Archive | courses

RSS feed for this section

Math 487, Nov. 16

  • Rational generating functions and linear recurrences
  • Expansions into elementary fractions and coefficients of rational generating functions
Read full story Comments { 0 }

ECE 515, Nov. 15

  • Finite horizon LQR, the Riccati Differential Equation
  • Hamiltonian dynamics of LQR
  • Riccati DE as dynamics on Lagrangian subspaces
Read full story Comments { 0 }

Math 487, Nov. 14

  • Rouche theorem.
  • Generating functions, examples.
Read full story Comments { 0 }

ECE 515, Nov. 13

  • Formulation of the optimal control problem for continuous time systems
  • Value function (action functional)
  • Hamilton-Jacobi-Bellmann Equation
Read full story Comments { 0 }

Math 487, homework 4

\(\def\Res{\mathtt{Res}}\)

  1. Which of the following functions is complex analytic in some region? (here \(z=x+iy\)):
    1. \(x+iy\mapsto (3x^2y-y^3)+i(x^3-3xy^2)\);
    2. \(z\mapsto x^2+i y^2\).
    3. \(x+iy\mapsto i(3x^2y-y^3)+(x^3-3xy^2)\);
    4. \(x+iy\mapsto (y-ix)/(x^2+y^2)\);
  2. Sketch the image of the circles \(|z|=1/3; |z|=3\) under the mapping \(z\mapsto z^3-\bar{z}\).
  3. Find the integral of
Read full story Comments { 0 }

Math 487, Nov. 12

  • Fresnel integral.
  • Logarithmic residues. Rouche theorem.
Read full story Comments { 0 }

midterm redo

Posted on compass. Reply here if any questions arise.

(Corrected) solutions are here.…

Read full story Comments { 6 }

ECE 515, homework 4

\(\def\Real{\mathbb{R}} \)

 

This is a somewhat computation-heavy homework; feel free to use your preferred software.

Consider the linear controlled system
$$
\dot{x}=Ax+Bu, y=Cx, x\in\Real^3, u\in\Real^2, y\in \Real,
$$
with
$$
A=\left(
\begin{array}{ccc}1&1&0\\0&1&0\\1&0&1\\\end{array}
\right);
B=\left(
\begin{array}{cc}1&0\\1&0\\0&1\\\end{array}
\right);
C=(1\quad 0\quad 1).…

Read full story Comments { 6 }

Math 487, midterm 2

\(\def\Comp{\mathbb{C}}\def\Real{\mathbb{R}}\)

Midterm 2.

    1. [15] Let \(u_1, u_2, u_3\in\Real^n\) are three unit vector-columns such that the angle between any two of them is \(120^\circ\). Find the spectrum of the \(n\times n\) matrix
      $$
      A=u_1\cdot u_1^*+u_2\cdot u_2^*+u_3\cdot u_3^*
      $$
      (here \(u*\) is
Read full story Comments { 0 }

Math 487, Nov. 9

  • Integrals involving multivalued functions.
Read full story Comments { 0 }