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RSS feed for this sectionECE 515, week of Dec. 3
\(\def\Real{\mathbb{R}}\)
 Pontryagin maximum principle
 Examples
 Lagrangians to Hamiltonians: FenchelLegendre transform.
Legendre (sometimes called LegendreFenchel) transform \(H\) of a convex function \(L:\Real\to\Real\)) is defined as
\[
H(p):=\sup_x pxL(x).
\]
Both functions, \(L\) and \(H\) can take value \(+\infty\).If \(L\) is strictly convex, the gradient mapping
\[
G:x\mapsto \frac{dL}{dx}
\]
is onetoone, 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
\(\def\Res{\mathtt{Res}}\)
 Find function with Laplace transform equal to \[\frac{s^3 – 1}{(1 + s)^4}.\]
 Find Fourier coefficients of the function \(t^3\pi^2 t\) on \([\pi,\pi]\).
Use Parseval’s identity to find \[\sum_{n=1}^\infty 1/n^6.\]  Find general formula for the coefficients of the generating function \(W(z)\) given by \(W=1+zW^4\).
 Find asymptotic frequency of the numbers \(n\) such that the first digits of \(2^n\) and \(3^n\) coincide.
ECE 515, Homework 5
\(\def\Res{\mathtt{Res}}\def\Real{\mathbb{R}}\)

 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
\[
V(x,0)=\min_X C(x,X,T), \mathrm{where} C(x,X,T)=\frac{(xX)^2}{2T}+\cos(X)
\]
(here \(X=x(T)\) is the potential position of the end of the trajectory, and \((xX)^2/2T\) is the minimal cost of reaching it from \(x\)).
 [20]
 Consider optimal control problem for \(x\in\Real, u\in\Real\):
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.
ECE 515, Nov. 27
 Infinite horizon LQR
 Algebraic Riccati Equation
 Spectral properties of optimal feedback and Hamiltonian matrix of LRQ
Math 487, Nov. 26
 Using residues to estimate coefficients of generating functions
 Algebraic generating functions: Catalan numbers
Exercises:
 Using residues, find coefficients of generating function \(H(z)=1+z+3z^2+\ldots\) satisfying
$$
H(z)=1+zH(z)^3.
$$  Consider linear recurrence given by
\[
f_n+f_{n1}+\ldots+f_{nk+2}+f_{nk+1}=0.
\]
Solve it (i.e. find the general formula for its terms \(f_n\)).