趙 堅(jiān),高 夯

(1.中央廣播電視大學(xué)數(shù)學(xué)系,北京 100031; 2.東北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,吉林長(zhǎng)春 130024)

關(guān)于極值點(diǎn)的必要條件

趙 堅(jiān)1,高 夯2

(1.中央廣播電視大學(xué)數(shù)學(xué)系,北京 100031; 2.東北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,吉林長(zhǎng)春 130024)

介紹了函數(shù)極值問(wèn)題、泛函極值問(wèn)題和最優(yōu)控制問(wèn)題,給出極值點(diǎn)與最優(yōu)控制的必要條件,闡述了最優(yōu)控制問(wèn)題的新進(jìn)展及它們之間的聯(lián)系.

極值點(diǎn);最優(yōu)控制;必要條件

1 函數(shù)的極值點(diǎn)問(wèn)題

2 泛函極值問(wèn)題

泛函極值問(wèn)題如同函數(shù)極值問(wèn)題一樣,具有大量的實(shí)際問(wèn)題作為背景,如捷線問(wèn)題、最小曲面問(wèn)題、等周問(wèn)題等.設(shè)集合

可以證明,集合Y(Y?C2[a,b])是一開(kāi)集.設(shè)f是關(guān)于其各變量連續(xù)可導(dǎo)的函數(shù),在Y上定義如下的泛函:

3 最優(yōu)控制問(wèn)題

其中,U?R是緊集.

假設(shè)f滿(mǎn)足如下的條件:

(C1)f關(guān)于變量t可測(cè),關(guān)于變量y連續(xù)可微,關(guān)于變量u連續(xù).

注8在(3.1)式中,若A是有界算子,則方程(3.1)是常微分方程,且X=Rn.若A是無(wú)界算子,則方程(3.1)可以是偏微方程.例如A=Δ(拉普拉斯算子).則可選擇X=W2,p0(Ω).對(duì)于任意的u∈U ad,在條件(C1)下,方程(3.1)存在唯一解.因此,可以在U ad上定義如下的泛函:

以及Uad=L2(Ω)等情形下,利用變量置換與懲罰泛函等手段,得到了相應(yīng)的最優(yōu)性條件.

在文獻(xiàn)[12-13]中,作者在問(wèn)題(3.9)中U是有限點(diǎn)集的情形,通過(guò)構(gòu)造拋物控制系統(tǒng),得到拋物系統(tǒng)最優(yōu)控制的必要條件,證明了擾動(dòng)的拋物系統(tǒng)的最優(yōu)對(duì)收斂到原橢圓系統(tǒng)的最優(yōu)對(duì),從而得到橢圓系統(tǒng)最優(yōu)對(duì)的必要條件.

在文獻(xiàn)[14]中,作者為了克服U ad不凸的困難,采用了松弛方法[15].將松弛方法用到多解控制系統(tǒng)是作者的一個(gè)重要?jiǎng)?chuàng)新之處.同時(shí),我們也看到了松弛控制方法也很難使用到很一般的情形.

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On the necessary conditions of m in imal element

ZHAO Jian1,GAO Hang2

(1.Department of Mathematics,China Central Radio and TV University,Beijing 100031,China; 2.School of Mathematics and Statistics,Northeast Normal University,Changchun 130024,China)

In this paper,discuss the p roblem s of function minimal element,functional m inimal element and p roblem of op timal control.Give the necessary conditions of m inimal elements and op timal control.On one hand,the new p rogress is introduced on op timal control research,on the other hand, give the relation of minimal elements and op timal control.

m inimal element;op timal control;necessary condition

O 231.2

120·30

A

1000-1832(2010)04-0001-05

2010-09-15

國(guó)家自然科學(xué)基金資助項(xiàng)目(10871039).

趙堅(jiān)(1958—),女,碩士,副教授,主要從事應(yīng)用數(shù)學(xué)研究;通訊作者:高夯(1956—),男,博士,教授,博士研究生導(dǎo)師,主要從事控制理論研究.

(責(zé)任編輯:陶 理)