# Interface optimization problems for parabolic equations

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Control and Cybernetics vol. 23 (1994) No. 3 Interface optimization problems for parabolic equations by Karl-Heinz Hoffmann Lehrstuhl fiir Angewandte Mathematik Institut fiir Angewandte Mathematik und Statistik Technische Universitiit Miinchen Dachauer Strasse 9a, D-8000 Miinchen 2 Germany J an Sokolowski Systems Research Institute, Polish Academy of Sciences Warsaw, ul. Newelska 6 Poland and Universite de Nancy I Departement de Mathematiques, B.P. 239 U.R.A-C.N.R.S. 750, Projet NUMATH, INRIA-Lorraine Vandoeuvre- Les-Nancy France An optimization problem for the heat equation with respect to the geometrical domain is considered. The existence of an optimal solution is obtained and the first order necessary optimality conditions are derived. 1. Introduction In this paper an optimization problem for the heat equation with respect to the geometrical domain is considered. Two C 1 curves are selected in an optimal way, the curves being the interfaces between the subsets of the lateral boundary of parabolic cylinder where the Neumann and Dirichlet boundary conditions are prescribed for the equation. The existence of an optimal solution is obtained by adding a regularizing term to the cost functional and the first order necessary optimality conditions are derived.

2 446 K.-H. HOFFMANN, J. SOKOLOWSKI The proofs of the results presented in the paper as well as some applications and numerical results are given in the forthcoming paper by Hoffmann and Sokolowski (1991). Let Q = 0 X (0, T), T > 0, denote a cylinder in ffi 3, where 0 E ffi 2 is a bounded domain with the smooth boundary ao. We assume that there are given compact, simply connected subsets K; C ao, i = 1, 2, such that K, 1 n K 2 = 0 and two sufficiently smooth curves X;(-) : [0, T]--. K; c ffi 3 on I;= an x [0, T]. The set I: consists of two subsets I:;, i = 1, 2, with I;"l n I:2 = u {X1(t) X t} U {X2(t) x t}. te[o,t] We assume that X;(-) E H 2 (0,T;ffi 2 ) fori= 1,2, and denote X= {X1,X2} E U = H 2 (0,T;ffi 4 ). ThesetofadmissiblecurvesUad ={X E H 2 (0,T;ffi 4 )1X(t) E K 1 x K 2 } obviously is not convex. The domain optimization problem can be formulated as follows inf J(X) XEUad with J(X) = ~ fo (y(x)- Yd) 2 dq + ~IIXII~ (1) where Yd E L 2 ( Q) is given, a ~ 0 is a regularization parameter. The function y(x)(x, t), X E Uad, (x, t) E Q, satisfies for a given function F E L 2 (Q) the heat equation ay at - -1\y = F ' in Q = 0 X (0, T) with given initial condition in 0 and mixed boundary conditions depending on X E Uad prescribed on the lateral boundary of Q. The boundary conditions are prescribed as follows: y = y(x) satisfies the nonhomegenuous Dirichlet condition on I;2 (resp. nonhomogenuous Neumann condition on I;1). In order to obtain the existence of an optimal domain i.e. the existence of an element X* E Uad such that J(X*) ; J(X) for all X E Uad, it is sufficient to select a family of admissible domains which is compact in an appropriate sense. For the problem under consideration it is sufficient to assume that for a minimizing sequence of domains { Qm} there exists a subsequence, still denoted { Qm}, such that the sequence of characteristic functions X m = characteristic function of I;~, converges in L 2 (I;) to a characteristic function X In order to have the existence of an optimal domain the standard approach of control theory is used, namely a regularizing term is introduced. We refer the reader to Sokolowski and Zolesio (1992) for a description of the material derivative method in the shape sensitivity analysis. The related results on the shape sensitivity analysis of optimal control problems can be found in Sokolowski (1987, 1988).

3 Inierfa.ce opiimiza.iion problems for pa.ra.bolic equa.iions Parabolic equation in variable domain Let us consider the following parabolic equation if- Ay = F, in Q = n x (0, T) -?,;- = f, on E1 y = g, y(x, 0) = Yo(x), on E2 inn where J, g E L 2 (E) are given, and E; = { ( x, t) E f i ( t) x { t}, t E ( 0, T) } i = 1, 2 f 1(t) n f2(t) = X1(t) U X1(t) for all t E (0, T]. Here n is a given domain, an= f1(0) U f2(0) U {X1(0)} U {X2(0)}. We denote fo = fo(o), f1 =: f1(0). In order to derive the first order necessary optimality conditions for the optimization problem defined in the variable domain, we assume X1(t) = Tt(V)(XI(O)), X2(t) = Tt(V)(X2(0)), 'Vt E (0, T] for a given vector field V(.,.) E C(O, T + 81; C 1 (ffi.2, ill?)), 81 > 0 with the support in a compact of an. First, we investigate the differential stability of the solution to the parabolic equation with respect to the perturbations of the curves X 1 (t), X2(t) E an C rn.2, t E (0, T]. Let Xf (.),,q (.) be perturbed curves defined below, where c: E (0, o) is a parameter. Denote by Ye a solution to the parabolic equation in the perturbed domain, ~ -Aye = F, in Q = n x (0, T) ~ = J, on Ei Ye = g, on Et Ye(x, 0) = Yo(x), inn here f = fe, g = ge denote the restriction to E2 (resp. Ei) offunctions f (resp. g) defined on E. We assume that for sufficiently small t the curves X1(.), X2(.) are not perturbed

4 448 K.-H. HOFFMANN, J. SOKOLOWSKI Furthermore, we assume that X{(t), X2(t) E on, X{(t) ::j; X:2(t), Vt E [0, T ], VE: E [0, b) and there exist fori = 1, 2 the limits in H 2 (0, T; IR?) h; = lim ~(Xt - xn, e!o E; where h 1 (t), h 2 (t), t E (0, T), are the tangent vectors on :E = 80 x (0, T). In this section we derive the form of the derivative y 1 of the solution Ye = y(xe) E L 2 (Q) to the parabolic equation, with respect toe:, ate: = 0, assuming that the derivative exists in the space L 2 (Q). For the proof of the existence we refer the reader to Hoffmann and Sokolowski (1991). Using the transposition method it follows that the parabolic equation is equivalent to the following integral identit,y V<p E H 2, 1 (Q), t.p(t) = 0 in D, <p = 0 on :E2 Let us select a sufficiently smooth test function <p and let us assume that the test function is independent of the parameter E: i.e., <p = 0 in a small neighbourhood of I;~ C IR 3. If Ye is differentiable with respect to E:, at E: = 0, the derivative y 1 satisfies the following integral identity y'( - -- o<p D.<p)dxdt 0 n ot 1T 1 = r g ~<p.c(h)df + r Jt.p.C(h)df } ar-2 un } ar-1 V<p E H 2, 1 (Q), <p(t) = 0 in D, <p = 0 on :E2, where we denote h = { h 1 = lime!o ~(X{ - Xo), h2 = lime!o t(x2- xn, dl = (1 +I d~; (t) l2)1/2at, on X;, i = 1, 2 1.C(h)(t) = a(t) [hi(t) - (h;(t), V(t, X;(t))JR2/IIV(t, X;(t))ilnvL on X;, i = 1, 2

5 Interface optimization problems for pa.ra.bolic equations 449 REMARK 1 If the test function r.p is not sufficiently smooth, the right hand side of the integral identity for y' is not well defined. In such a case, assuming the differentiability of c: ---* Ye, the form of the derivative y' is defined by an auxiliary parabolic problem. PROPOSITION 1 There exist distributions g 0, g 1 supported on 8E 2 ( resp. on 8E 1 ) such that 1T r y 1 (- or.p- /:ir.p)dxdt 0 ln at \fr.p E H 2 1(Q), r.p(t) = 0 in fl, r.p = 0 on E2 = Eg 3. Optimization problem Finally, the necessary optimality conditions are derived for the following optimization probleum! Minimize the cost functional J(X) = ~it l (y,- Yd) 2 dxdt +~IIXIIu + /3 1 { g 2 de + f3 2 { f 2 de 2 2 }E 2 2 je 1 subject to the following nonconvex constraints X(t) = (X1(t), X2(t)) E Uad, for all t E [0, T], where a > 0, {3 1, {32 :::: 0 are given, U = H 2 (0, T; IR 4 ). PROPOSITION 2 There exists an optimal solution X E Uad for the optimization problem. In order to derive the first order necessary optimality conditions we assume that there exists a vector field V such that X;(t) =: 1t(V)(X;(O)) Vt E [0, T], i = 1, 2 A vector field H(.,.) defines an admissible perturbation of the optimal solution X provided that for c: > 0, c: small enough, Tt(V.)(X!(O)) E K-1, Tt(V.)(X2(0)) E K-2, Vt E [0, T] where V.= V+ c:h. For simplicity, we denote an optimal solution of the optimization problem by X. THEOREM 1 An optimal solution X E Uad satisfies the following optimality system State equation:

6 450 K.-H. HOFFMANN, J. SOKOLOWSKJ %? - D.y = F, in Q = n X (0, T)?, = J, on ~1 y=g, on~ 2 y(x, 0) = Yo(x), in fl Adjoint state equation: -%? - D.p = Y- Yd, in Q = fl x (0, T) y = 0, on ~2 p( X, T) = 0, in n Optimality conditions: for any admissible vector field H(.,.). Acknowledgement. Partially supported by Deutsche Forschungsgemeinschaft, SPP "Anwendungsbezogene Optimierung und Steuerung", and by the grant of the State Comittee for Scientific Research of the Republic of Poland. References HOFFMANN, K.-H. AND SOKOLOWSKI, J. (1991) Domain optimization problem for parabolic equation, submitted for publication, and DFG Report No. 342, Augsburg. LIONS, J.L. AND MAGENES, E. (1968) Problemes aux Limites Non Homogenes et Applications. Vol.l, Dunod, Paris. SOKOLOWSKI, J. (1987) Sensitivity analysis of control constrained optimal control problems for-distributed parameter systems. SIAM J. Control and Op~ timization 25,

7 Interfa.ce optimization problems for pa.ra.bolic equa.tions 451 SoKOLOWSKI, J. (1988) Shape sensitivity analysis of boundary optimal control problems for parabolic systems. SIAM Journal on Control and Optimization 26, SOKOLOWSKI, J. AND ZoLESIO, J.-P. (1992) Introduction to Shape Optimization. Shape sensitivity analysis. Springer Ve,rlag.

8

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for Higher-Order Adjacent Derivative

### Shape Optimization Problems over Classes of Convex Domains

Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY e-mail: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore

### Limits and Continuity

Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.

5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2

### ON A GLOBALIZATION PROPERTY

APPLICATIONES MATHEMATICAE 22,1 (1993), pp. 69 73 S. ROLEWICZ (Warszawa) ON A GLOBALIZATION PROPERTY Abstract. Let (X, τ) be a topological space. Let Φ be a class of realvalued functions defined on X.

### Class Meeting # 1: Introduction to PDEs

MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x

### A Domain Embedding/Boundary Control Method to Solve Elliptic Problems in Arbitrary Domains

A Domain Embedding/Boundary Control Method to Solve Elliptic Problems in Arbitrary Domains L. Badea 1 P. Daripa 2 Abstract This paper briefly describes some essential theory of a numerical method based

### Practice Final Math 122 Spring 12 Instructor: Jeff Lang

Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series

1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,

### Econ 8105 MACROECONOMIC THEORY DYNAMIC PROGRAMMING FOR MACRO. Prof. L. Jones. Fall 2010

Econ 8105 MACROECONOMIC THEORY DYNAMIC PROGRAMMING FOR MACRO Prof. L. Jones Fall 2010 These notes are a condensed treatment of the chapters in SLP that deal with Deterministic Dynamic Programming used

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Perron vector Optimization applied to search engines

Perron vector Optimization applied to search engines Olivier Fercoq INRIA Saclay and CMAP Ecole Polytechnique May 18, 2011 Web page ranking The core of search engines Semantic rankings (keywords) Hyperlink

### Convexity in R N Main Notes 1

John Nachbar Washington University December 16, 2016 Convexity in R N Main Notes 1 1 Introduction. These notes establish basic versions of the Supporting Hyperplane Theorem (Theorem 5) and the Separating

### OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov

DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics

### MATH 425, HOMEWORK 7, SOLUTIONS

MATH 425, HOMEWORK 7, SOLUTIONS Each problem is worth 10 points. Exercise 1. (An alternative derivation of the mean value property in 3D) Suppose that u is a harmonic function on a domain Ω R 3 and suppose

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION

Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7-669. UL: http://ejde.math.txstate.edu

### Solutions to Practice Problems for Test 4

olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,

### A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY

Guidetti, D. and Lorenzi, A. Osaka J. Math. 44 (27), 579 613 A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY DAVIDE GUIDETTI and ALFREDO LORENZI (Received January 23, 26,

### M.L. Sampoli CLOSED SPLINE CURVES BOUNDING MAXIMAL AREA

Rend. Sem. Mat. Univ. Pol. Torino Vol. 61, 3 (2003) Splines and Radial Functions M.L. Sampoli CLOSED SPLINE CURVES BOUNDING MAXIMAL AREA Abstract. In this paper we study the problem of constructing a closed

### Scheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates

Scheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates Donatas Elvikis, Horst W. Hamacher, Marcel T. Kalsch Department of Mathematics, University of Kaiserslautern, Kaiserslautern,

### Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.

Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian

### On a comparison result for Markov processes

On a comparison result for Markov processes Ludger Rüschendorf University of Freiburg Abstract A comparison theorem is stated for Markov processes in polish state spaces. We consider a general class of

### Robust Geometric Programming is co-np hard

Robust Geometric Programming is co-np hard André Chassein and Marc Goerigk Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany Abstract Geometric Programming is a useful tool with a

### Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

### 15 Limit sets. Lyapunov functions

15 Limit sets. Lyapunov functions At this point, considering the solutions to ẋ = f(x), x U R 2, (1) we were most interested in the behavior of solutions when t (sometimes, this is called asymptotic behavior

### 5. Convergence of sequences of random variables

5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

### OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem")

OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem") BY Y. S. CHOW, S. MORIGUTI, H. ROBBINS AND S. M. SAMUELS ABSTRACT n rankable persons appear sequentially in random order. At the ith

### An optimal transportation problem with import/export taxes on the boundary

An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint

### x + 5 x 2 + x 2 dx Answer: ln x ln x 1 + c

. Evaluate the given integral (a) 3xe x2 dx 3 2 e x2 + c (b) 3 x ln xdx 2x 3/2 ln x 4 3 x3/2 + c (c) x + 5 x 2 + x 2 dx ln x + 2 + 2 ln x + c (d) x sin (πx) dx x π cos (πx) + sin (πx) + c π2 (e) 3x ( +

### Fuzzy Differential Systems and the New Concept of Stability

Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

### Sequences of Functions

Sequences of Functions Uniform convergence 9. Assume that f n f uniformly on S and that each f n is bounded on S. Prove that {f n } is uniformly bounded on S. Proof: Since f n f uniformly on S, then given

### S = k=1 k. ( 1) k+1 N S N S N

Alternating Series Last time we talked about convergence of series. Our two most important tests, the integral test and the (limit) comparison test, both required that the terms of the series were (eventually)

### EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### MATHEMATICAL BACKGROUND

Chapter 1 MATHEMATICAL BACKGROUND This chapter discusses the mathematics that is necessary for the development of the theory of linear programming. We are particularly interested in the solutions of a

### An interval linear programming contractor

An interval linear programming contractor Introduction Milan Hladík Abstract. We consider linear programming with interval data. One of the most challenging problems in this topic is to determine or tight

### PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS

PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,

### Functional Optimization Models for Active Queue Management

Functional Optimization Models for Active Queue Management Yixin Chen Department of Computer Science and Engineering Washington University in St Louis 1 Brookings Drive St Louis, MO 63130, USA chen@cse.wustl.edu

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### 88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

### Section 3.1 Calculus of Vector-Functions

Section 3.1 Calculus of Vector-Functions De nition. A vector-valued function is a rule that assigns a vector to each member in a subset of R 1. In other words, a vector-valued function is an ordered triple

### Infinite series, improper integrals, and Taylor series

Chapter Infinite series, improper integrals, and Taylor series. Introduction This chapter has several important and challenging goals. The first of these is to understand how concepts that were discussed

### m = 4 m = 5 x x x t = 0 u = x 2 + 2t t < 0 t > 1 t = 0 u = x 3 + 6tx t > 1 t < 0

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 21, Number 2, Spring 1991 NODAL PROPERTIES OF SOLUTIONS OF PARABOLIC EQUATIONS SIGURD ANGENENT * 1. Introduction. In this note we review the known facts about

### COLLAGE-TYPE APPROACH TO INVERSE PROBLEMS FOR ELLIPTIC PDES ON PERFORATED DOMAINS

Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 48, pp. 1 11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu COLLAGE-TYPE APPROACH

### Adaptive Numerical Solution of State Constrained Optimal Control Problems

Technische Universität München Lehrstuhl für Mathematische Optimierung Adaptive Numerical Solution of State Constrained Optimal Control Problems Olaf Benedix Vollständiger Abdruck der von der Fakultät

### Numerisches Rechnen. (für Informatiker) M. Grepl J. Berger & J.T. Frings. Institut für Geometrie und Praktische Mathematik RWTH Aachen

(für Informatiker) M. Grepl J. Berger & J.T. Frings Institut für Geometrie und Praktische Mathematik RWTH Aachen Wintersemester 2010/11 Problem Statement Unconstrained Optimality Conditions Constrained

### General Equilibrium Theory: Examples

General Equilibrium Theory: Examples 3 examples of GE: pure exchange (Edgeworth box) 1 producer - 1 consumer several producers and an example illustrating the limits of the partial equilibrium approach

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### Chapter (AB/BC, non-calculator) (a) Write an equation of the line tangent to the graph of f at x 2.

Chapter 1. (AB/BC, non-calculator) Let f( x) x 3 4. (a) Write an equation of the line tangent to the graph of f at x. (b) Find the values of x for which the graph of f has a horizontal tangent. (c) Find

### General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1

A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions

### ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series

ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don

### TECHNICAL NOTE. Proof of a Conjecture by Deutsch, Li, and Swetits on Duality of Optimization Problems1

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 102, No. 3, pp. 697-703, SEPTEMBER 1999 TECHNICAL NOTE Proof of a Conjecture by Deutsch, Li, and Swetits on Duality of Optimization Problems1 H. H.

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### 14. Dirichlet polygons: the construction

14. Dirichlet polygons: the construction 14.1 Recap Let Γ be a Fuchsian group. Recall that a Fuchsian group is a discrete subgroup of the group Möb(H) of all Möbius transformations. In the last lecture,

### 4 Lyapunov Stability Theory

4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We

### MVE041 Flervariabelanalys

MVE041 Flervariabelanalys 2015-16 This document contains the learning goals for this course. The goals are organized by subject, with reference to the course textbook Calculus: A Complete Course 8th ed.

### Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Chapter 1: Initial value problems in ODEs Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical

### DEGREE THEORY E.N. DANCER

DEGREE THEORY E.N. DANCER References: N. Lloyd, Degree theory, Cambridge University Press. K. Deimling, Nonlinear functional analysis, Springer Verlag. L. Nirenberg, Topics in nonlinear functional analysis,

### Section 12.6: Directional Derivatives and the Gradient Vector

Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### Regularity of solution maps of differential inclusions under state constraints

Regularity of solution maps of differential inclusions under state constraints Piernicola Bettiol, Hélène Frankowska Abstract Consider a differential inclusion under state constraints x (t) F (t, x(t)),

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### x 2 x 2 cos 1 x x2, lim 1. If x > 0, multiply all three parts by x > 0, we get: x x cos 1 x x, lim lim x cos 1 lim = 5 lim sin 5x

Homework 4 3.4,. Show that x x cos x x holds for x 0. Solution: Since cos x, multiply all three parts by x > 0, we get: x x cos x x, and since x 0 x x 0 ( x ) = 0, then by Sandwich theorem, we get: x 0

### MATH PROBLEMS, WITH SOLUTIONS

MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These

### Lecture 10. Finite difference and finite element methods. Option pricing Sensitivity analysis Numerical examples

Finite difference and finite element methods Lecture 10 Sensitivities and Greeks Key task in financial engineering: fast and accurate calculation of sensitivities of market models with respect to model

### Quasi-static evolution and congested transport

Quasi-static evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed

### SOLVING DYNAMICAL SYSTEMS WITH CUBIC TRIGONOMETRIC SPLINES

Applied Mathematics E-Notes, 5(005), 6-3 c ISSN 607-50 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ SOLVING DYNAMICAL SYSTEMS WITH CUBIC TRIGONOMETRIC SPLINES Athanassios Nikolis,

### CURRENTS FOLIATIONS CURVATURES. Pawe l Walczak ( Lódź, Poland) Kraków, 2008

CURRENTS FOLIATIONS CURVATURES Pawe l Walczak ( Lódź, Poland) Kraków, 2008 Here: M a compact oriented manifold, n = dim M F an oriented foliation on M, k = dim F g a Riemannian structure on M X(F, g) a

### Figure 2.1: Center of mass of four points.

Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would

### k=1 k2, and therefore f(m + 1) = f(m) + (m + 1) 2 =

Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1.1. Prove that 1 2 +2 2 + +n 2 = 1 n(n+1)(2n+1) for all n N. 6 Put f(n) = n(n + 1)(2n + 1)/6. Then f(1) = 1, i.e the theorem

### By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

### The inflation of attractors and their discretization: the autonomous case

The inflation of attractors and their discretization: the autonomous case P. E. Kloeden Fachbereich Mathematik, Johann Wolfgang Goethe Universität, D-654 Frankfurt am Main, Germany E-mail: kloeden@math.uni-frankfurt.de

### Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space

Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space Virginie Debelley and Nicolas Privault Département de Mathématiques Université de La Rochelle

### Date: April 12, 2001. Contents

2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........

### be a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that

Problem 1A. Let... X 2 X 1 be a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that i=1 X i is nonempty and connected. Since X i is closed in X, it is compact.

### College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions

College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use

### SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

### Finite covers of a hyperbolic 3-manifold and virtual fibers.

Claire Renard Institut de Mathématiques de Toulouse November 2nd 2011 Some conjectures. Let M be a hyperbolic 3-manifold, connected, closed and oriented. Theorem (Kahn, Markovic) The fundamental group

### Solving Time-Dependent Optimal Control Problems in Comsol Multiphysics

Excerpt from the Proceedings of the COMSOL Conference 2008 Hannover Solving Time-Dependent Optimal Control Problems in Comsol Multiphysics Ira Neitzel,1, Uwe Prüfert 2, and Thomas Slawig 3 1 Technische

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### The Annals of Mathematics, 2nd Ser., Vol. 149, No. 1. (Jan., 1999), pp

A Counterexample to the "Hot Spots" Conjecture Krzysztof Burdzy; Wendelin Werner The Annals of Mathematics, 2nd Ser., Vol. 149, No. 1. (Jan., 1999), pp. 309-317. Stable URL: http://links.jstor.org/sici?sici=0003-486x%28199901%292%3a149%3a1%3c309%3aactt%22s%3e2.0.co%3b2-b

### Notes on weak convergence (MAT Spring 2006)

Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p

### How to Manage the Maximum Relative Drawdown

How to Manage the Maximum Relative Drawdown Jan Vecer, Petr Novotny, Libor Pospisil, Columbia University, Department of Statistics, New York, NY 27, USA April 9, 26 Abstract Maximum Relative Drawdown measures

### A class of infinite dimensional stochastic processes

A class of infinite dimensional stochastic processes John Karlsson Linköping University CRM Barcelona, July 7, 214 Joint work with Jörg-Uwe Löbus John Karlsson (Linköping University) Infinite dimensional

### Convergence of Feller Processes

Chapter 15 Convergence of Feller Processes This chapter looks at the convergence of sequences of Feller processes to a iting process. Section 15.1 lays some ground work concerning weak convergence of processes

### 1 Conservative vector fields

Math 241 - Calculus III Spring 2012, section CL1 16.5. Conservative vector fields in R 3 In these notes, we discuss conservative vector fields in 3 dimensions, and highlight the similarities and differences

### Solution: APPM 2360 Exam1 Spring 2012

APPM 60 Exam1 Spring 01 Problem 1: (0 points) Consider the following initial value problem where y is the radius of a raindrop and y = y / ; y 0 (1) y(0) = 1, () models the growth of the radius (as water

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### Extremal equilibria for reaction diffusion equations in bounded domains and applications.

Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal Rodríguez-Bernal Alejandro Vidal-López Departamento de Matemática Aplicada Universidad Complutense de Madrid,

### ffi I Q II MM I VI I V2 I V3 I V 4 I :11 a a a e b Math 101, Final Exam, Term IANSWER KEY I : 1 a e b c e 2 a b a c d!

Math 0, Final Exam, Term 3 IANSWER KEY I I Q II MM I VI I V I V3 I V 4 I : a e b c e a b a c d! 3 a d e d c i 4 a a a e a 5 a e d e c! 6 a a d e c 7 a e c d c 8 a d d b b! 9 a c d b e 0 a c a c a : a a

### Sequences and Series

Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will

### General theory of stochastic processes

CHAPTER 1 General theory of stochastic processes 1.1. Definition of stochastic process First let us recall the definition of a random variable. A random variable is a random number appearing as a result

### CALCULUS 2. 0 Repetition. tutorials 2015/ Find limits of the following sequences or prove that they are divergent.

CALCULUS tutorials 5/6 Repetition. Find limits of the following sequences or prove that they are divergent. a n = n( ) n, a n = n 3 7 n 5 n +, a n = ( n n 4n + 7 ), a n = n3 5n + 3 4n 7 3n, 3 ( ) 3n 6n

### 6.8 Taylor and Maclaurin s Series

6.8. TAYLOR AND MACLAURIN S SERIES 357 6.8 Taylor and Maclaurin s Series 6.8.1 Introduction The previous section showed us how to find the series representation of some functions by using the series representation