(uh)∥1,Ω2dτ≤C∫0t‍∑τ‍hτ2∫τ‍(∫0t‍yht-div(A∇yh)0000000000000000-∫0t‍div(ψ(t,τ)∇yh)dτ0000000000000000+ϕ(yh)-f-Buh∫0t‍)2dτ+∫0t‍∑τ‍hl∫∂τ‍[∫0t‍(A∇yh)·n0000000000000000+∫0t‍(ψ(t,τ)∇yh(τ))·ndτ]2dτ+δ∫0t‍∥yh-y(uh)∥1,Ω2dτ+C∫0t‍∫τ‍∥(yh-y(uh))(s)∥1,Ω2dsdτ+∥yh(x,0)-y0(x)∥L2(Ω)2.
Since δ is small enough, then from (49) and the Gronwall inequality, we have
(50)∫0t‍∥yh-y(uh)∥1,Ω2dτ≤C∫0t‍∑τ‍hτ2∫τ‍(∫0t‍yht-div(A∇yh)0000000000000000-∫0t‍div(ψ(t,τ)∇yh)dτ0000000000000000+ϕ(yh)-f-Buh∫0t‍)2dτ+C∫0t‍∑τ‍hl∫∂τ‍[∫0t‍(A∇yh)·n+∫0t‍(ψ(t,τ)∇yh(τ))000000000000000000·ndτ∫0t‍]2dτ+∥yh(x,0)-y0(x)∥L2(Ω)2.
So, by using the inequality (50) we obtain
(51)∥yh-y(uh)∥L2(J;H1(Ω))2≤C∑i=24‍ηi2.
This completes the proof.Analogous to Theorem3, we can prove the following estimates.Theorem 4.
Let(y(uh),p(uh)) and (yh,ph) be the solutions of (23)–(26) and (18)–(22), respectively. Then
(52)∥p(uh)-ph∥L2(J;H1(Ω))2≤C∑i=26‍ηi2,
where
(53)η52=∫0T‍∑τ∈𝒯h‍hτ2∫τ‍(∫tT‍yh-y0+pht+div(A*∇ph)00000000000000+∫tT‍div(ψ*(τ,t)∇ph(τ))dτ00000000000000-ϕ′(yh)ph∫tT‍)2dτdt,η62=∫0T‍∑τ∈𝒯h‍hl∫∂τ‍[∫0t‍(A*∇ph)·n00000000000000+∫0t‍((ψ*(t,τ)∇ph(τ))·n)dτ]2dldt,
where η2–η4 are defined in Theorem 3, l is a face of an element τ, and [(A*∇ph)·n] is the A-normal derivative jump over the interior face l, defined by
(54)[(A*∇ph)·n]l=(A*∇ph∣τl1-A*∇ph∣τl2)·n,
where n is the unit normal vector on l=τ-l1∩τ-l2 outwards τl1. For convenience, one defines [(A∇ph)·n]l=0 when l⊂∂Ω.Proof.
Letep=p(uh)-ph, and let eIp=π^hep, where π^h is the Clément-type interpolator defined in Lemma 1. Then, from (20) and (25), we obtain
(55)-(qh,(ph-p(uh))t)+a(qh,ph-p(uh))+∫tT‍ψ(τ,t;qh(t),(ph-p(uh))(τ))dτ+(ϕ′(yh)ph-ϕ′(y(uh))p(uh),qh)=(yh-y(uh),qh),∀qh∈Vh.
Namely,
(56)-(qh,(ph-p(uh))t)+a(qh,ph-p(uh))+∫tT‍ψ(τ,t;qh(t),(ph-p(uh))(τ))dτ+(ϕ′(yh)(ph-p(uh)),qh)=(yh-y(uh),qh)-((ϕ′(yh)-ϕ′(y(uh)))p(uh),qh).
By using (56), we obtain
(57)-(ep,(ph-p(uh))t)+a(ep,ph-p(uh))+∫tT‍ψ(τ,t;ep(t),(ph-p(uh))(τ))dτ+(ϕ′(yh)(ph-p(uh)),ep)=-(ep-eIp,(ph-p(uh))t)+a(ep-eIp,ph-p(uh))+∫tT‍ψ(τ,t;(ep-eIp)(t),(ph-p(uh))(τ))dτ+(ϕ′(yh)(ph-p(uh)),ep-eIp)-(eIp,(ph-p(uh))t)+a(eIp,ph-p(uh))+∫tT‍ψ(τ,t;eIp(t),(ph-p(uh))(τ))dτ+(ϕ′(yh)(ph-p(uh)),eIp)=-(ep-eIp,(ph-p(uh))t)+a(ep-eIp,ph-p(uh))+∫tT‍ψ(τ,t;(ep-eIp)(t),(ph-p(uh))(τ))dτ+(ϕ′(yh)(ph-p(uh)),ep-eIp)+(yh-y(uh),eIp)-((ϕ′(yh)-ϕ′(y(uh)))p(uh),eIp)=-(ep-eIp,pht)+a(ep-eIp,ph)+∫tT‍ψ(τ,t;(ep-eIp)(t),ph)(τ)dτ+(ϕ′(yh)(ph),ep-eIp)+(ep-eIp,pt(uh))-a(ep-eIp,p(uh))-∫tT‍ψ(τ,t;(ep-eIp)(t),p(uh)(τ))dτ-(ϕ′(yh)(p(uh)),ep-eIp)+(yh-y(uh),eIp)-((ϕ′(yh)-ϕ′(y(uh)))p(uh),eIp)=∑τ‍∫τ‍(∫tT‍-yh+y0-pht-div(A*∇ph)0000000000-∫tT‍div(ψ*(τ,t)∇ph(τ))dτ+ϕ′(yh)ph)000000×(ep-eIp)+∑τ‍∫∂τ‍((A*∇ph)·n+∫0t‍((ψ*(t,τ)∇ph(τ))·n)dτ)00000000×(ep-eIp)+(yh-y(uh),eIp)-((ϕ′(yh)-ϕ′(y(uh)))p(uh),ep).
Then, we have
(58)-12ddt∥ph-p(uh)∥0,Ω2+c∥ph-p(uh)∥1,Ω2≤-(ep,(ph-p(uh))t)+a(ep,ph-p(uh))+(ϕ′(yh)(ph-p(uh)),ep)≤∑τ‍∫τ‍(∫tT‍-yh+y0-pht-div(A*∇ph)0000000000-∫tT‍div(ψ*(τ,t)∇ph(τ))dτ0000000000+ϕ′(yh)ph∫tT‍)(ep-eIp)+∑τ‍∫∂τ‍(∫tT‍(A*∇ph)·n000000000000+∫tT‍((ψ*(t,τ)∇ph(τ))·n)dτ)00000000×(ep-eIp)+(yh-y(uh),eIp)-((ϕ′(yh)-ϕ′(y(uh)))p(uh),ep)+∫tT‍ψ(τ,t;ep(t),(ph-p(uh))(τ))dτ.
By integrating time fromttoTin the above inequality, combining (6) and the Schwartz inequality, we have
(59)12∥ph-p(uh)∥0,Ω2+c∫tT‍∥ph-p(uh)∥1,Ω2dτ≤C∫tT‍∑τ‍hτ2∫τ‍(∫tT‍yh-y0+pht+div(A*∇ph)0000000000000000+∫tT‍div(ψ*(τ,t)∇ph(τ))dτ0000000000000000-ϕ′(yh)ph∫tT‍)2dτ+∫tT‍∑τ‍hl∫∂τ‍[∫tT‍(A*∇ph)·n00000000000000000+∫tT‍((ψ*(t,τ)∇ph(τ))·n)dτ]2dτ+δ∫tT‍∥ph-p(uh)∥1,Ω2dτ+C∫tT‍∫τ‍∥(ph-p(uh))(s)∥1,Ω2dsdτ+C∫tT‍∫τ‍∥yh-y(uh)∥0,Ω2dτ.
Sinceδis small enough, then from (59) and the Gronwall inequality, we have
(60)∫tT‍∥ph-p(uh)∥1,Ω2dτ≤C∫tT‍∑τ‍hτ2∫τ‍(∫tT‍yh-y0+pht+div(A*∇ph)0000000000000000+∫tT‍div(ψ*(τ,t)∇ph(τ))dτ0000000000000000-ϕ′(yh)ph∫tT‍)2dτ+∫tT‍∑τ‍hl∫∂τ‍[∫tT‍(A*∇ph)·n00000000000000000+∫tT‍((ψ*(t,τ)∇ph(τ))·n)dτ]2dτ+∥y(uh)-yh∥L2(J;L2(Ω))2≤C∫tT‍∑τ‍hτ2∫τ‍(∫tT‍yh-y0+pht+div(A*∇ph)0000000000000000+∫tT‍div(ψ*(τ,t)∇ph(τ))dτ0000000000000000-ϕ′(yh)ph∫tT‍)2dτ+∫tT‍∑τ‍hl∫∂τ‍[∫tT‍(A*∇ph)·n00000000000000000+∫tT‍((ψ*(t,τ)∇ph(τ))·n)dτ]2dτ+∥y(uh)-yh∥L2(J;H1(Ω))2.
Finally, combine inequality (60) and Theorem 3 to obtain
(61)∥p(uh)-ph∥L2(J;H1(Ω))2≤C∑i=26‍ηi2.
This completes the proof.Hence, we combine Theorems2–4 to conclude.Theorem 5.
Let(y,p,u) and (yh,ph,uh) be the solutions of (10)–(14) and (18)–(22), respectively. Then
(62)∥u-uh∥L2(J;L2(Ω))2+∥y-yh∥L2(J;H1(Ω))2+∥p-ph∥L2(J;H1(Ω))2≤C∑i=16‍ηi2,
where η1–η6 are defined in Theorems 2–4, respectively.Proof.
From (10)–(14) and (23)–(26), we obtain the error equations
(63)(yt-yt(uh),w)+a(y-y(uh),w)+∫0t‍ψ(t,τ;(y-y(uh))(τ),wh)dτ+(ϕ(y)-ϕ(y(uh)),w)=(B(u-uh),w),-(pt-pt(uh),q)+a(q,p-p(uh))+∫tT‍ψ(τ,t;qh(t),(p-p(uh))(τ))dτ+(ϕ′(y)p-ϕ′(y(uh))p(uh),q)=(y-y(uh),q)
for all w∈V and q∈V. Thus, it follows from (63) that
(64)(yt-yt(uh),w)+a(y-y(uh),w)+∫0t‍ψ(t,τ;(y-y(uh))(τ),wh)dτ+(ϕ(y)-ϕ(y(uh)),w)=(B(u-uh),w),-(pt-pt(uh),q)+a(q,p-p(uh))+∫tT‍ψ(τ,t;qh(t),(p-p(uh))(τ))dτ+(ϕ′(y(uh))(p-p(uh)),q)=(y-y(uh),q)+(ϕ~′′(y(uh))(y(uh)-y)p,q).
By using the stability results in [17, 19], then we can prove that
(65)∥y-y(uh)∥L2(J;H1(Ω))2≤C∥u-uh∥L2(J;L2(Ω))2,∥p-p(uh)∥L2(J;H1(Ω))2≤∥y-y(uh)∥L2(J;H1(Ω))2≤C∥u-uh∥L2(J;L2(Ω))2.
Finally, combining Theorems 2–4 and (65) leads to (62).


## 4. An Adaptive Algorithm

In this section, we introduce an adaptive algorithm to guide the mesh refinement process. A posteriori error estimates which have been derived in Section3 are used as an error indicator to guide the mesh refinement in adaptive finite element method.Now, we discuss the adaptive mesh refinement strategy. The general idea is to refine the mesh such that the error indicator likeηis equally distributed over the computational mesh. Assume that an a posteriori error estimatorηhas the form of η2=∑τiητi2, whereτiis the finite elements. At each iteration, an average quantity of all ητi2 is calculated, and each ητi2 is then compared with this quantity. The element τi is to be refined if ητi2 is larger than this quantity. As ητi2 represents the total approximation error over τi, this strategy makes sure that higher density of nodes is distributed over the area where the error is higher.Based on this principle, we define an adaptive algorithm of the semilinear parabolic integrodifferential optimal control problems (1)-(2) as follows: starting from initial triangulations 𝒯h0 of Ω, we construct a sequence of refined triangulation 𝒯hj as follows. Given 𝒯hj, we compute the solutions (yh,ph,uh) of the system (18)–(22) and their error estimator as follows:
(66)ητ2=∫0T‍∑τ∈𝒯h‍hτ1+s∥αuh+B*ph∥Hs(τ)1+sdt+∫0T‍∑τ∈𝒯h‍hτ2∫τ‍(∫0t‍f+Buh-yht+div(A∇yh)0000000000000000+∫0t‍div(ψ(t,τ)∇yh(τ))dτ0000000000000000-ϕ(yh)∫0t‍)2dτdt+∫0T‍∑τ∈𝒯h‍hl∫∂τ‍[∫0t‍(A∇yh)·n0000000000000000+∫0t‍((ψ(t,τ)∇yh(τ))·n)dτ]2dldt+∥yh(x,0)-y0(x)∥L2(Ω)2+∫0T‍∑τ∈𝒯h‍hτ2∫τ‍(∫tT‍yh-y0+pht+div(A*∇ph)0000000000000000+∫tT‍div(ψ*(τ,t)∇ph(τ))dτ0000000000000000-ϕ′(yh)ph∫tT‍)dτdt+∫0T‍∑τ∈𝒯h‍hl∫∂τ‍[∫0t‍(A*∇ph)·n0000000000000000+∫0t‍((ψ*(t,τ)∇ph(τ))000000000000000000000·n(ψ*(t,τ)∇ph(τ)))dτ∫0t‍]2dldt,Ej=∑τ∈𝒯h‍ητ2.Then, we adopt the following mesh refinement strategy: all the trianglesτ∈𝒯hj satisfying ητ2≥ρEj/n are divided into four new triangles in 𝒯hj+1 by joining the midpoints of the edges, where n is the number of the elements of 𝒯hj and ρ is a given constant. In order to maintain the new triangulation 𝒯hj+1 to be regular and conformal, some additional triangles need to be divided into two or four new triangles depending on whether they have one or more neighbors which have been refined. Then, we obtain the new mesh 𝒯hj+1. The above procedure will continue until Ej≤tol, where tol is a given tolerance error.


## 5. Numerical Example

In this section, we will give a numerical example to illustrate our theoretical results. Our numerical example is the following semilinear parabolic integrodifferential optimal control problem:(67)minu(t)∈K{∫01‍(12∥y-y0∥2+12∥u-u0∥2)dt}yt-div(∇y(x,t))-∫0t‍div(ψ(t,τ)∇y(x,τ))dτ+ϕ(y)=f+u,x∈Ω,t∈J,y(x,t)=0,x∈∂Ω,t∈J,y(x,0)=0,x∈Ω,-pt-div(∇p(x,t))-∫t1‍div(ψ*(τ,t)∇p(x,τ))dτ+ϕ′(y)p=y-y0,x∈Ω,t∈J,p(x,t)=0,x∈∂Ω,t∈J,p(x,1)=0,x∈Ω.In this example, we choose the domainΩ=[0,1]×[0,1]. Let Ω be partitioned into 𝒯h as described Section 2. For the constrained optimization problem:
(68)minu∈K⊂U∫01‍S(u)dt,
where S(u)=(1/2)∥y-y0∥2+(1/2)∥u-u0∥2is a convex functional on U and K={u∈U:u≥0a.e.inΩ×J}; the iterative scheme reads (n=0,1,2,…)
(69)b(un+1/2,v)=b(un,v)-ρn(S′(un),v),∀v∈U,un+1=PKb(un+1/2),
whereb(·,·)is a symmetric and positive definite bilinear form such that there exist constants c0 and c1 satisfying
(70)|b(u,v)|≤c1∥u∥U∥v∥U,∀u,v∈U,b(u,u)≥c0∥u∥U2,
and the projection operator PKbU→K is defined: for given w∈U find PKbw∈K such that
(71)b(PKbw-w,PKbw-w)=minu∈Kb(u-w,u-w).
The bilinear form b(·,·) provides suitable preconditioning for the projection algorithm. An application of (69) to the discretized semilinear parabolic integrodifferential optimal control problem yields the following algorithm:
(72)b(un+1/2,vh)=b(un,vh)-ρn(un+pn,vh),∀vh∈Kh,∫01‍((yt,w)+a(y,w)+∫0t‍ψ(t,τ;y(τ),w)dτ+(ϕ(y),w)∫0t‍)dt=∫01‍(f+u,w)dt,∀w∈V,∫01‍(-(pt,q)+a(q,p)+∫t1‍ψ(τ,t;q,p(τ))dτ+(ϕ′(y)p,q)∫t1‍)dt=∫01‍(y-y0,q)dt,∀q∈V,un+1=PKb(un+1/2),un+1/2,un∈Kh.
The main computational effort is to solve the state and costate equations and to compute the projection PKbun+1/2. In this paper, we use a fast algebraic multigrid solver to solve the state and costate equations. Then, it is clear that the key to saving computing time is how to compute PKbun+1/2 efficiently. For the piecewise constant elements, Kh={uh:uh≥0} and b(u,v)=(u,v)U; then
(73)PKbun+1/2∣τ=max(0,avg(un+1/2)∣τ),
where avg(un+1/2)|τ is the average of un+1/2 over τ.In solving our discretized optimal control problem, we use the preconditioned projection gradient method withb(u,v)=(u,v)U and a fixed step size ρ=0.9. We now briefly describe the solution algorithm to be used for solving the numerical example in this section as follows.(1)
Solve the discretized optimization problem with the projection gradient method on the current meshes and calculate the error estimatorsηi.(2)
Adjust the meshes using the estimators and update the solution on new meshes, as described.Now, we give a numerical example to illustrate our theoretical results.Example 1.
Letψ(t,τ)=1, ϕ(y)=y5. We choose the state function by
(74)y(x1,x2)=2sinπx1sinπx2sinπt.
The function f is given by f(x)=yt-div(∇y(x,t))-∫0t‍div(∇y(x,τ))dτ+y5-u. The costate function can be chosen as
(75)p(x1,x2)=sinπx1sinπx2sinπt.
The function y0 is given by y0(x)=y+pt+div(∇p(x,t))+∫t1‍div(∇p(x,τ))dτ-5y4p. We assume that
(76)λ={0.8,x1+x2>1.0,0.3,x1+x2≤1.0,u0(x1,x2)=1-sinπx12-sinπx22+λ.
Thus, the control function is given by
(77)u(x1,x2)=max(u0-p,0).
In this example, the control functionuhas a strong discontinuity introduced by u0. The control function u is discretized by piecewise constant functions, whereas the stateyand the costate p were approximated by piecewise linear functions. In Table 1, numerical results ofu,y, andpon uniform and adaptive meshes are presented. It can be found that the adaptive meshes generated using our error indicators can save substantial computational work, in comparison with the uniform meshes. On the other hand, for the discontinuous control variableu, the accuracy has become better from the uniform meshes to the adaptive meshes in Table 1.Table 1
Numerical results on uniform and adaptive meshes.
On uniform mesh
On adaptive mesh
u
y
p
u
y
p
Nodes
8097
8097
8097
1102
1969
1969
Sides
23968
23968
23968
3143
5744
5744
Elements
15872
15872
15872
2042
3776
3776
Dofs
15872
15872
15872
2042
3776
3776
TotalL2 error
4.312
e
-
03
5.457
e
-
3
2.869
e
-
3
4.018
e
-
03
5.365
e
-
3
2.768
e
-
3


## 6. Conclusion and Future Works

In this paper, we discuss the semi-discrete finite element methods of the semilinear parabolic integrodifferential optimal control problems (1)-(2). We have established a posteriori error estimates for each the state, the costate, and the control approximation. The posteriori error estimates for those problems by finite element methods seem to be new.In our future work, we will use the mixed finite element method to deal with nonlinear parabolic integrodifferential optimal control problems. Furthermore, we will consider a posteriori error estimates and superconvergence of mixed finite element solution for nonlinear parabolic integrodifferential optimal control problems.



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*Source: 302935-2013-09-26.xml*