Hamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t

Size: px
Start display at page:

Download "Hamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t"

Transcription

1 M ah Fall L ecure 1 0 O c. 7, Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n 0, 1 u = g on R n { = 0. 2 To avoid confusion, we use he following noaion: x D u x, u z, u p = p p n Then we can re-wrie he equaion o where F D u, u, u, x, = 0 4 F p, z, x p n+ 1 + H p, x. 5 The characerisics ODEs hen are x D = x = D p F = p H, 6 1 D ż = D p F p = p H p = p 1 p n+ 1 + D p H p = D p H p H p, x, 7 n+ 1 ṗ Dx H = p = D ṗ z F p D x F =. 8 n Mehod of characerisics. We ry o solve he characerisic ODEs. Firs noice ha, since = 1, we can simply use as he parameer insead of s. Thus he equaions become x = D p H, 9 ż = D p H p H p, x, 1 0 ṗ = D x H, 1 1 p n+ 1 = p n+ 1 = I is clear he all we need o do is o solve he firs 3 equaions. Losing a bi rigor, we assume for now only H is differeniable and sricly convex. We also assume H grows super-linearly a infiniy: H p lim = +, 1 3 p p Noicing ha We can define p 0 = argmax p R n { D p H p 0, x p H p, x. 1 4 q D p H p, x 1 5 which also give p as a funcion of q as D p 2 H is non-singular due o he convexiy of H. We wrie As a consequence he z equaion becomes where q saisfies L q, x sup p R n { q p H p, x. 1 6 ż = L q, x 1 7 q = D p H p, x. 1 8

2 Therefore he soluion u is given by u x = u x 0 + where x and x 0 are relaed by where X solves To furher simplify he sysem, we noice ha implies which implies ha q, x minimizes 0 L q τ, x τ dτ. 1 9 x = X s 20 d d X = D ph = q, X 0 = x wih x 0, x fixed. See Evans for deails. To see his, wrie L q, x = q p q, x H p q, x, x, and compue x = D p H, ṗ = D x H 22 d ds D ql + D x L = L q τ, x τ dτ 24 D q L = p + q D q p D p H D q p = p, 25 D x L = q D x p D x H D p H D x p = D x H, 26 where we have used q = D p H. Now he equaion ṗ = D x H gives wha we wan. Thus we see ha he Hamilon-Jacobi equaion can be solved as soon as we find ou he rajecories x and q. Below we will see ha in a special case, his can indeed be done in some sense. 2. The Hopf-Lax formula. This special case is when H is independen of x, ha is H = H D u. The characerisic equaions can hen be furher simplified o x = D p H, 27 ż = D p H p H p = L q, 28 ṗ = D x H = 0, 29 p n+ 1 = p n+ 1 = We see ha p is a consan vecor along he characerisic curve, and as a consequence x = D p H is a consan vecor, and herefore he characerisics x are sraigh lines. Furhermore we know ha he velociy q = x is consan. Thus if x 0 = y and x = x, we mus have As a consequence d z = L q = L d q =. 31 z = z 0 + L = g y + L. 32 Now he only problem is ha y is no known. Now hink of g y as no merely an iniial funcion, bu as an inermediae record. In oher words, insead of saring a = 0, imagine our sysem sars from =, say, 1. We consider all possible rajecories emanaing from some poin a = 1, passing y a = 0, and finally reach ime a x. Think of g y as he record of work done from = 1 o = 0. Obviously he correc rajecory should be he one ha is he minimizer among hem all.

3 Following his idea, we reach he following Hopf-Lax formula: { u x, = z = inf L + g y. 33 y R n Remark 1. I can be shown ha L grows superlinearly a infiniy. As a consequence, if we assume g o be Lipschiz coninuous, hen he infimum is acually a minimum. Remark 2. The relaion L q = H q sup p R n { q p H p 34 is called Legendre ransform and is very useful. I can be shown ha he following heorem holds Evans p Theorem 3. Le H = H p be convex, and saisfies hen i. H q is also convex, ii. H = H. H p lim = +, 35 p p H q lim = +, 36 q q Inspecing he proof, one sees ha i sill holds even if H is no convex, bu convexiy is necessary for ii If H is no convex, hen i canno be he same as H, which is convex by i. Remark 4. Noe ha convex funcions are coninuous. The proof can go roughly as follows. Firs one can show ha f he convex funcion is bounded, le he bound be denoed M. Then using he definiion of convexiy we have, for any fixed x, y, Leing α 0 we see ha u y + α u y + α u x u y u y + 2 α M. 37 On he oher hand, for any x n x we have, by convexiy This gives limsup u x n u x. 38 x n x u x 1 2 [ u x n + u 2 x x n ]. 39 u x 1 2 liminf [ u x n + u 2 x x n ]. 40 x n x Coninuiy hen follows. One can in fac prove ha any convex funcion is Lipschiz coninuous, see e. g. B. Dacorogna Direc Mehods in he Calculus of Variaions, 2nd ed., Springer, 2008, Soluion of he Hamilon-Jacobi equaion. Now we show ha he Hopf-Lax formula { u x, = inf L y R n indeed solves he Hamilon-Jacobi equaion, albei only almos everywhere. + g y. 41 Remark 5. I is easy o see ha in general one canno expec he exisence of classical soluions due o possible inersecions of characerisics.

4 There are hree hings o show. 1. u = g on R n { = 0, 2. u, D u exis almos everywhere, 3. u + H D u = 0 a. e. We show hem one by one. 1. u = g on R n { = 0. Recall he formula: Taking y = x we have { u x, = min L y + g y. 42 u x, g x + L 0 limsup u x, g x On he oher hand, we compue { u x, = min L + g y y { = g x + min L + g y g x y { g x max Lip g y x L y = g x max { Lip g z L z z { = g x max max { w z L z w B L i p g z = g x max H w. 44 w B L i p g As H is coninuous, we have Thus ends he proof. 2. u, D u exis almos everywhere. I suffices o show ha u is Lipschiz wih respec o x and o. liminf u x, g x u is Lipschiz w. r.. x. We esimae u xˆ, u x,. Choose y such ha u x, = L + g y. 46 Then { xˆ z u xˆ, u x, = min L + g z L g y. 47 Taking z = xˆ x + y such ha xˆ z = we have Similarly we can show u xˆ, u x, g xˆ x + y g y Lip g xˆ x. 48 The Lipschiz coninuiy of u hen follows. u x, u xˆ, Lip g xˆ x. 49 u is Lipschiz w. r... This follows from he following propery of he Hopf-Lax formula: { u x, = min s L + u y, s. 50 y R n s Tha his should hold is inuiively very clear following our derivaion of he formula. For a proof see Evans p

5 Using his formula, we see ha esimaing u x, u x, s is no differen han esimaing u x, g x. Thus a similar argumen as in Sep 1. gives u x, u x, s C s u + H D u = 0 a. e. Fix any q R n, we compue u x + h q, + h = { x + h q y min h L + u y, h h L q + u x,. 52 This implies for all q R n. Therefore and u + q D u L q u D u q L q 53 u max { D u q L q = H D u 54 q u + H D u For he oher direcion ha is u + H D u 0, we only need o find one q such ha or more specifically u + q D u L q 56 u x, u y, s s L q 57 where is in he direcion of q. As u is a minimum, o ge he u x, u y, s somehing, we ge rid of he minimum in u x,. Take z such ha x z u x, = L + g z. 58 Now ha q = x z is already chosen, y has o be on he line segmen connecing x and z. Thus we ake s = h, y = s x + 1 s z. 59 Then we have x z As we ge and finish he proof. = q. We compue x z u x, u y, s L x z = s L = y z s u x, u y, s s u + x z ] y z s L + g z s. 60 [ + g z u + x z D u, 61 x z D u L 62

Math 527 Lecture 6: Hamilton-Jacobi Equation: Explicit Formulas

Math 527 Lecture 6: Hamilton-Jacobi Equation: Explicit Formulas Mah 527 Lecure 6: Hamilon-Jacobi Equaion: Explici Formulas Sep. 23, 2 Mehod of characerisics. We r o appl he mehod of characerisics o he Hamilon-Jacobi equaion: u +Hx, Du = in R n, u = g on R n =. 2 To

More information

Hamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:

Hamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation: M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno

More information

Hamilton Jacobi equations

Hamilton Jacobi equations Hamilon Jacobi equaions Inoducion o PDE The rigorous suff from Evans, mosly. We discuss firs u + H( u = 0, (1 where H(p is convex, and superlinear a infiniy, H(p lim p p = + This by comes by inegraion

More information

Differential Equations

Differential Equations Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding

More information

Math 334 Fall 2011 Homework 11 Solutions

Math 334 Fall 2011 Homework 11 Solutions Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then

More information

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,

More information

An Introduction to Malliavin calculus and its applications

An Introduction to Malliavin calculus and its applications An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214

More information

Solutions from Chapter 9.1 and 9.2

Solutions from Chapter 9.1 and 9.2 Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is

More information

Concourse Math Spring 2012 Worked Examples: Matrix Methods for Solving Systems of 1st Order Linear Differential Equations

Concourse Math Spring 2012 Worked Examples: Matrix Methods for Solving Systems of 1st Order Linear Differential Equations Concourse Mah 80 Spring 0 Worked Examples: Marix Mehods for Solving Sysems of s Order Linear Differenial Equaions The Main Idea: Given a sysem of s order linear differenial equaions d x d Ax wih iniial

More information

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still. Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in

More information

Predator - Prey Model Trajectories and the nonlinear conservation law

Predator - Prey Model Trajectories and the nonlinear conservation law Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories

More information

Chapter 2. First Order Scalar Equations

Chapter 2. First Order Scalar Equations Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.

More information

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities: Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial

More information

ODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004

ODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004 ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform

More information

t is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...

t is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t... Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger

More information

Announcements: Warm-up Exercise:

Announcements: Warm-up Exercise: Fri Apr 13 7.1 Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis hp//en.wikipedia.org/wiki/muli-comparmen_model Announcemens Warm-up Exercise Here's a relaively simple

More information

Homogenization of random Hamilton Jacobi Bellman Equations

Homogenization of random Hamilton Jacobi Bellman Equations Probabiliy, Geomery and Inegrable Sysems MSRI Publicaions Volume 55, 28 Homogenizaion of random Hamilon Jacobi Bellman Equaions S. R. SRINIVASA VARADHAN ABSTRACT. We consider nonlinear parabolic equaions

More information

Finish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!

Finish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details! MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his

More information

dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.

dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3. Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies

More information

Matlab and Python programming: how to get started

Matlab and Python programming: how to get started Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,

More information

Lecture 20: Riccati Equations and Least Squares Feedback Control

Lecture 20: Riccati Equations and Least Squares Feedback Control 34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he

More information

Chapter 6. Systems of First Order Linear Differential Equations

Chapter 6. Systems of First Order Linear Differential Equations Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh

More information

EXERCISES FOR SECTION 1.5

EXERCISES FOR SECTION 1.5 1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler

More information

LAPLACE TRANSFORM AND TRANSFER FUNCTION

LAPLACE TRANSFORM AND TRANSFER FUNCTION CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions

More information

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 301 Signals & Systems Prof. Mark Fowler EECE 31 Signals & Sysems Prof. Mark Fowler Noe Se #1 C-T Sysems: Convoluion Represenaion Reading Assignmen: Secion 2.6 of Kamen and Heck 1/11 Course Flow Diagram The arrows here show concepual flow beween

More information

Math Week 14 April 16-20: sections first order systems of linear differential equations; 7.4 mass-spring systems.

Math Week 14 April 16-20: sections first order systems of linear differential equations; 7.4 mass-spring systems. Mah 2250-004 Week 4 April 6-20 secions 7.-7.3 firs order sysems of linear differenial equaions; 7.4 mass-spring sysems. Mon Apr 6 7.-7.2 Sysems of differenial equaions (7.), and he vecor Calculus we need

More information

MATH 128A, SUMMER 2009, FINAL EXAM SOLUTION

MATH 128A, SUMMER 2009, FINAL EXAM SOLUTION MATH 28A, SUMME 2009, FINAL EXAM SOLUTION BENJAMIN JOHNSON () (8 poins) [Lagrange Inerpolaion] (a) (4 poins) Le f be a funcion defined a some real numbers x 0,..., x n. Give a defining equaion for he Lagrange

More information

INSTANTANEOUS VELOCITY

INSTANTANEOUS VELOCITY INSTANTANEOUS VELOCITY I claim ha ha if acceleraion is consan, hen he elociy is a linear funcion of ime and he posiion a quadraic funcion of ime. We wan o inesigae hose claims, and a he same ime, work

More information

Solutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore

Solutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore Soluions of Sample Problems for Third In-Class Exam Mah 6, Spring, Professor David Levermore Compue he Laplace ransform of f e from is definiion Soluion The definiion of he Laplace ransform gives L[f]s

More information

Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution

Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his

More information

ENGI 9420 Engineering Analysis Assignment 2 Solutions

ENGI 9420 Engineering Analysis Assignment 2 Solutions ENGI 940 Engineering Analysis Assignmen Soluions 0 Fall [Second order ODEs, Laplace ransforms; Secions.0-.09]. Use Laplace ransforms o solve he iniial value problem [0] dy y, y( 0) 4 d + [This was Quesion

More information

MATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence

MATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,

More information

Some Basic Information about M-S-D Systems

Some Basic Information about M-S-D Systems Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,

More information

Hybrid Control and Switched Systems. Lecture #3 What can go wrong? Trajectories of hybrid systems

Hybrid Control and Switched Systems. Lecture #3 What can go wrong? Trajectories of hybrid systems Hybrid Conrol and Swiched Sysems Lecure #3 Wha can go wrong? Trajecories of hybrid sysems João P. Hespanha Universiy of California a Sana Barbara Summary 1. Trajecories of hybrid sysems: Soluion o a hybrid

More information

KEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow

KEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering

More information

Section 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

Section 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous

More information

Example on p. 157

Example on p. 157 Example 2.5.3. Le where BV [, 1] = Example 2.5.3. on p. 157 { g : [, 1] C g() =, g() = g( + ) [, 1), var (g) = sup g( j+1 ) g( j ) he supremum is aken over all he pariions of [, 1] (1) : = < 1 < < n =

More information

2. Nonlinear Conservation Law Equations

2. Nonlinear Conservation Law Equations . Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear

More information

10. State Space Methods

10. State Space Methods . Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he

More information

Solutions to Assignment 1

Solutions to Assignment 1 MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we

More information

KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow

KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow 1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering

More information

(1) (2) Differentiation of (1) and then substitution of (3) leads to. Therefore, we will simply consider the second-order linear system given by (4)

(1) (2) Differentiation of (1) and then substitution of (3) leads to. Therefore, we will simply consider the second-order linear system given by (4) Phase Plane Analysis of Linear Sysems Adaped from Applied Nonlinear Conrol by Sloine and Li The general form of a linear second-order sysem is a c b d From and b bc d a Differeniaion of and hen subsiuion

More information

THE MYSTERY OF STOCHASTIC MECHANICS. Edward Nelson Department of Mathematics Princeton University

THE MYSTERY OF STOCHASTIC MECHANICS. Edward Nelson Department of Mathematics Princeton University THE MYSTERY OF STOCHASTIC MECHANICS Edward Nelson Deparmen of Mahemaics Princeon Universiy 1 Classical Hamilon-Jacobi heory N paricles of various masses on a Euclidean space. Incorporae he masses in he

More information

Two Coupled Oscillators / Normal Modes

Two Coupled Oscillators / Normal Modes Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own

More information

System of Linear Differential Equations

System of Linear Differential Equations Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y

More information

Math Wednesday March 3, , 4.3: First order systems of Differential Equations Why you should expect existence and uniqueness for the IVP

Math Wednesday March 3, , 4.3: First order systems of Differential Equations Why you should expect existence and uniqueness for the IVP Mah 2280 Wednesda March 3, 200 4., 4.3: Firs order ssems of Differenial Equaions Wh ou should epec eisence and uniqueness for he IVP Eample: Consider he iniial value problem relaed o page 4 of his eserda

More information

arxiv: v2 [math.ap] 16 Oct 2017

arxiv: v2 [math.ap] 16 Oct 2017 Unspecified Journal Volume 00, Number 0, Pages 000 000 S????-????XX0000-0 MINIMIZATION SOLUTIONS TO CONSERVATION LAWS WITH NON-SMOOTH AND NON-STRICTLY CONVEX FLUX CAREY CAGINALP arxiv:1708.02339v2 [mah.ap]

More information

( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is

( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires

More information

Let us start with a two dimensional case. We consider a vector ( x,

Let us start with a two dimensional case. We consider a vector ( x, Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our

More information

arxiv: v1 [math.ca] 15 Nov 2016

arxiv: v1 [math.ca] 15 Nov 2016 arxiv:6.599v [mah.ca] 5 Nov 26 Counerexamples on Jumarie s hree basic fracional calculus formulae for non-differeniable coninuous funcions Cheng-shi Liu Deparmen of Mahemaics Norheas Peroleum Universiy

More information

MA 366 Review - Test # 1

MA 366 Review - Test # 1 MA 366 Review - Tes # 1 Fall 5 () Resuls from Calculus: differeniaion formulas, implici differeniaion, Chain Rule; inegraion formulas, inegraion b pars, parial fracions, oher inegraion echniques. (1) Order

More information

1 Solutions to selected problems

1 Solutions to selected problems 1 Soluions o seleced problems 1. Le A B R n. Show ha in A in B bu in general bd A bd B. Soluion. Le x in A. Then here is ɛ > 0 such ha B ɛ (x) A B. This shows x in B. If A = [0, 1] and B = [0, 2], hen

More information

15. Vector Valued Functions

15. Vector Valued Functions 1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,

More information

Math Final Exam Solutions

Math Final Exam Solutions Mah 246 - Final Exam Soluions Friday, July h, 204 () Find explici soluions and give he inerval of definiion o he following iniial value problems (a) ( + 2 )y + 2y = e, y(0) = 0 Soluion: In normal form,

More information

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid

More information

IMPLICIT AND INVERSE FUNCTION THEOREMS PAUL SCHRIMPF 1 OCTOBER 25, 2013

IMPLICIT AND INVERSE FUNCTION THEOREMS PAUL SCHRIMPF 1 OCTOBER 25, 2013 IMPLICI AND INVERSE FUNCION HEOREMS PAUL SCHRIMPF 1 OCOBER 25, 213 UNIVERSIY OF BRIISH COLUMBIA ECONOMICS 526 We have exensively sudied how o solve sysems of linear equaions. We know how o check wheher

More information

Kinematics and kinematic functions

Kinematics and kinematic functions Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables and vice versa Direc Posiion

More information

SMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.

SMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15. SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a

More information

1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.

1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx. . Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.

More information

Designing Information Devices and Systems I Spring 2019 Lecture Notes Note 17

Designing Information Devices and Systems I Spring 2019 Lecture Notes Note 17 EES 16A Designing Informaion Devices and Sysems I Spring 019 Lecure Noes Noe 17 17.1 apaciive ouchscreen In he las noe, we saw ha a capacior consiss of wo pieces on conducive maerial separaed by a nonconducive

More information

Inventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions

Inventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.

More information

The Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales

The Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions

More information

Homework sheet Exercises done during the lecture of March 12, 2014

Homework sheet Exercises done during the lecture of March 12, 2014 EXERCISE SESSION 2A FOR THE COURSE GÉOMÉTRIE EUCLIDIENNE, NON EUCLIDIENNE ET PROJECTIVE MATTEO TOMMASINI Homework shee 3-4 - Exercises done during he lecure of March 2, 204 Exercise 2 Is i rue ha he parameerized

More information

Notes for Lecture 17-18

Notes for Lecture 17-18 U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up

More information

Sections 2.2 & 2.3 Limit of a Function and Limit Laws

Sections 2.2 & 2.3 Limit of a Function and Limit Laws Mah 80 www.imeodare.com Secions. &. Limi of a Funcion and Limi Laws In secion. we saw how is arise when we wan o find he angen o a curve or he velociy of an objec. Now we urn our aenion o is in general

More information

Math 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11.

Math 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11. 1 Mah 334 Tes 1 KEY Spring 21 Secion: 1 Insrucor: Sco Glasgow Daes: Ma 1 and 11. Do NOT wrie on his problem saemen bookle, excep for our indicaion of following he honor code jus below. No credi will be

More information

Chapter 7: Solving Trig Equations

Chapter 7: Solving Trig Equations Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions

More information

THE WAVE EQUATION. part hand-in for week 9 b. Any dilation v(x, t) = u(λx, λt) of u(x, t) is also a solution (where λ is constant).

THE WAVE EQUATION. part hand-in for week 9 b. Any dilation v(x, t) = u(λx, λt) of u(x, t) is also a solution (where λ is constant). THE WAVE EQUATION 43. (S) Le u(x, ) be a soluion of he wave equaion u u xx = 0. Show ha Q43(a) (c) is a. Any ranslaion v(x, ) = u(x + x 0, + 0 ) of u(x, ) is also a soluion (where x 0, 0 are consans).

More information

3, so θ = arccos

3, so θ = arccos Mahemaics 210 Professor Alan H Sein Monday, Ocober 1, 2007 SOLUTIONS This problem se is worh 50 poins 1 Find he angle beween he vecors (2, 7, 3) and (5, 2, 4) Soluion: Le θ be he angle (2, 7, 3) (5, 2,

More information

Linear Response Theory: The connection between QFT and experiments

Linear Response Theory: The connection between QFT and experiments Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and

More information

Ordinary Differential Equations

Ordinary Differential Equations Ordinary Differenial Equaions 5. Examples of linear differenial equaions and heir applicaions We consider some examples of sysems of linear differenial equaions wih consan coefficiens y = a y +... + a

More information

ON THE DEGREES OF RATIONAL KNOTS

ON THE DEGREES OF RATIONAL KNOTS ON THE DEGREES OF RATIONAL KNOTS DONOVAN MCFERON, ALEXANDRA ZUSER Absrac. In his paper, we explore he issue of minimizing he degrees on raional knos. We se a bound on hese degrees using Bézou s heorem,

More information

u(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x

u(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x . 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih

More information

Second Order Linear Differential Equations

Second Order Linear Differential Equations Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous

More information

Lecture 10: The Poincaré Inequality in Euclidean space

Lecture 10: The Poincaré Inequality in Euclidean space Deparmens of Mahemaics Monana Sae Universiy Fall 215 Prof. Kevin Wildrick n inroducion o non-smooh analysis and geomery Lecure 1: The Poincaré Inequaliy in Euclidean space 1. Wha is he Poincaré inequaliy?

More information

An random variable is a quantity that assumes different values with certain probabilities.

An random variable is a quantity that assumes different values with certain probabilities. Probabiliy The probabiliy PrA) of an even A is a number in [, ] ha represens how likely A is o occur. The larger he value of PrA), he more likely he even is o occur. PrA) means he even mus occur. PrA)

More information

k 1 k 2 x (1) x 2 = k 1 x 1 = k 2 k 1 +k 2 x (2) x k series x (3) k 2 x 2 = k 1 k 2 = k 1+k 2 = 1 k k 2 k series

k 1 k 2 x (1) x 2 = k 1 x 1 = k 2 k 1 +k 2 x (2) x k series x (3) k 2 x 2 = k 1 k 2 = k 1+k 2 = 1 k k 2 k series Final Review A Puzzle... Consider wo massless springs wih spring consans k 1 and k and he same equilibrium lengh. 1. If hese springs ac on a mass m in parallel, hey would be equivalen o a single spring

More information

Oscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson

Oscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,

More information

From Complex Fourier Series to Fourier Transforms

From Complex Fourier Series to Fourier Transforms Topic From Complex Fourier Series o Fourier Transforms. Inroducion In he previous lecure you saw ha complex Fourier Series and is coeciens were dened by as f ( = n= C ne in! where C n = T T = T = f (e

More information

Fishing limits and the Logistic Equation. 1

Fishing limits and the Logistic Equation. 1 Fishing limis and he Logisic Equaion. 1 1. The Logisic Equaion. The logisic equaion is an equaion governing populaion growh for populaions in an environmen wih a limied amoun of resources (for insance,

More information

Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles

Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance

More information

Lie Derivatives operator vector field flow push back Lie derivative of

Lie Derivatives operator vector field flow push back Lie derivative of Lie Derivaives The Lie derivaive is a mehod of compuing he direcional derivaive of a vecor field wih respec o anoher vecor field We already know how o make sense of a direcional derivaive of real valued

More information

Dual Representation as Stochastic Differential Games of Backward Stochastic Differential Equations and Dynamic Evaluations

Dual Representation as Stochastic Differential Games of Backward Stochastic Differential Equations and Dynamic Evaluations arxiv:mah/0602323v1 [mah.pr] 15 Feb 2006 Dual Represenaion as Sochasic Differenial Games of Backward Sochasic Differenial Equaions and Dynamic Evaluaions Shanjian Tang Absrac In his Noe, assuming ha he

More information

23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes

23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion

More information

KEY. Math 334 Midterm III Fall 2008 sections 001 and 003 Instructor: Scott Glasgow

KEY. Math 334 Midterm III Fall 2008 sections 001 and 003 Instructor: Scott Glasgow KEY Mah 334 Miderm III Fall 28 secions and 3 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering

More information

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

t 2 B F x,t n dsdt t u x,t dxdt

t 2 B F x,t n dsdt t u x,t dxdt Evoluion Equaions For 0, fixed, le U U0, where U denoes a bounded open se in R n.suppose ha U is filled wih a maerial in which a conaminan is being ranspored by various means including diffusion and convecion.

More information

Review - Quiz # 1. 1 g(y) dy = f(x) dx. y x. = u, so that y = xu and dy. dx (Sometimes you may want to use the substitution x y

Review - Quiz # 1. 1 g(y) dy = f(x) dx. y x. = u, so that y = xu and dy. dx (Sometimes you may want to use the substitution x y Review - Quiz # 1 (1) Solving Special Tpes of Firs Order Equaions I. Separable Equaions (SE). d = f() g() Mehod of Soluion : 1 g() d = f() (The soluions ma be given implicil b he above formula. Remember,

More information

Let ( α, β be the eigenvector associated with the eigenvalue λ i

Let ( α, β be the eigenvector associated with the eigenvalue λ i ENGI 940 4.05 - Sabiliy Analysis (Linear) Page 4.5 Le ( α, be he eigenvecor associaed wih he eigenvalue λ i of he coefficien i i) marix A Le c, c be arbirary consans. a b c d Case of real, disinc, negaive

More information

ME 391 Mechanical Engineering Analysis

ME 391 Mechanical Engineering Analysis Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of

More information

Basilio Bona ROBOTICA 03CFIOR 1

Basilio Bona ROBOTICA 03CFIOR 1 Indusrial Robos Kinemaics 1 Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables

More information

Exercises: Similarity Transformation

Exercises: Similarity Transformation Exercises: Similariy Transformaion Problem. Diagonalize he following marix: A [ 2 4 Soluion. Marix A has wo eigenvalues λ 3 and λ 2 2. Since (i) A is a 2 2 marix and (ii) i has 2 disinc eigenvalues, we

More information

FEEDBACK NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION

FEEDBACK NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION Differenial and Inegral Equaions Volume 5, Number, January 2002, Pages 5 28 FEEDBACK NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION Mihai Sîrbu Deparmen of Mahemaical Sciences, Carnegie Mellon Universiy

More information

4 Sequences of measurable functions

4 Sequences of measurable functions 4 Sequences of measurable funcions 1. Le (Ω, A, µ) be a measure space (complee, afer a possible applicaion of he compleion heorem). In his chaper we invesigae relaions beween various (nonequivalen) convergences

More information

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms

More information

Optimality Conditions for Unconstrained Problems

Optimality Conditions for Unconstrained Problems 62 CHAPTER 6 Opimaliy Condiions for Unconsrained Problems 1 Unconsrained Opimizaion 11 Exisence Consider he problem of minimizing he funcion f : R n R where f is coninuous on all of R n : P min f(x) x

More information

KINEMATICS IN ONE DIMENSION

KINEMATICS IN ONE DIMENSION KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec

More information

) were both constant and we brought them from under the integral.

) were both constant and we brought them from under the integral. YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha

More information

Traveling Waves. Chapter Introduction

Traveling Waves. Chapter Introduction Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from

More information

THE 2-BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI

THE 2-BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI THE 2-BODY PROBLEM ROBERT J. VANDERBEI ABSTRACT. In his shor noe, we show ha a pair of ellipses wih a common focus is a soluion o he 2-body problem. INTRODUCTION. Solving he 2-body problem from scrach

More information