楊繼明，朱先睿

(湖南工程學(xué)院 理學(xué)院，湘潭 411104)

一類非線性最優(yōu)控制問題的間斷有限元方法

楊繼明，朱先睿

(湖南工程學(xué)院 理學(xué)院，湘潭 411104)

對(duì)于一類非線性最優(yōu)控制問題，采用間斷有限元方法進(jìn)行離散求解.數(shù)值實(shí)驗(yàn)表明，間斷有限元方法能有效求解所研究的最優(yōu)控制問題.

最優(yōu)控制；間斷有限元方法；非線性

0 引言

偏微分方程最優(yōu)控制在科學(xué)和工程領(lǐng)域中發(fā)揮著重要的作用，這類問題數(shù)值方法的研究受到許多學(xué)者的關(guān)注.

嚴(yán)寧寧和劉文斌等采用有限元法對(duì)橢圓最優(yōu)控制問題進(jìn)行了研究[1].陳艷萍教授等在最優(yōu)控制問題的混合有限元方法上做了大量工作[2-3].馬和平教授等人研究了最優(yōu)控制問題的Chebyshev-Legendre配置法[4].陳艷萍教授、周建偉教授在文獻(xiàn)研究了最優(yōu)控制問題的譜方法[5-7].Meyer等研究了線性二次橢圓最優(yōu)控制問題有限元法的先驗(yàn)誤差和超收斂性質(zhì)[8-9].Merino研究了Burgers方程最優(yōu)控制問題的有限元方法[10].

間斷有限元方法是利用完全間斷的分片多項(xiàng)式空間作為近似解和試驗(yàn)函數(shù)空間的一種非協(xié)調(diào)有限元方法.它已經(jīng)在許多含有激波、接觸斷面、大梯度、大變形等間斷問題上顯示出強(qiáng)大的功效[11-14].但關(guān)于最優(yōu)控制問題間斷有限元方法的研究工作很少.

本文采用間斷有限元方法求解一類非線性最優(yōu)控制問題.

1 問題描述

設(shè)Ω和ΩU是有界凸多邊形區(qū)域，其邊界分別為?Ω和?ΩU.定義相應(yīng)的控制集：

Uad={v∈L2(ΩU),v≥0},U=L2(ΩU).

考慮一類非線性最優(yōu)控制問題：

s.t.-Δy+φ(y)=f+Bu,x∈Ω,

y=0,x∈?Ω.

(1)

上式中α>0是已知的量，函數(shù)φ(y)為關(guān)于y的非線性函數(shù)，φ(y)∈W(Ω)，并滿足對(duì)任意的φ′(y)≥0.假設(shè)f∈L2(Ω)，yd∈L2(Ω),B是從ΩU到Ω上的一個(gè)連續(xù)線性算子.

2 變分形式和最優(yōu)性條件

問題(1)相應(yīng)的變分形式為：

(2)

a(y,v)+b(u,v)=(f,v),?v∈V,

(3)

(y,u)∈Y×Uad.

a(y,v)+b(u,v)=(f,v),?v∈V,

(4)

a(ψ,p)=-(y-yd,ψ),?ψ∈V,

(5)

(αu-B*p,w-u)U≥0,?w∈Uad,

(6)

其中B*是B的伴隨算子[15]，(·,·)U表示U中的內(nèi)積.

3 離散格式

Vh=Yh={y∈L2(Ω),y|K∈P1(K),K∈Th}.

[y]=yK1|E·nK1+yK2|E·nK2，

Uh= {u∈L2(Ω),u|KU∈P1(KU),

其中σ>0是內(nèi)罰項(xiàng).

那么，最優(yōu)控制問題(4)～(6)的間斷有限元逼近為：尋找三元組(yh,uh,ph)使得：

ah(yh,vh)+bh(uh,vh)=(fh,vh),

?vh∈Vh,

(7)

a(ψh,ph)=-(yh-yd,ψh),?ψh∈Vh,

(8)

(9)

4 數(shù)值實(shí)驗(yàn)

在問題(1)中取Ω=ΩU=(0,1),φ(y)=y3,B=I，其中I為恒等算子.f=(π2-1)sinπx+sin3πx，yd=(1-π4)sinπx+3sin3πx.狀態(tài)變量、對(duì)偶變量和控制變量的真解為y=sinπx，u=max(p,0),p=-sinπx.

利用間斷有限元方法(7)-(9)進(jìn)行求解，σ=6.對(duì)非線性方程組采用牛頓迭代法求解.表1中給出了數(shù)值解的誤差情況.

實(shí)驗(yàn)數(shù)據(jù)表明,間斷有限元方法能有效求解最優(yōu)控制問題(1).

表1 間斷有限元方法求解的數(shù)值解誤差

[1] W. Liu and N. Yan, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Beijing: Science Press, 2008.

[2] Y. Chen and Z. Lu, High Efficient and Accuracy Numerical Methods for Optimal Control Problems[J].Beijing: Science press, 2015.

[3] Y. Chen, Y. Huang, W. Liu and N. Yan, Error Estimates and Superconvergence of Mixed Finite Element Methods for Convex Optimal Control Problems[J].J. Sci. Computing, 2010,42(3):382-403.

[4] W. Zhang and H. Ma, The Chebyshev-Legendre Collocation Method for a Class of Optimal Control Problems[J]. Int. J. Comp. Math., 2008,85(2):225-240.

[5] Y. Chen and F. Huang, Galerkin Spectral Approximation of Elliptic Optimal Control Problems with H1-norm State Constraint[J]. J. Sci. Comp., 2016(67):65-83.

[6] Y. Chen, N. Yi, and W. Liu, A Legendre Galerkin Spectral Method for Optimal Control Problems Governed by Elliptic Equations[J]. SIAM J. Numer. Anal., 2008,46(5):292-315.

[7] J. Zhou, Improved Optimal Conditions and Iterative Parameters for the Optimal Control Problems with an Integral Constraint in Square[J].J.Comp.Appl.Math.,2016,307(1):367-373.

[8] C.Meyer and A. Rosch,Lestimates for Approximated Optimal Control Problems[J]. SIAM J. Control. Optim., 2005(44):1636-1649.

[9] C. Meyer and A. Rosch,Superconvergence Properties of Optimal Control Problems[J]. SIAM J. Control. Optim., 2004,43(3):970-985.

[10] P. Merino, Finite Element Error Estimates for an Optimal Control Problem Governed by the Burgers Equation[J]. Comp. Opt. Appl., 2016,63(3):793-824.

[11] B. Rivière, J. Tan and T. Thompson, Error Analysis of Primal Discontinuous Galerkin Methods for a Mixed Formulation of the Biot Equations[J]. Comp. Math. Appl., 2017,73(4):666-683.

[12] A. Ern, I. Mozolevski and L. Schuh, Discontinuous Galerkin Approximation of Two-phase Flows in Heterogeneous Porous Media with Discontinuous Capillary Pressures[J]. Comp. Meth. Appl. Mech. Eng., 2010,199(23-24):1491-1501.

[13] J. Kou and S. Sun, Analysis of a Combined Mixed Finite Element and Discontinuous Galerkin Method for Incompressible Two Phase Flow in Porous Media[J]. Math. Meth. Appl. Sci., 2014,37(7):962-982.

[14] J. Zhang, J. Zhu, R. Zhang, D. Yang and A. Loula, A Combined Discontinuous Galerkin Finite Element Method for Miscible Displacement Problem[J]. J. Comp. Appl. Math., 2017,309(1):44-55.

[15] J. Lions, Optimal Control of Systems Governed by Partial Differential Equations[M]. Berlin: Springer, 1971.

ADiscontinuousGalerkinMethodForaKindofNonlinearOptimalControlProblems

YANG Ji-ming, ZHU Xian-rui

(College of Science, Hunan Institute of Engineering, Xiangtan 411104, China)

For a kind of nonlinear optimal control problem, a discontinuous Galerkin method is investigated for discretization. The numerical experiments show that the discontinuous Galerkin approximation is effective to solve the optimal control problem under consideration.

optimal control; discontinuous Galerkin methods; nonlinear

O241.8

A

1671-119X(2017)03-0037-03

2017-04-01

湖南省教育廳科研資助項(xiàng)目(14A034);湖南工程學(xué)院2017年大學(xué)生研究性學(xué)習(xí)和創(chuàng)新性實(shí)驗(yàn)項(xiàng)目(校教字[2017]18號(hào)文69號(hào)).

楊繼明(1975- )，男，博士，教授，研究方向：微分方程數(shù)值解法.