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)

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)值實驗表明，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式空間作為近似解和試驗函數(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.

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)值實驗

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.

實驗數(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)新性實驗項目(校教字[2017]18號文69號).

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