 We will see the proof of the Lasallian variance principle okay and you will see I am basically whenever we do these proofs one of the things that we learnt much later and this is how we catch people doing bad proofs is that all the elements of the all the assumptions that you made in your proofs should be used. All the assumptions that you made while stating the theorem must always be used in the proof. If they are not then you know there is some weakness in it okay or you have just made too many assumptions alright. So this is one of the things always keep in mind that we do this. I will not be giving complete proof of everything and I am referring to Vidya Sagar for couple of steps it is your call and not your call actually it is an exercise. So you will have to go back and check those you have to read those proofs that is one of the exercises. So we will start and we will see how the assumptions get used okay. So I hope you remember what was the Lasallian variance principle we have done this we have read this a few times we have seen all the sets that are being constructed and all that. So that is it you have a domain you have an omega compact invariance you have an E set which is closed and invariant sorry which is closed and bounded which is compact not necessarily invariant and then you are finally M which is invariant and compact again these are the sets okay great. So started with omega compact which implies it is closed and bounded and also that it is invariant which means that if I start in omega then my entire trajectory by the way this is the notation complete notation for the trajectory Vidya Sagar uses this notation so I have also used this now not this is I have been using the shorthand this is the notation why this notation I am bringing it back is because initial condition is very important okay in reality whenever you write this sets it is a function of the initial condition okay limit sets and if all the limit sets limit points these are defined as a function of the initial condition okay remember this so this is omega of x0 so this is just the definition of omega being invariant further we have bounded trajectories because omega is bounded correct I already said that if you start in omega you are remaining in omega and the entire Lassal invariant states that you have to start in the omega set okay great. So let us denote the limit set by omega bar okay we are trying to find the limit set now because what do we know we know that trajectories go to the limit set okay by the definition of the limit set okay so we are denoting a limit set omega bar as omega bar x0 okay okay I know you might have used this notation also to indicate closure of a set in your analysis but this is not closure of omega set this is a different set okay I am just denoting the limit set as this okay just a notation. Now by Vidya Sagar's theorem 5.230 this is non-empty closed and bounded okay this is an exercise why you can just have to look at Vidya Sagar's proof and copy paste if you want hopefully you will understand it also while copy pasting but this is the so what I have said is I have asked you to show the proof of that lemma 5.230 for autonomous systems Vidya Sagar's proofs are all for periodic systems okay you have to convert it to autonomous system the time invariant system very should be very easy you have to remove the time okay but Vidya Sagar's theorem says that the limit set is non-empty closed and bounded if the trajectories that you started with were bounded okay if the system has bounded trajectories then limit set is also non-empty closed and bounded okay great further you also have two other lemmas okay which says that again if your trajectories solution trajectories are bounded which they are then you will converge to the limit set and the limit set is invariant you already had non-empty closed bounded you further have invariance and of course this result that you will approach the limit set this notation I do not know if you follow but this is just the distance metric okay we defined this right distance from a set for example if you have a circle and a point you find the distance between the circle and a point by drawing the you know it will be something like this right we have a circle and a point then you will find the distance between them by just drawing the normal okay so this would be the distance so this is the distance this is the way of typically defining distance metric okay so you can use any metric so this is basically distance between a point and a set okay and we claim that as t goes to infinity this distance is 0 which means that the trajectories this is again I have used shorthand again this is the trajectories approach the set for large time okay so this is again from Vidya Sagar so these two results are more like enough for us to prove everything it says that the limit set is non-empty compact and further you will approach the limit set as time goes to infinity and the limit set is invariant now the only thing we have to do is relate the limit set to these sets that we have constructed E and M okay because we have until now talked about sets omega E and M but now we have to discuss the connection between omega bar that is the limit set and E and M okay that is our job now now it should it is not too complicated you already I hope you already are convinced that omega bar is inside omega right I hope you it is clear to you that omega bar is in omega right because omega bar is omega is where you are starting so omega bar has to be inside omega because all trajectories remain inside omega for all time therefore your limit set cannot go outside so omega bar has to be inside omega so this is fine okay great now we use a result called the monotone convergence theorem okay this I discuss in adaptive control but I have not discussed it here but it basically says that if the if a function is lower bounded and non-increasing just like our v function function is lower bounded by 0 and it is non-increasing right just by our LaSalle invariance assumptions function is lower bounded at 0 function is non-increasing then the function has a limit as t goes to infinity in fact 0 is not required it has a limit as t goes to infinity okay basically function is lower bounded so there is a lower bound it is non-increasing which means that wherever it starts it can either stay constant or go down stay constant go down stay constant go down okay if this happens then the function has a limit as t goes to infinity this is called the monotone convergence theorem okay and that is this result so we have used a bunch of results known in mathematics and applied control okay and we proceed all right so we do not know what the value of the limit is it is actually not necessarily 0 I should not say it is 0 it is some constant limit exists it is a constant okay great now what do we what are we going to claim I will already tell you what we are planning to prove that omega bar is inside okay not just you constructed these sets right I know that omega bar is somewhere inside this but now I am claiming that it is going to be inside e okay how do I claim that look at this you already know this yeah I also put a nice justification it says since omega is closed omega bar is the limit set limit set contains limit points and a close set contains all its limit points one of the points you have noted down that you have to memorize okay so therefore omega bar has to be inside omega this is also just nice justification okay now suppose I take some point limit point inside this some point limit point p inside this okay arbitrary then by the definition of limit point there is a time sequence such that you have this convergence right this is just the definition of limit point yeah that is there is a time sequence is that if you keep computing x t1 x t2 x t3 and so on in this time sequence it is going to go to this point that is what it means to be a limit point okay now I use the continuity of this function v right how do I use the continuity of the function v I will write this as v of limit i because v is continuous because x is continuous I can move the limits outside okay this is again a property of continuous function if the limit if so I can move the limit outside or wherever I want I can move it here move it here and so on and so forth okay result does not change so all I have done is if the functions involved are not continuous this is not okay remember this you can get very wrong result by doing that so I move this outside okay and what do I know that from this result here limit as t goes to infinity that is v of p right this is actually v of p right just take the limit here and this is this p inside by the definition here and then v of p is this guy and this limit i goes to infinity v x t i okay just by moving it outside okay all right and this is equal to some constant right because because of this result right by monotone convergence theorem this is actually a constant by monotone convergence theorem now notice p was arbitrary okay remember p was arbitrary okay we have taken an arbitrary p and I have basically concluded that v of p is a constant okay now what do I say so so what can I claim because my p was arbitrary I can claim that for all x in this limit set this is equal to some constant right I have just proved it right because I have taken an arbitrary p I can take any other p it does not matter so for all p in this limit set v of p is actually a constant and the same constant same constant huh so whatever I prove that v is a constant function in omega bar in the set omega bar v is a constant function v does not change okay this is is this clear okay because I took an arbitrary p from that and I use the definition and I use the moving limits inside and outside using continuity of v I prove that v of p is exactly a constant and the same constant so v does not change in omega bar so if I take a trajectory starting in this guy consider a trajectory with initial condition in the limit set okay yes okay again let me go back to this it is okay if we do not complete it today yeah we will look at half of it later no problem so you are saying p is not infinity right I hope you understand p is just a point in this set right it is a finite or infinite set whatever just if you are confused about what this p's are then keep going back to this example all points in this circle are the p's huh all points in this this set this circle is the omega bar okay so if you have any confusion in this proof always go back to this this is the omega bar set okay so all p is just some point here some point here okay now what did I do I started with some point on that circle for example all right I started with some point on that circle and then I use the definition of the limit point which said that I will converge to that point as time goes to infinity right as time goes to infinity I converge to that point okay in this case I will always remain here yeah in this particular case but in general I will converge to this point that is the definition of the limit point yeah it's the definition of this point p okay so I have just written the definition I have not done anything more than write the definition of the limit point here okay now you forget this side okay I know that you can forget this side don't worry about it I know that limit i goes to infinity v x t i is actually equal to this yeah I can move the limits around because of continuity and once I move it around you see that as i goes to infinity t i goes to infinity so x of this quantity goes to p this guy is just p just from here just by this definition right x of t i goes to p as t i goes to infinity yes so this thing whatever I have written is actually v of t and I have proven what that v of p so I have just proven that v of p is actually equal to this guy correct but this guy by monotone convergence theorem is just a constant yeah by two different results yeah this right hand side became v of t for some point in the limit set and the left hand side became a constant and the same constant it's not a changing constant because there exists a limit okay and limit cannot be multiple points one point as t goes to infinity this function has to go to this point c it can't go to something else okay therefore v of p for all p is actually equal to that same constant okay okay again for those who understand analysis is well enough for them this is basically somehow saying that subsequent sub sequence convergence and sequence convergence becomes the same in this case okay because this is like saying I have many many limit points but I am but all of them map to the same limit point in once you put it inside the v function all of them map to the same point which is the c okay so all this this entire set omega bar maps to the same point c okay in fact this is like a sort of like a level set argument okay we have not talked about definition of level sets in fact Vidya Sagar uses that notation also but that's the idea okay there are this set omega omega bar could be anything okay but the function v maps it to one single point c okay all these points yeah subsequence is irrelevant see notice this the subsequence is only used to construct this set okay what will your typical trajectories look like I am telling you it could be like this could be this could be this could be this okay so this is some p1 this is p2 this is p3 this is p4 okay I have not drawn a very nice picture I can tell you this is not a unique solution it doesn't look nice and it's not exactly like this but the idea is the subsequences are used only to confind these individual points okay but one after that the work of subsequence is done exactly you got this entire set and because of monotone convergence theorem these can only map to one single point and not my typical point and this is the magic of this result okay just it's okay we will stop here but go through this go through Vidya Sagar's proof also yeah once you see it again you will see how cool this is in fact very powerful that you took one result on the left hand side and then another set of results on the right hand side to conclude that the entire limit set will map to one single point okay and that's pretty cool it's not very evident in this example okay the problem is this is this is the transient set limit set everything because once I start here I will only be on this circle okay and in this case talking about this point and this point is there is no differentiation you don't understand you can't say that I will converge to this point not easy to say that I will converge to this point or this point or this point yeah this is where you will need the subsequence idea okay so this is a little bit complicated it is easier to see in the van der paal oscillator type examples but when you have these continuous limit sets huh things are not that easy okay things are not that easy to visualize is what I'm saying that is why we have to do this analysis type proof which is beyond visualization you don't you don't visualize anything it's very algebraic yeah because if I had to visualize this and do this proof it would be very complicated okay that's why I give you this example you have the subsequence it goes here it goes here it goes here this helps you construct the omega bar okay so and so this is what gives you this sequence that goes here but the but what but what a result on the left hand side is saying is that all of these map to the same point C okay and what we will do we will use this I'm not doing it now but we use this to say that v dot is 0 on omega bar right because v is constant in omega bar therefore v dot is 0 in omega bar which means omega e omega bar lies inside e because e is exactly the set where v dot is 0 omega bar has to be inside e okay so this is what we'll do okay all right we'll stop you thank you