MA20221 Week 2
Definitions
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 wellposed, then $x(t)$ must either be a steady state solution, $x^*$, or strictly monotonic.