Week 2: 16/02/22

Table of Contents

Outline of lecture content

What makes a well posed problem?

• solution exists (for at least as long as the prediction is required)
• the solution is unique
• the solution depends continuously on the initial condition and model parameters

What is a positively invariant set?

A set $Y\subset X$ is positively invariant if $x(0)\in Y$ implies that $x(t)\in Y$ for all $t\in T$ with $t>0$. (It is negatively invariant if the same is true for all $t\in T$ with $t<0$.)

Stability

• Steady state: $x^{\ast }$, is where $f(x^{\ast })=0$
• Stable: for any $ϵ>0$ there exists $\delta >0$ such that $|x(t)-x^{\ast }|<ϵ$ for all positive $t\in T$ whenever $|x_0-x^{\ast }|<\delta$, and unstable otherwise.
• Asymptotically stable: if it is stable and there exists $\delta >0$ such that $|x-x^{\ast }|\to 0$ as $t\to \mathrm{\infty }$ whenever $|x_0-x^{\ast }|<\delta$.
• stable: start close, stay close.
• asym. stable: start close, converge to steady state.

Time dependent solution

If $\frac{dx}{dt}=f(x)$ and $f(x_0)\ne 0$, then $$t={\int }_{x_0}^{x(t)}\frac{ds}{f(s)}$$

• If $x(t)$ is not a steady state, $x^{\ast }$,then it either tends to a steady state or it tends to $\pm \mathrm{\infty }$
• If the model system is well-posed, then $x(t)$ must either be a steady state solution, $x^{\ast }$, or strictly monotonic.