# Archive | courses

## ECE 490, week of Jan 21

Following Chapter 2 of Guler.

Class notes (pretty incomplete, to be used just as a study guide), here and here.

Exercises:

• Does there exist a differentiable function on unit interval with infinitely many global maxima and and global minima?
• Does there exist a function (on real plane) with infinitely many global maxima but no local minima?
• If a function on real line has a local minimum at $$x$$; does it imply that for some $$\epsilon>0$$ the function increases on $$[x,x+\epsilon)$$, decreases on $$(x-\epsilon,x]$$?

## ECE 490, week of Jan. 14

$$\def\Real{\mathbb{R}}$$

#### Taylor formula

Spaces of differentiable functions on an interval $$I=[a,b]\subset\Real$$:
$C(I,\Real), C^1(I,\Real),\ldots, C^n(I,\Real),\ldots$

For a function in $$C^1(I,\Real)$$,
$f(y)=f(x)+\int_x^yf'(s)ds.$
Iterating (for functions in $$C^n(I,\Real)$$, we obtain
$f(y)=f(x)+f'(x)(y-x)+\frac{f”(x)}{2!}(y-x)^2+\ldots+\frac{f^{(n-1)}(x)}{(n-1)!}(y-x)^{n-1}+ \int_{x<s_1<s_2<\ldots<s_n<y}f^{(n)}(s_1)\frac{(y-s_1)^{n-1}}{(n-1)!}ds.$

Implications:

• Mean value theorem
• If $$f”\geq 0$$ on $$I$$ and $$f'(x)=0$$, then $$x$$ is a global minimum of $$f$$.

#### Several variables

Domain: $$U\subset\Real^n$$, an open set.

Reminder: open, closed, bounded, compact sets.

Exercise: is the set $$\{|x|\leq 1, |y| \lt 1\}$$ open?…

Here.…

## ECE 515, week of Dec. 3

$$\def\Real{\mathbb{R}}$$

• Pontryagin maximum principle
• Examples
• Lagrangians to Hamiltonians: Fenchel-Legendre transform.

Legendre (sometimes called Legendre-Fenchel) transform $$H$$ of a convex function $$L:\Real\to\Real$$) is defined as
$H(p):=\sup_x px-L(x).$
Both functions, $$L$$ and $$H$$ can take value $$+\infty$$.

If $$L$$ is strictly convex, the gradient mapping
$G:x\mapsto \frac{dL}{dx}$
is one-to-one, and it is clear that the argmax in the definition of $$H$$ solves $$p=G(x)$$. Denote the functional inverse to $$G$$ as $$F: p=G(x) \Leftrightarrow x=F(p)$$. In this case,
$H(p)=pF(p)-L(F(p)). ## Math 487, week of Dec 3 • Fourier series: smoothness and decay of coefficients. • Fourier transform (following chapter 18 of Zorich, Mathematical Analysis, 2). • Motivation • Examples • Properties • Kotelnikov formula • Asymptotics of integrals (chapter 19 of ibid.) • Laplace method • Localization principle • Standard integrals • Applications: Stirling formula • Tauberian theorems: regularly varying functions ## Math 487, Homework 5 $$\def\Res{\mathtt{Res}}$$ 1. Find function with Laplace transform equal to \[\frac{s^3 – 1}{(1 + s)^4}.$
2. Find Fourier coefficients of the function $$t^3-\pi^2 t$$ on $$[-\pi,\pi]$$.
Use Parseval’s identity to find $\sum_{n=1}^\infty 1/n^6.$
3. Find general formula for the coefficients of the generating function $$W(z)$$ given by $$W=1+zW^4$$.
4. Find asymptotic frequency of the numbers $$n$$ such that the first digits of $$2^n$$ and $$3^n$$ coincide.

## ECE 515, Homework 5

$$\def\Res{\mathtt{Res}}\def\Real{\mathbb{R}}$$

1. Consider optimal control problem for $$x\in\Real, u\in\Real$$:
$\dot{x}=u,\quad J_0^T=\int_0^T\frac{u^2}{2}dt +\cos(x(T))\to\min.$
• [20]
Find maximal $$T$$ for which the cost function (referred to as $$S$$ in the course notes) at time $$0$$,
$V(x, 0)=inf_{\{u(t)\}_{0\leq t\leq T}, x(0)=x}\ J_0^T$
is smooth as a function of the initial point $$x$$ (i.e. has continuous derivative).

Solution. As we discussed, the cost function for the problem above is given by
$V(x,0)=\min_X C(x,X,T), \mathrm{where} C(x,X,T)=\frac{(x-X)^2}{2T}+\cos(X)$
(here $$X=x(T)$$ is the potential position of the end of the trajectory, and $$(x-X)^2/2T$$ is the minimal cost of reaching it from $$x$$).

## Math 487, Nov. 30

• Inverting Fourier transform.
• “Uncertainty principle”.
• Introduction to duality – asymptotics vs. local behavior

## ECE 515, Nov. 29

• Kalman identity and some corollaries
• Introduction to the minimum principle

## Math 487, Nov. 28

• Fourier series
• Elements of Fourier transform
• Parseval theorem.