2
$\begingroup$

Consider the following time-varying system:

$\dot{x} = f(x,t)$

The solution of this system which starts from the point $x_0$ at time $t_0 \geq 0$ is denoted as $x(t;x_0,t_0)$ with $x(t_0;x_0,t_0)=x_0$. The solution $x(t;x_0,t_0)$ of above system is:

  1. Uniform stable, if for each $\epsilon >0$ there exists a $\delta(\epsilon)>0$ such that: $|\tilde{x}_0-x_0|<\delta \Rightarrow |x(t;\tilde{x}_0,t_0)-x(t;x_0,t_0)|<\epsilon, \forall t \geq t_0$
  2. Uniform attractive, if there exist an $r>0$ and, for each $\epsilon >0$, a $T(\epsilon)>0$ such that: $|\tilde{x}_0-x_0|<r \Rightarrow |x(t;\tilde{x}_0,t_0)-x(t;x_0,t_0)|<\epsilon, \forall t \geq t_0+T$

I find it very difficult to understand the differences between these definitions.

$\endgroup$
2
+50
$\begingroup$

Let us interpret the initial datum $\tilde{x}_0$ as a perturbation of $x_0$. The "uniform stable" condition means that, if the perturbation is small enough, then the solution with perturbed data stays near the original solution at all times.

On the other hand, the "uniform attractive" condition means that there exists an attractive region, namely the ball $B(x_0, r)$. Every time the perturbed datum lies in this region, the corresponding solution is asymptotically close to the original. This means that, if we go far enough in the future, the two solutions are as close as we want.

Note that the "uniform stable" condition does not imply that the solution with perturbed data is asymptotically close to the original. An example that might be illuminating is a simple harmonic oscillator $$ \begin{cases} \dot{v}+\omega^2 x=0 \\ \dot{x}=v. \end{cases} $$ Here $x$ can be interpreted as the position and $v$ as the velocity of a mass point attached to a spring. Clearly, the initial condition $(x_0, v_0)=(0,0)$ gives rise to the stationary solution $(x, v)=(0,0)$, corresponding to the point being motionless. But a small perturbation $(\tilde{x}_0, \tilde{v}_0)$ will give rise to the oscillating solution $$ (\tilde{x}(t), \tilde{v}(t))=\left( \tilde{x}_0\cos(\omega t)+\frac{\tilde{v}_0 \sin(\omega t)}{\omega} , -\omega \tilde{x}_0 \sin(\omega t)+\tilde{v}_0 \cos(\omega t)\right), $$ which is close to $(0,0)$ if $\sqrt{ \tilde{x}_0^2+\tilde{v}_0^2 }$ is small enough, but is not asymptotic to $(0,0)$ since it keeps oscillating.

Therefore the initial datum $(0,0)$ is for this system uniform stable but not uniform attractive.

$\endgroup$
  • $\begingroup$ Thanks a lot for your answer. The idea of a stable and uniform attractive solution is clear now, though I'd like to grasp these concepts by understanding the "mathematical" definitions in my question. Because from these definitions I cannot see the difference. $\endgroup$ – Pietair Feb 24 '14 at 12:26
  • $\begingroup$ The mathematical differences between the two definitions are two. The first one is the following. In the definition of "uniform stable" solution (US), the region involved is the ball $B(x_0, \delta(\epsilon))$. Note that the radius $\delta(\epsilon)$ depends on $\epsilon$. On the other hand, the region involved in the definition of "uniform attractive" (UA) solution is fixed: the ball $B(x_0, r)$. Note that $r>0$ does not depend on $\epsilon$. $\endgroup$ – Giuseppe Negro Feb 24 '14 at 15:04
  • $\begingroup$ The second one is the presence, in the definition (UA), of a later time $T(\epsilon)$. This means that the effects of the property are not seen immediately, but only at a later time and far enough in the future (asymptotically). On the other hand, in the definition (US) the effect is immediate. $\endgroup$ – Giuseppe Negro Feb 24 '14 at 15:06
1
$\begingroup$

Your notion of uniform attractive is also known as asymptotic stability. The notion of uniform stability is often called just the stability (or Lyapunov stability). In my experience, the notations in italics are much more widespread.

The difference between these stabilities lies in the behaviour of the solutions. In the case of stability, your solution changes by at most a constant for a sufficiently small change of initial data.

In the case of asymptotic stability, if the change of initial data is suffciently small, then the distance between the solutions tends to zero (some sort of convergence).

You can try to look at the system

$$\dot x(t)=\begin{pmatrix}0&1\\-1&0\end{pmatrix}x(t)$$ The zero solution is stable (in fact, you can prove that $\|x(t)\| = \|x(0)\|$), but not asymptotically.

However, in the case of $$\dot x(t)=\begin{pmatrix}-2&1\\0&-2\end{pmatrix}x(t)$$ you have asymptotical stability of zero solution. Whatever the initial data, the solution converges to zero as $t\to\infty$.

$\endgroup$
  • $\begingroup$ Thanks a lot for your answer. The idea of a stable and uniform attractive solution is clear now, though I'd like to grasp these concepts by understanding the "mathematical" definitions in my question. Because from these definitions I cannot see the difference. $\endgroup$ – Pietair Feb 24 '14 at 12:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.