Class meets: MW / 11:50AM-01:10PM LS 102

Office hours: W, 1:30-2:30 PM or by appointment

Telephone: 581-6822

E-mail: cherk@math.utah.edu

* *** ***David Hilbert *

Syllabus

- Introduction
- Stationarity condition 1. Euler equation
- Geometric optics, brachistochrone, minimal surface of revolution
**Second Variation I (1d). Legendre, Weierstrass, Jacobi tests. Examples**- Constrained problems: Lagrange
multipliers, Isoperimentric problems. Functionals

- Isoperimetric
and geodesics problems

- Constraints and Hamiltonian. Lagrangian mechanics
**Legendre Duality: Dual Variational Principles****Approximation with penalty**- Two body problem in celestial mechanics
- Numerical methods

**Irregular solutions: Sketch**- Reminder. Vector and matrix differentiation, Integral formulas
- Stationarity condition 2. Multiple integrals.One minimizer.
- Stationarity condition 3. Multiple integrals. Several minimizers. Examples: Elasticity, Complex conductivity
- Optimal design: Problems with differential constraints
- Second Variation 2 (Multivariable). Legendre, Weierstrass, Jacobi tests.
- Variation of Domain. Applications to geometry

Inequalities
that Imply the Isoperimetric Inequality: an article by
Andrejs Treibergs:
http://www.math.utah.edu/~treiberg/isoperim/isop.pdf

HW2 2018

Sources for Numerical methods

Shooting methods

https://en.wikipedia.org/wiki/Shooting_method

Lecture

https://www.mathworks.com/matlabcentral/fileexchange/32451-shooting-method?focused=5194030&tab=function&s_tid=gn_loc_drop

Matlab

https://www.mathworks.com/matlabcentral/fileexchange/32451-shooting-method?focused=5194030&tab=function&s_tid=gn_loc_drop

Matematica

https://www.mathworks.com/matlabcentral/fileexchange/32451-shooting-method?focused=5194030&tab=function&s_tid=gn_loc_drop

Approximation method (see also HW5)

https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Ritz_method

HW5 2018

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Ref for formulation of the control problems:

by Daniel Liberzon

University of Illinois at Urbana-Champaign

http://liberzon.csl.illinois.edu/teaching/cvoc/node45.html

by Lawrence C. Evans

University of California, Berkeley

math.berkeley.edu/~evans/control.course.pdf

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Relaxation of nonconvex variational problems:

http://www.math.utah.edu/~cherk/teach/12calcvar/150existance.pdf

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Math 5500-001 Review Session:

4/27/18 - 1:30PM - 3:30PM Scheduled LCB 121

R

Notice: an extra-credit problem is added.

HW2 - approximates

HW3 - constraints

HW3 - the new file

HW4 - numerical solutions

HW5 - Hamiltonian and Legendre transform

** *******************************
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HW5
- duality
NW6 - see the note
HW
8 (PDE)
Final
HW 2017
**