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

Time dependent solution

If $\frac{dx}{dt}=f(x)$ and $f(x_0)\neq0$, then $$ t = \int_{x_0}^{x(t)}\frac{ds}{f(s)} $$

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