 So, one of the reasons for not one of the reasons, the reason for not including this term in trying to compute delta is of course it would have made the mathematics messier. But the most important thing is that if I had included this term in trying to compute delta, my delta would potentially depend on capital T because this term depends on capital T. So, if I try to compute delta using this term, it could delta would almost certainly depend on capital T ok. And that is just not allowed ok. Because remember first came delta ok, there is no T here when we started talking about delta ok. So, we cannot have dependence on T ok. So, that is sort of critical to remember that is why we got rid of this term just by yeah I mean just by saying that you are always going to go become smaller in some sense ok great. So, now we have been able to choose a delta. The important thing to remember is that there are a few elements here. So, delta does depend on T0 and R because of V is dependence on T0 and of course R is on the right hand side and you can almost never get rid of the dependence on R. But as you can imagine for uniform convergence or uniform attractivity, you will need this dependence to go away ok. For uniform results you will need this guy to go away that is pretty much the key aspect here excellent. Now that we have chosen a delta and we figured out that it depends on T0 and R, but independent of whatever independent of epsilon independent of T which it has to be now we go on to find our capital T. So, we just then we use this small idea ok. What was the idea? We we already know that this is upper bounded by gamma R. The term inside this is upper bounded by gamma of R ok just because of this guy ok that is what we will use ok. So, once you have chosen a delta you have V T0 x0 which is on the left hand side here is smaller than phi epsilon 1 plus gamma of R goes out and the integral is just capital T right integral is just capital T. So, what do I have? I just have V T0 x0 is less than phi epsilon 1 plus capital T times gamma R. Of course, we always already stated by choice of delta norm of x is less than R whenever norm x is less than delta ok. I hope that is clear Q yeah from here ok. All right great. So, once I have this kind of an expression it is pretty straightforward. I need if you just compute T from here you just get this expression T has to be greater than this guy ok and it is it is very natural for you to get a get a expression with T greater than right because any T larger than that is allowed your conditions are not violated for any time greater than this capital T. So, anything beyond this time is ok and you can see this time depends on epsilon right initial time state whatever so on and so forth yeah depends on a lot of things ok. Now, if you want to see dependence on delta directly you can of course use the fact that this is greater than phi norm x0 squared sorry phi of norm x0 not norm x0 square. So, this is greater than phi of norm x0 right just by positive definiteness. So, that will also bring in some kind of delta dependence as you can see yeah because phi of norm x0 so norm x0 is less than delta you will not get an exact result in equality like that but you can see that there is a dependence on delta also just by virtue of this guy just by virtue of this guy because the way we have chosen delta is exactly this yeah. So, there will be some kind of connection with delta here alright ok great anyway so there is epsilon and R alright. So, like I said in order to prove I mean the second exercise of course to prove uniform asymptotic stability yeah. So, for that the first exercise is anyway useful because you proved uniform stability to prove uniform attract sorry asymptotic stability you have to prove uniform attractivity. So, like I said this dependence on t0 has to go ok the delta dependence on t0 has to go. So, therefore, you have to think how you get rid of this dependence ok what is the additional assumption that you use here alright. So, that is the idea ok good. So, these are the only two theorems that we prove ok, but you can see that the other ones will also be very very closely connected to this yeah if I wanted to as you can imagine if I wanted to go to the global version I do not need you know I do not have any ball of radius R ok. So, for all the global versions of the result the ball of radius R business goes away ok. So, for all the uniformity the t0 dependence goes away exponential stability is the only thing that I have not actually even approached ok. That will require a little bit more not this kind of analysis that requires more because you have to like I said you have to use the same order of magnitude ideas and so on and so forth yeah. So, that I am not going into yeah you can check out the proof in textbooks like you know Khalil and Vidya Sagar and so on ok you will have some proof there ok if you are interested alright great. If there is nothing no questions here then I will move on to our next set of lectures which are on Lasal invariance yeah. So, anyway this is again notescribed by some students in the past. So, what we want to do is we want to discuss the Lasal's invariance principle today ok. Now what is the motivation I will simply stated. So, motivation for several asymptotically stable systems v is c1 v positive definite, but v dot comes out to be negative semi definite ok. This happens a lot especially for all the dynamic systems specialists yeah because typically we are I count myself more as a control non-linear controls right. So, for me I put some effort into choosing v which is not necessarily the energy of the system ok. So, a large part of my work is actually choosing good functions to work with yeah. So, we typically do not choose the energy of the system as we may start there, but then we modify it ok. But typical dynamical systems guys they are they want to analyze the system just with the energy ok and when you do that in a lot of cases this will happen that you will have a positive definite and continuous v which is obvious because it is the energy of the system or the Hamiltonian or whatever I mean you can use the Hamiltonian use the energy all of these are energy like quantities. But v dot will invariably come out to be 0 or negative semi definite you will never get a negative definite v dot yeah this happens a lot. And then but you know again like the pendulum case yeah you know I mean I modified the system of course I played with the system. But you know very well that the pendulum the system itself is asymptotically stable just because you use the energy of the system and the v dot turned out to be 0 does not mean that the system is not asymptotically stable it is you can see yeah. So, obviously we need some more mechanism to prove asymptotic stability in such cases this was the first motivation the other motivation was anyway this is the first motivation. The second one is there are systems limit cycle behavior yeah another obvious system is the Van der Waal oscillator we have already seen that in an assignment I hope yeah. So, you the trajectory tend to converge to you know this close bounded set yeah I mean another example is just this oscillator here it is a bit more dull than the Van der Waal oscillator. Because in the Van der Waal oscillator I do not remember it just whatever it has something like this I am making it very badly like this I think yeah. So, trajectory is actually converge to this do this this is a bit more dull because wherever it starts it just continues in that circle it does not converge to it starts converge yeah there is nothing to converge to it just keeps circling like this this is the standard linear oscillator X1 dot to the X2 X2 dot is minus X1. But then the Van der Waal oscillator which is a little bit more nonlinear oscillator tends to do have behavior like this and such systems are very very important I already told you nonlinear oscillators are very critical and are used in a lot of a bio rhythm applications. Okay so obviously we want to study such systems so in for such systems also we want to have a little bit more general definition of convergence okay we want to have a more general definition of convergence not just something as basic as going to a point we may want something more in fact there are more modern problems in controls like you have the say you have the platooning problem do you know what is platooning so it is a very modern you know I would say transportation theory idea in which basic idea there are a lot of these transportation trucks yeah that are carrying a lot of good logistics and it so turns out that if they continue to move in a straight line with uniform distance between each other that is optimal in the sense of fuel efficiency and so on and so forth okay so this is called a platooning problem however you start you maintain this straight line formation with uniform distance between these vehicles very sounds very straightforward but of course here the point is there may be traffic on the street and whatever I mean you may have to change lanes so once you do a lane change maneuver you have to redo the platooning and so on and so so these platooning things are automated in some so this is actually a formation a specific example of a formation control problem okay another formation could be you have a bunch of defense vehicles going to you know carrying whatever supplies arms ammunition whatever yeah and you want to guard them yeah and there is a lot of applications now where you have aerial drones which are circling around them okay and so if they are circling around so you have a bunch of you know bunch of drones which are circling this you know arm armor trucks yeah and how do you do that this is also then you are sort of trying to converge to a you know circular pattern in some sense right and this is also convergence to it because obviously you cannot start in the circular pattern and even if you started in the circular pattern and you exactly positioned them you let them off in this very nice circular pattern around them but the vehicles armored vehicles are moving right and they are doing whatever they are doing depending on road conditions especially in India right and and in border areas right there may be no roads yeah so they are doing whatever they are doing so but these guys have to keep you know converging again to another formation right so every time the formation requirement changes the center changes and so on and so forth so obviously they have to reconverse so this is also a sort of limit cycle behavior you are looking at if you want to study it in that sense many people do not most people do not study it in that sense they think of it as a coordination problem but most coordination problems can lead you formation problems can lead you to limit cycle type behaviors okay this is very important set of results actually Lassarian variants all right excellent after this mighty introduction we will look at the systems that we are interested these results work for nonlinear autonomous system going the ones I am stating yeah there are extensions it is a topic of great research of trying to do this for non-autonomous systems also yeah and there are results there and you can look at it but it is still an active topic of research yeah it is not like you know it is not textbook material yet yeah and most of what we do here is textbook material okay so we start with as usual a nice continuous lip sheets locally lip sheets continuous vector field with some initial conditions we succinctly denote the solutions as x of t right we have been doing that okay and we make some definitions okay so remember we start with autonomous systems yeah that is the main thing to remember all right first we define what is an invariant set what is an invariant set a set omega is said to be invariant if you start in this set you remain in this set for all time okay that is what is an invariant set if you start in the set you remain in the set okay great so I mean in this case it is pretty obvious I hope it is obvious that for system like this what is the invariant set it is a circle yeah or whatever radius you started with if you started with a circle of radius square root of c remain in that circle of radius square root of c yeah obviously we are not discussing cases where there are disturbances and errors and sensing and all that stuff yeah this is the precise case yeah this is the ideal case if you may okay great so omega is this okay it is a set of all points in the circle this is how you write yeah omega is exactly this yeah notice it is just the circle not inside the circle not the disc it is just the circle okay so this is what is a typical limit cycle yeah if you are inside a disc that is not a limit cycle that is the stability okay that is what we do if you are within if you are given an epsilon then you start within delta you remain within epsilon that is not nothing to do with limit cycle or anything it is you are in a disc yeah here you have a circle that is a limit cycle okay remember that okay limit point what is a limit point a point p is said to be a limit point of this function x of t again whenever I write x of t it is the solution of this equation okay so a point is said to be a limit point of x of t there exists a time sequence a sequence of time such that as t n goes to infinity such that t n goes to infinity as n goes to infinity and x of t n goes to this point whatever point we are denoting as the limit point okay so basically there is a time sequence such that if you keep keep writing terms x t1 x t2 x t3 x t4 x tk but the important thing is this tk has to go to infinity as k goes to infinity remember whenever I say sequence it is an infinite it is infinite size not finite okay so as k goes to infinity tk has to go to infinity the basic idea is it is limit point I mean infinite we want to look at behavior asymptotically therefore as k goes to infinity tk has to go to infinity and if in that case x tk converges to some point p then it is a limit point okay a very simple example I have constructed just to illustrate how things are different if you look at this series or sequence sorry not series yeah it is half 1 half 1 half 1 and so on and so forth so if I so if I take the time sequence and I have denoted this as 0 1 2 t0 t1 t2 and so on if I take the time sequence t0 t2 t4 okay where what is the limit point correct if I take t1 t3 t5 t7 1 okay so remember any one function or a sequence can have multiple limit points not just one this and this is what constitutes a limit set okay this is what constitutes a limit set okay you can have multiple limit points and the set of all those limit points is the limit set okay just like half the set containing half and one is the limit set in this case and if this limit set is a cycle okay and then the question is how do you define a cycle you can define it mathematically I do not want to get it to the mathematical definition but think like a circle it is somehow closed in some sense closed compact those are the requirements so if the limit set is a cycle then it is a limit cycle then you have a limit cycle behavior again Van der Poel also little okay that weird looking set is a limit set but it is a limit cycle because it has it is a cycle here it is it has a cyclical thing and there is a periodicity here basically it has periodicity okay great I hope those points are clear once those points are clear to us we can state the most general form of the Lasals invariance principle so are these three definitions relatively clear to you yeah invariant set limit point limit set and limit cycle is just an extension of limit set okay okay great let omega subset of D subset of Rn be compact that is closed and bounded in the case of reals compact and closed and bounded are equivalent and invariant what is D in this case D is the domain just like your ball of radius R right D is that ball of radius R type of this is the domain in which you are working so you want the existence of an omega which is compact and invariant inside this domain let V mapping D to R be a C1 function okay such that Vx is greater than equal to 0 yeah look at the interesting things that are already happening we are denoting it as V against scalar valued C1 function looks like a Lyapunov candidate at least but it is not necessarily because you only require semi-definiteness positive definiteness not required so it is not a Lyapunov candidate okay does not have to be and further you want that V dot is negative semi definite in this compact invariant set omega okay then you define E as the set of points in omega such that V dot is exactly 0 yeah it is evident that V dot is 0 somewhere otherwise no need of saying V is only semi definite right so there are points other than the origin inside omega such that V dot is exactly 0 okay and so E essentially captures those points where V dot is exactly 0 yeah of course origin is also there but there are potentially points beyond the origin because V is semi definite V dot is sorry yeah V dot is semi definite and let M be the largest invariant set inside this okay so lot of technical terms coming up now okay we will try to clarify this okay so we have used many sets yeah so we define E as the set of points where V dot is 0 we define M as the largest invariant set inside E then we can claim that if your initial conditions start in this omega then as T goes to infinity your solutions will lie in this largest invariant set M okay and M is obviously a positive limit set okay not necessarily a cycle yeah but it is a positive limit set so the immediate obvious thing is the Lassal invariance principle does not require Lyapunov candidates let us look at the obvious things then we look at the more non-obvious things okay does not require Lyapunov candidates because it is starting with V only greater than equal to 0 second thing it gives a actually seems to give a more general result than Lyapunov theorems right because it talks about convergence to positive limit set yeah it basically at the end of a Lassal invariance analysis you could potentially say that my solutions go to go to say if the limit set contains 10 points or 10 equilibria yeah then you can actually say that the solutions can go to any one of these 10 equilibria okay so that is stronger than not stronger but more general than saying it goes to this one point okay that is a more obviously if it is a limit cycle then Lassal invariance actually gives you a way of saying that you converge to this limit cycle there are hardly any other results which will talk about how you go to limits I mean there is also the Poincaré's theorem but there are very few results which tell you that you will actually converge to a limit cycle or a limit set because or converge to a set for that matter because until now we are only talking about converging to a point yeah we even we have not even said anything about convergence to a set okay in fact if you guys notice this notation is in itself not very obvious if m is a finite set of points suppose m is a finite set of points then this is okay you are just saying that as n time goes to infinity your states go to one of these points so you can analyze any all the points but if m is suppose m is a continuous set like a circle then how do you even I mean this is not a very simple notion to understand okay when I when I use this terminology this terminology is difficult all your it is in a sense what you are saying actually I would put it this way if any of you knows this notation okay this would be the more precise notation that we are used to for what do you know sorry do you understand this norm of xt subscript m what is an m norm mean like L0, L1, L2, m norm no no it is not that although I have used similar subscript notation I agree but it is not that this notation is or maybe I know there is a overloaded operator here but this notation is also used for and if it is not a number and a symbol and or a letter it is definitely used for norm with respect to set or distance from a set okay this is a distance from a set this is defined with a infimum okay this will be defined as inf over all x0 in m norm of x okay it is basically telling you shortest distance from the set yeah so this is actually a set norm because we are until now we are used to measuring so all of this is nice and easy to do in Rn so yeah we can do it in more complicated topological spaces well I will say it is their problem and whoever is working with it they have to figure out all this so this is actually a more special norm it tells you distance not from a point like origin or something it tells you distance from a set so that is rather interesting