# Archive | courses

## Math 487, Nov. 16

• Rational generating functions and linear recurrences
• Expansions into elementary fractions and coefficients of rational generating functions

## ECE 515, Nov. 15

• Finite horizon LQR, the Riccati Differential Equation
• Hamiltonian dynamics of LQR
• Riccati DE as dynamics on Lagrangian subspaces

## Math 487, Nov. 14

• Rouche theorem.
• Generating functions, examples.

## ECE 515, Nov. 13

• Formulation of the optimal control problem for continuous time systems
• Value function (action functional)
• Hamilton-Jacobi-Bellmann Equation

## 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

## Math 487, Nov. 12

• Fresnel integral.
• Logarithmic residues. Rouche theorem.

## midterm redo

Posted on compass. Reply here if any questions arise.

(Corrected) solutions are here.…

## 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
$$A=\left( \begin{array}{ccc}1&1&0\\0&1&0\\1&0&1\\\end{array} \right); B=\left( \begin{array}{cc}1&0\\1&0\\0&1\\\end{array} \right); C=(1\quad 0\quad 1).… ## Math 487, midterm 2 $$\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

## Math 487, Nov. 9

• Integrals involving multivalued functions.