\documentclass{article}
\usepackage{graphicx}
\usepackage{amsmath}
\begin{document}
\title{557 Homework 3}
\author{Niranjana Prasannakumar | npkumar}
\maketitle
\section*{Question 2}
Scalar ODE:
\begin{equation*}
\frac{dx}{dt} = \frac{1}{(x+1)^2}-0.5
\end{equation*}
Equilibrium points are those at which $\frac{dx}{dt} = 0$\newline
\[\implies \frac{1}{(x+1)^2}-0.5 = 0 \implies x_e = \sqrt{2} - 1 \in (-1,\infty)\]
For the problem of congestion control wth one flow(scalar problem), each user uses gradient descend to adjust its flow according to $\frac{dx}{dt} = k(x(t))(U'(x(t))-q) $, \newline
where $U(x(t))$ is a strictly concave fucntion and q is the sum of link prices.The equilibrium point for this equation is also the maxizing solution of the primal problem
\[W(x) = max_{x\geq0} U(x) - \sum_{l \in L}B_l(x,C_l)\]where $\sum_{l\in L}\frac{dB_l}{dx} = q$.By inspection of the given ODE, we see that
\[W(x) = U(x) - \sum_{l \in L}B_l(x,C_l) = -\frac{1}{x+1} - 0.5x + m\] where m is a constant of integration. To prove the existsnece if GAS equilibrium point, we formulate the lagrangian as
\[V(x) = W(x^*) - W(x) = W(x^*) + \frac{1}{x+1} + 0.5x - m\] where $x^* = x_e = \sqrt{2}-1$
To check if $x_e = x^*$ is GAS, we check the conditions of Lyapunov Globally Asymptotically Stable Theorem. Domain is assumed to be $(-1,\infty)$
\begin{enumerate}
\item{Radially unbounded:} If $||x|| \rightarrow \infty \implies V(x) \rightarrow \infty$ since the linear term $0.5x \rightarrow \infty$
\item V(x) is continuously differentiable in domain
\item $V(x_e) = V(x^*) = 0$\newline
$\forall x \neq x_e, V(x) > 0$ since $x_e$ is the unique maximizer of W(x) as W is strictly concave in its domain
\item $\dot{V}(x) = \nabla V(x)^T \frac{dx}{dt} = (U'(x) - q)(-(U'(x) - q)) = -(U'(x)-q)^2 \leq 0 \phantom \forall x \in \Omega$
\end{enumerate}
Therefor by theorem, $x_e$ is Global asymptotically stable.
\section*{Question 3}
\section*{Question 4}
\end{document}
About Online Latex Compiler
Try our Online Latex Compiler (Version TeX Live 2016) to Edit, Run, and Share your Latex Code directly from your browser. This online development environment provides you the latest version TeX Live 2016.
How to use Online Latex Compiler?
Write and Execute Code
- Write your program (or, paste it) directly under the "Source Code" tab.
- If you want to save your program, go to the "Project" menu and save it.
- You can directly execute your program without saving it by clicking on on "Execute" button.
User Input
The latest version of Coding Ground allows to provide program input at run time from the termnial window exactly the same way as you run your program at your own computer. So simply run a program and provide your program input (if any) from the terminal window available in the right side.
Online Latex Compiler: Keyboard Shortcuts
The following are the keyword shortcut of this Online Latex Compiler:
Shortcut | Description |
⌘ + Enter | Run the program |
⌘ + S | Save Project (Login Required) |
⇧ + ⌘ + S | Save As Project |
⌘ + P | New Project |
⌘ + G | Share Project |
⌘ + Z | Undo Editing |
⌘ + Y | Redo Editing |
⌘ + A | Select All Text |
⌘ + X | Cut Selected Text |
⌘ + C | Copy Selected Text |
⌘ + V | Paste Copied Text |
⌘ + F | Search Text |
⌘ + ⌥ + F | Replace Text |
Shortcut | Description |
Ctrl + Enter | Run the program |
Ctrl + S | Save Project |
Shift + Ctrl + S | Save As Project |
Ctrl + G | Share Project |
Ctrl + Z | Undo Editing |
Ctrl + Y | Redo Editing |
Ctrl + A | Select All Text |
Ctrl + X | Cut Selected Text |
Ctrl + C | Copy Selected Text |
Ctrl + V | Paste Copied Text |
Ctrl + F | Search Text |
Ctrl + H | Replace Text |
Online Latex Compiler: Save and Share Latex Code (Project)
Save Latex Project Online
You can save your Latex Project with us so that you can access this project later on. To save a project you will need to create a login Id with us. So before you save a project, please create a login Id using a link given at the top right corner of this page.
Share Latex Project Online
You can use this feature to share your Latex Code with your teachers, classmates and colleagues. Just click Share Button and it will create a short link, which can be shared through Email, WhatsApp or even through Social Media. A shared link will be deleted if it has been passive for almost 3 months.
More Features of Online Latex Compiler
- Theme – You can change the current editor's theme from the "Editor Theme" option under "Settings" menu.
- Font Size – You can change the font size of the editor /compiler from from the "Font Size" option under "Settings" menu.
- Tab Size – You can change the tab size from the "Tab Size" option under "Settings" Menu.
- Show/Hide Line Numbers – You can show/hide the line number with the code from the "Show Line Numbers" or "Hide Line Numbers" option under "Settings" Menu.
- And, many more.
Benefits of Using Online Latex Compiler
There are several benefits of using the Online Latex Compiler to run your Latex code:
- Platform independence: You can run your code from any device without taking care of operating systems.
- Convenience: You don't need to install anything for using this.
- No setup required: There is no need for additional setup to run your code.
- Updated version: Our online compiler/editors/terminals are the latest up-to-date.