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2.035: SelectedTopics inMathematicswithApplicationsFinalExamSpring2007

Everyproblem in thecalculusofvariationshasasolution,providedthewordsolution issuitablyunderstood.

DavidHilbert(1862-1943)

Workany5problems.

Pick-upexam: 12:30PMonTuesdayMay8,2007Turn-insolutions: 11:00AMonTuesdayMay15,2007

Youmayusenotesinyourownhandwriting(takenduringand/orafterclass)andallhandouts(includinganythingIemailedtoyou)andmyboundnotes. Donotuseanyother

sources.

Donotspendmorethan2hoursonanyoneproblem.

Pleaseinclude,onthefirstpageofyoursolutions,asignedstatementconfirmingthatyouadheredtoalloftheinstructionabove.

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Problem1: Usingfirstprinciples(i.e. dontusesomeformulaliked/dx(F/)F/=0butrathergo throughthestepsofcalculatingF andsimplifying itetc.) determinethefunction Athatminimizesthefunctional

1

1

F{}= ()2 + + dx

20overtheset

A={|:[0,1]R, C2[0,1]}.Notethatthevaluesofarenotspecifiedateitherend. Problem 2: A problem of some importance involves navigation through a network ofsensors. Suppose that the sensors are located at fixed positions and that one wishes tonavigate in such a way that the navigating observer has minimal exposure to the sensors.Considerthefollowingsimplecaseofsuchaproblem. Seefigureonlastpage.

A single sensor is located at the origin of the x, y-plane and one wishes to navigatefrom the point A = (a,0) to the point B = (bcos, bsin). Let (x(t), y(t)) denote thelocation of the observer at time t so that the travel path is described parametrically byx=x(t), y=y(t),0 t T. Theexposureoftheobservertothesensorischaracterizedby

TE{x(t), y(t)}= I(x(t), y(t))v(t)dt

0wherev(t)isthespeedoftheobserverandthesensitivityI isgivenby

1I(x, y) = .

x2 +y2DeterminethepathfromAtoB thatminimizestheexposure.Hint: Workinpolarcoordinatesr(t), (t)andfindthepathintheformr=r().Remark: Forfurtherbackgroundonthisproblemincludinggeneralizationtonsensors,seethe paper Qingfeng Huang, Solving an Open Sensor Exposure Problem using VariationalCalculus,TechnicalReportWUCS-03-1,WashingtonUniversity,DepartmentofComputerScienceandEngineering,St. Louis,Missouri,2003.

Problem3: ConsideradomainDofthex, y-planewhoseboundaryD issmooth. LetAdenotethesetofallfunctions(x, y)thataredefinedandsuitablysmoothonDandwhichvanishontheboundaryofD: = 0for(x, y)D. Definethefunctional

1 22 1F{}=

D 2 xy + 22 +q dA2

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for all A where q = q(x, y) is a given function onD. You are asked to minimize thefunctionalF overthesetA. Determinetheboundary-valueproblemthattheminimizermustsatisfy. (YoudoNOTneedtosolveit.) Problem4: Derivethetwocornerconditions

f , ,

x=s+f f f

f = f =

x=s x=s+ x=sthataminimizerofthefunctional

F{}=01

f(x,,)dxmust satisfy if the minimizer is continuous and has piecewise continuous derivatives; herex=sdenotesthelocationofadiscontinuityin andthevalueofsisnotknownapriori.Remark: ThisquestionsimplyasksyoutoderivetheresultsthatIquoted inclasswithoutproofonThursdayMay3. Problem5: Considertheeigenvalueproblem(1), (2) forfindingtheeigenfunctions(x)andeigenvaluesoftheboundary-valueproblem

a (x) b (x) +c (x) = 0 for 0 x 1, (1)(0)=0, (1)=0, (2)

wherea,b,candareconstants. (Remark: There isNOtypo intheequationsabove,justincaseyouexpectedittobeb(x)insteadofb(x).)

a) Showthatminimizingthefunctional 1

F{}= a()2 + (bc) ()2 dx0

overthesetA={|:[0,1]R, C2[0,1], (0)=(1)=0}leadstoequation(1)asitsEulerequation.

b) SincethefunctionalF involvestheunknowneigenvalues,theprecedingisnotaparticularlyusefulvariationalstatement,whichmotivatesustolookforothervariationalformulationsoftheeigenvalueproblem. Showthatminimizingthefunctional

1

G{}= a()2 +b()2 dx0

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overthesetAsubject to theconstraint 1

H{}= c()2dx=constant0

leadsto

equation

(1)

as

its

Euler

equation

with

as

aLagrange

multiplier.

c) Showthatminimizingthefunctional

1

a()2 +b()2 dx0

I{}= 1c 2dx

0overthesetA leadstoequation(1)as itsEulerequation,andthatthevaluesofI attheminimizersaretheeigenvalues.

Problem6: Supposethatanairplanefliesinthex, y-planeataconstantspeedvo relativeto the wind. The flight path begins and ends at the same location so that the path is aloop. The flight takes a given duration T. Assume that the wind velocity has a constantmagnitudec(< vo)andaconstantdirection(which,withoutlossofgenerality,youcantaketobethex-direction).

Alongwhatclosedcurveshouldtheplanefly iftheflightpathistoenclosethegreatestarea?

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A

B

a

b

r(t)

(t)t = 0

t = T

(x(t), y(t))

Sensor

Observer

Problem2:Anobservermovesalongapathinthex, y-planesuchthatitsrectangularcartesiancoordinatesattimetare(x(t), y(t))(oritspolarcoordinatesare(r(t), (t)).Asensor islocatedattheorigin.

Problem3:

A

domain

D

in

the

x, y-plane

with

a

smooth

boundary

D.

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