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week 1

Introduction

Lange, Ch. 1.

Basic notions: feasible points; criterion of optimality; maxima and minima.

Existence and nonexistence of optimal points; uniqueness and nonuniqueness.

Univariate problems. Critical points. Classical inequalities.

Multivariate problems (unconstrained).

Constrained problems; Lagrange multipliers.


Sample problems:

  • On
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manifolds

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\torus{\mathbb{T}}
\)

Manifolds are topological spaces that locally look like Euclidean spaces.

Formalism: charts and atlases. Beware of pathologies (long line; double point…)! Smooth manifolds vs. topological ones.

Inverse function theorem (for a …

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final

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)
Due by noon 12.18.

Numbers in brackets indicate the points for each (sub)problem.

Questions? Misprints suspected? Post as a comment here!

  1. Consider the vector field \(\bv\) given by
    \[
    \dot{x}=-xy^2-x^5;\\
    \dot{y}=-x^2y-y^5.
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december 5

  1. Solve the Dirichlet problem in the disk \(D=\{x^2+y^2\leq 4\}\) (i.e. find a harmonic function \(u\) such that on the boundary \(u\) is equal to
    1. \[
      x^3;
      \]
    2. \[
      2x^2+y^2;
      \]
    3. \[
      y^3-y.
      \]
  2. Let \(u\) be a harmonic function in
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parabola construction

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)

Consider the optimal control problem
\[
\dot{x}=u; x(0)=x_o; \int_0^T u^2/2dt+M(x(T))\to\min.
\]
The task is to find the cost function \(S(x_o)\) (the optimal achievable value).

It can be solved as follows. Assume for …

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ece515, homework 5

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Due by midnight of december 4.

Use MATLAB or octave if needed.

  1. Consider the optimal control problem
    \[{x}”’=u; \int_0^\infty u^2+x^2 dt\to\min.
    \]
    • Write this system as a LQR;
    • Solve the ARE;
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november 21

  1. Find sin transform of the function on \([0,1]\)
    given by
    1. \(1\);
    2. \(\cos(\pi x)\);
    3. \(x\).
  2. Solve the heat equation
    \[
    u_t=\frac{1}{2} u_{xx},
    \]
    on \([0,1]\) with boundary conditions \(u(0,t)=u(1,t)=0\) and initial values
    1. u(x,0)=\(1\);
    2. u(x,0)=\(\cos(\pi x)\);
    3. u(x,0)=\(x\).
  3. Solve the heat equation
    \[
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math285, week of november 17

Read the textbook, chapters 9.5 and 9.7.

Videos to watch.

Homework (due by Monday, 12.1):

  1. Find the cosine transform of \(\sin(x)\) on \([0,\pi]\).
  2. Using separation of variables, solve the heat equation on the interval \([0,\pi]\):
    \[u_{t}=u_{xx}; \quad u_x(0,t)=u_x(\pi,t)=0; u(x,0)=\sin(x)
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november 14

  1. Find the solutions of the wave equation at time \(t=1/6,1/3,1/2\)
    \[
    u_{tt}=u_{xx}, u(0,t)=u(1,t)=0,
    \]
    on \([0,1]\)
    with initial data \(u_t(x,0)=0\) and \(u(x,0)\) is as shown:

    wave_1a wave_1b wave_1c

  2. Same question, but the initial data are
    \(u(x,0)=0\) and \(u_t(x,0)\) is as shown:

    wave_1a wave_1b wave_1c

  3. Sketch the
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math 285, week of november 10

Read the textbook, chapters 9.4 and 9.6.

Videos to watch.

Homework (due by Monday, 11.17):

  1. Find sine transform of the function (\T(x)\) on \([0,1]\) equal to \(x\) for \(0\leq x\leq 1/2\) and to \(1-x\) for \(1/2\leq x \leq1\).
  2. Using
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