I came across the following alternate definition for Lyapunov stability of continuous linear time varying (CLTV) systems in a textbook:

A CLTV system is sait to be stable in the sense of Lyapunov (isL) if for every initial condition $x(t_0)=x_0\in\mathbb{R}^n$, the homogeneous state response $$ x(t)=\Phi(t,t_0)x_0, \forall t \geq 0$$ is uniformly bounded.

But what about the CLTV (time invariant even) system $$\dot x (t) = 0,$$ which is stable isL according to the original defn. If $M$ is it's upper bound then choose $x_0=2M$ then $x(t)=2M, \forall t>0$, which is a contradiction, so these can't be equivalent definitions. What am I missing?

  • 3
    $\begingroup$ Perhaps, the textbook meant that the response is uniformly bounded over time for any fixed $x_0$, not uniformly bounded over all possible initial conditions (that would be impossible since $x(t_0)=x_0$). The bound then is allowed to depend on $x_0$, and your example is stable. $\endgroup$ – Conifold Jun 9 '14 at 22:03
  • $\begingroup$ Thats a good point. If that's the case whats the difference between a single function that is uniformly bounded vs bounded? $\endgroup$ – user1816847 Jun 9 '14 at 22:05
  • $\begingroup$ I don't see much of a difference in this context, but sometimes people talk about locally bounded (in some neighborhood of $t_0$), so maybe "uniformly" was added for clarity. Alternatively, it may refer to uniform over $x$ but only in some neighborhood of $x_0$. Depends on the context in the textbook. $\endgroup$ – Conifold Jun 9 '14 at 22:12
  • $\begingroup$ sigh, this is the first mention of uniformly bounded in the text book and it comes right at the start of a section so there's not much context. I guess the only (correct) way to interpret it is as applying to the response. $\endgroup$ – user1816847 Jun 9 '14 at 22:52

Usually these definitions include M(x(0)), and thus your example is fulfilled, since you cannot choose the bound before the initial condition. The same applies to the original definition of Lyapunov stable for LTI systems. Think about local stability, where you have to restrict your set of initial conditions. Then extend the notion to a global definition.


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