general form of First-Order Differential Equations

This will depend on both unknown function \(x\), and the independent variable \(t\). These could and could not be separable.

\begin{equation} \dv{x}{t} = F(t,x),\ x(t_{0}) = x_{0} \end{equation}

Let’s imagine \(F\) is “bounded” and “continuous” on \(I \times \omega\), where \(I\) is an open interval about \(t_{0}\) and \(\omega\) is an open subset of \(\mathbb{R}^{n}\), containing \(x_{0}\). \(F\) is bounded; the results are bounded??

functions embedded in vector spaces

We understand that such First-Order Differential Equations will describe a subset of an infinite dimensional vector space.

Systems of Linear Equations

\begin{equation} T v = v' \end{equation}

every system of linear equations is decomposed into this. Classically, there’s either a unique solution, no solution, infinite solutions—

problems with zero

“zero” is really hard to define. For instance:

\begin{equation} 6.23423 \times 10^{192} - 1 \times 10^{7} = 6.23423 \times 10^{192} \end{equation}

so in this case \(10^{7}\) literally behaves like zero. (small numbers have the opposite problem)

so, we use elementary row operations to make sure that enormous numbers are essentially standardized—if a row has huge numbers, we may want to scale it down to smaller numbers to make them nice.

see linear temporal logic

An exact solution for a dynamic system with quadratic costs and linear differential equation describing the dynamics.

For some non-linear function, we can use its first Jacobian to create a linear system. Then, we can use that system to write the first order Taylor:

\begin{equation} y’ = \nabla F(crit)y \end{equation}

where \(crit\) are critical points.

Phase Portrait stability

• if all \(Re[\lambda] < 0\) of \(\qty(\nabla F)(p)\) then \(p\) is considered stable—that is, points initially near \(p\) will exponentially approach \(p\)

• if at least one \(Re[\lambda] > 0\) of \(\qty(\nabla F)(p)\) then \(p\) is considered unstable—that is, points initially near \(p\) will go somewhere else