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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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november 6

  • Find Fourier series for \(2\pi\)-periodic function defined on \(-\pi,\pi\) by
    1. \[
      \cos(2x)\sin^2(x);
      \]
    2. \[
      \cos^3(x);
      \]
    3. \[
      f(x)=\left\{\begin{array}{rcl} 1& \mathrm{if}& -\pi\leq x<0;\\
      -1 & \mathrm{if}& 0\leq x \leq \pi.\\
      \end{array}\right.
      \]
  • Find the Fourier series for the function which is
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    notes for lectures 10-14

    here

    Many thanks to James!…

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    solutions to midterm

    here

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    ece515, homework 4

    \(\def\Real{\mathbb{R}}
    \def\Comp{\mathbb{C}}
    \def\Rat{\mathbb{Q}}
    \def\Field{\mathbb{F}}
    \def\Fun{\mathbf{Fun}}
    \def\e{\mathbf{e}}
    \def\f{\mathbf{f}}
    \)
    Due by midnight of november 8.

    1. Consider the system \(\dot{x}=Ax+bu\),
      \[A=\left(
      \begin{array}{ccc}
      1&1&0\\
      1&1&1\\
      0&1&1\\
      \end{array}\right);
      b=\left(
      \begin{array}{c}
      b_1\\
      b_2\\
      b_3\\
      \end{array}\right).
      \]
      • Derive conditions on \(b\) making the system controllable.
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