 Rational generating functions and linear recurrences
 Expansions into elementary fractions and coefficients of rational generating functions
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RSS feed for this sectionMath 487, Nov. 16
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)
 HamiltonJacobiBellmann Equation
Math 487, homework 4
\(\def\Res{\mathtt{Res}}\)
 Which of the following functions is complex analytic in some region? (here \(z=x+iy\)):
 \(x+iy\mapsto (3x^2yy^3)+i(x^33xy^2)\);
 \(z\mapsto x^2+i y^2\).
 \(x+iy\mapsto i(3x^2yy^3)+(x^33xy^2)\);
 \(x+iy\mapsto (yix)/(x^2+y^2)\);
 Sketch the image of the circles \(z=1/3; z=3\) under the mapping \(z\mapsto z^3\bar{z}\).
 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 computationheavy 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.

 [15] Let \(u_1, u_2, u_3\in\Real^n\) are three unit vectorcolumns 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
 [15] Let \(u_1, u_2, u_3\in\Real^n\) are three unit vectorcolumns such that the angle between any two of them is \(120^\circ\). Find the spectrum of the \(n\times n\) matrix
Math 487, Nov. 9
 Integrals involving multivalued functions.