Let's talk about non-autonomous systems. All the proofs you've talked about neighborhoods which are balls, L2 norm balls. Lyapunov stable basically meant that for any final neighborhood Epsilon, non-zero neighborhood epsilon, I can come up with an initial neighborhood delta. To end up with one degree, if I start within 10 degrees, I can guarantee my final errors will be less than one degree, that's what we did. Asymptotic stability was the same, it goes to zero, but you can come up with a neighborhood that if you start within ten degrees, I am guaranteed I can converge and asymptotically and be good. If your system is all of a sudden time-dependent, you can still make the same arguments, but the one difference who's eagle-eyed, who sees a difference in this and this definition here, compared to our earlier one. I'll give you a tip. One of the characters is a comma, and there's two more characters. [inaudible] The delta of a sum depends on your initial time. That wasn't the case before, it said for if I want to be within one degree of a sudden, it says, ''If you start at noon within 10 degrees, I can guarantee you converge.'' How did things become time-dependent? You guys are orbits people as well, and how to do rendezvous, you have so much fuel. You can guarantee I can reach that asteroid if I launch in September, it's got enough fuel. My doctoral work, I will converge, the control will find, the guidance control and everything. But if you launched in January, that's not going to work. You still argue you have Lyapunov stability of a time-dependent system, but all of a sudden, the stability is now a function of initial time. You can say at this time or within this time window, that's where you are okay. Can we argue that something is stable regardless of when I start? Again, explicit time stuff that has to do with these asteroids. You can put your spacecraft in any state, any attitude, as much fuel, find out the amount of fuel you want to give it, but just find out. The asteroid does what it does, two weeks from now, it'll be there, two more weeks, it's going to be here, it's the attitude, is going to be this, that is a time prescribed behavior. You don't get to control how that asteroid behaves, that's just a naturally occurring thing, so that's what comes in. If its uniform is stable, and that means now that the system was not autonomy, it depends explicitly on time. Instead of having a delta that also depended on did you launch at noon, or in January, you just say you will get there regardless of when you start. You can see that's a much stronger stability argument all of a sudden, and that's what we call uniformly Lyapunov stable. If you can argue uniform stability, you're typically, well, you could argue uniform stability of a time-invariant system. But the people who know what they're talking about will know you're basically trying to just put on airs. Any time-independent system is uniform and stable, if it's stable, it's stable, and if it doesn't depend on time, it does you know, spring-mass damper system will converge if you ping it today or five weeks from now, or 50,000 years from now, it's going to converge, so calling a time-independent system uniform and stable is a little bit overkill, hopefully you agree. But if it's a two-part time-dependent system, calling it just stable versus uniformly stable has two distinct meanings. Stable means it's okay if you go at noon, uniform, it says, it's just okay, go whenever you want to, and that's a stronger argument. Every time we have stronger arguments, there's little more math, and we'll look at that next. But this was basically for Lyapunov stable, and this is uniform asymptotic stability. Again, the delta, that region doesn't depend on initial time. How did we prove that? I'm showing you a way with Lyapunov functions how we can do this. This is what we'll do next.
