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october 2

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

Quadratic forms as Lyapunov functions

If the operator defines an asymptotically stable system, it is Hurwitz. For a Hurwitz operator, a quadratic form exists which is a (strict) Lyapunov function. A strict Lyapunov function implies …

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solutions

for homework 1 and homework 2.

Thanks to Cheng!…

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math285: the week of september 29

Please read the text book, Chapter 3.1-2.

The lectures are here and here.

The homework (due by Monday night):

  1. Solve
    \[
    y”+y’-2y=0.
    \]
    Answer:
    \(c_1e^t+c_2e^{-2t}\).
  2. Find the bounded solutions of
    \[
    y”=y
    \]
    such that \(y(0)=1\).

    Answer:

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september 30

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

Lyapunov’s direct method, cont’d

For complex spaces quadratic forms are not really suitable (if one wants just a real number as a result, the signatures are all \((n,n)\)

Sylvester criterion for positive definiteness.
(Bonus: Rayleigh …

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midterm 1, september 29

  1. Solve
    \[
    y’=\cos^2(2y-x).
    \]
  2. Sketch the slope field
    \[
    y’=y-x^2.
    \]
    Find where the solutions to this differential equation have horizontal inflection points.
  3. \[
    x^3yy’=x^4+x^2y^2+y^4;
    \]
  4. Reduce to a linear DE and solve:
    \[
    y’=y^4\cos(x)+y\tan(x).
    \]
  5. Solve
    \[
    yy”-2(y’)^2=0.
    \]
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september 26

  • A brine tank of volume \(V\) get an influx of \(p\%\) brine at the rate \(r\), which mixes instantaneously and flows away at the same rate \(r\). If the tank was filled with fresh water initially, when it will have
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old (mock) midterm

  1. Consider the force function
    \[
    f(y)=\sin(y)+1/2.
    \]
    1. Draw the graph of the potential function
      \[
      U(y)=-\int_0^y f(\eta)d\eta.
      \]
    2. Sketch the level sets of \( v^2/2+U(y)=C\) for \(C=-1,0,1\) on \((y,v)\) plane.
    3. Sketch the solutions of
      \[
      y”=\sin(y)+1/2,
      \]
      passing through \((0,0)\).
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september 25

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

Lyapunov’s direct method, cont’d

Convenient Lyapunov functions: homogeneous ones.

Strict homogeneous Lyapunov function for a linear system remains one after a perturbation of the system.

Examples.

Good homogeneous functions: quadratic forms.

Quadratic forms

Coordinate forms. …

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more on humanitarian disaster

Bug in the system

Handling of all – not just Salaita’s – cases by U of I is deficient, with the final approval coming only after the move to campus is done. We all are culpable, registering this issue when …

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september 23

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

Global asymptotic stability for LTI.

Equivalent to local stability.

Hurwitz operators. Multiple eigenvalues and Jordan normal forms.

Lyapunov’s direct method

Positive definite functions.

Lyapunov functions, definition.

Lyapunov’s theorem.

Convenient Lyapunov functions: homogeneous ones.

Strict homogeneous …

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