 Welcome everyone to this lecture in non-linear dynamical systems. Today, we are going to cover about some norms of vectors. In particular, we are interested in signals and then norms of operators. Operators are in our context systems. Of course, this is not standard. There can be various types of operators, but we are speaking in this context. So, we will also speak about eventually finite gain stable. That is the objective for this lecture, finite gain stable. We are going to see the definition of what does it mean for a system to be finite gain stable. For defining that, we need these other things also and eventually, we will speak about feedback interconnection being stable. Under what conditions do we speak about? Feedback interconnection is stable. Feedback interconnection is stable. In order to define this, we need these concepts. So, today's lecture is titled norms of vectors, signals, operators and the notion of stability of a system possibly non-linear. Of course, this will also require us to understand linear systems better because I personally believe that one requires a good understanding of linear systems before one goes ahead and understands non-linear systems. Just like one should know how to build a one-floor building well before one goes ahead and starts building a five-story building. So, we have vector spaces. These are called vector spaces. There are various types of vector spaces, but these are the simplest. These are m-tuples. There are m components of real numbers. So, real numbers are also called as r and hence m components are called rm. One can think of as v1, v2 up to vm. So, one could choose to write it as a row vector like we have done here or one can write it as a column vector. In any case, it is m components and each component can be chosen independently as any real number. Hence, these all together constitute rm. So, now one can speak of a norm for a vector v, v in rm. Suppose, v is equal to this, one can speak of v norm. So, norm we might have seen in the beginning of this course already. So, norm is required to satisfy a few conditions, but we can think of already start seeing some examples. So, two norm of v is defined as, so we use this symbol for define. When we have a column on any of the two sides, it means one is being defined as the other. Since the column is on the left side, the left side is being defined by what we will write now, v1 square plus v2 square up to vmth component square and then together square root. So, this is the most common notion of distance which we call as the two norm, which is also called Euclidean distance. This is our conventional notion of distance. Euclidean distance or Euclidean norm, as I said this is also called as two norm. Why? Because instead of the second power and then taking square root, one could take any power, one could take the pth power and still it will satisfy the notion of a norm v1 to the pth power plus v2 to the pth power plus up to vmth to the pth power. But then if p is odd, this does not ensure that these are all positive quantities. So, we will take absolute values of each of these real numbers and we will add after taking, we are going to add after taking the pth power of the absolute values. If p is 2 or if it is any even number, then this absolute value is not required to be taken. They are all real numbers positive or negative and for even powers, they will always be positive after taking the even pth power and after adding them all, we will take the pth root. So, for any p greater than or equal to 1, this is how the pth norm of a vector v is defined. One can also take the so-called infinity norm. What is infinity and how is that related to p? One can check that if you have m fixed components and you go on raising p, you make p2, 3, 4, 5 and you take this absolute value and then take the pth power and add them and then take the 1 by pth power, then it will eventually converge to the maximum of the absolute value of the ith component where i varies from 1 to m. When you say max over i of the absolute value of v i, it will turn out to be the value that pth norm of v converges as p tends to infinity. So, this is i which is same as saying where i ranges from 1 up to m as take different values of i from 1 to m and for each of these cases look at the absolute value of the ith component of v and look at the maximum over these m components of the absolute value and that is defined as the infinity norm and how is that related to p? It also turns out to become equal to vp as p tends to infinity. So, this is another notion of norm. All these norms have a notion of distance. They all satisfy our conventional feeling of distance, so norm. So, why do we not take p strictly less than 1? Because a norm of a vector has to satisfy three conditions, 1, 2 and 3. What are those conditions? For any vector v, the norm has to be greater than or equal to 0. We are not comfortable with a distance which is equal to minus 2. Any norm if somebody says this is the norm of this vector, it had better be a positive quantity and if the norm is equal to 0 that is only when the vector v is equal to 0. If somebody tells that look here is a vector whose norm is equal to 0 that should happen only when v is equal to 0. But for the 0 vector it obviously happens because of the second rule that we will write. If somebody writes, if somebody scales a vector that is nothing but the norm of the same vector multiplied by the absolute value. Alpha is a scalar, it is a real number from what we are doing in this case. So, if somebody multiplies that vector by 5 then gets scaled by 5 or by minus 5. Minus 5 means you just multiply scale it by 5 times and reverse the direction. Both of this should just result in magnification of the vector by 5 times independent of plus 5 or minus 5 that is for the absolute value. So, this automatically means that when v is equal to 0 then the norm is equal to 0 that is why this only when is playing a role in item 1 and the third one is so called triangular inequality. This cannot be greater than v1 norm plus v2 norm. All these three are satisfied for any v, v1, v2 alpha. For any vectors v1, v2 and v and for any real number for any scalar alpha these three inequalities have to be satisfied. This one just says that the norm cannot be a negative quantity and it is norm is equal to 0 only when v is equal to 0. Second one says about the scaling by a scalar alpha and the third one speaks about the so called triangular inequality. What is triangle and what is inequality about it? This is vector v1, this is vector v2 then v1 plus v2 by the so called parallelogram rule is this. So, the length of this cannot be greater than length of this plus length of this. That is what is triangular inequality. This one is v1 plus v2. So, these three together are required for any notion of norm and the pth norm satisfy all these three conditions. But for that we require p to be greater than or equal to 1. So, how do these norms play a role? We are actually concerned with signals. So, consider function f that from real numbers for any real number it has m components. In what does that mean? f the independent variable r means in our course because we are dealing with the systems course this independent variable we like to think of as time. One can also think of it as space but in any case there is only one independent variable f of t. At any time t f of t is a vector in rm. This is the meaning that f is a map from r to rm. So, we can speak of that vectors norm, we can speak of its two norm. Now, we are going to be interested in looking at all so called square integrable functions. We will take any function f from r to rm. So, this integrable set of all such integrable functions is also called as L. L I think stands for Lebesque thanks to one friend of mine called Ajay who told me that L that we use so often in L as in L2, Lp. All that L stands for Lebesque and the person who came up with a very concise very systematic definition of integration after Riemann. So, L2 we are going to define now. L2 is a, we are going to say what is this set? It is a set of, we define it as a set of all functions from r to rm such that some property is satisfied. What property from minus infinity to infinity? Just take the two norm, this should be finite. Let me write this little more slowly and clearly. So, what is the L2 norm defined and what is the L2 space defined? It is defined as, consider the space of all functions from r to rm which all functions will be taken put into this set. Take all those functions r to rm and check. Take all those f for which this vertical bar should be read as for which. It can also be read as such that all those f from r to rm such that the two norm, this two norm stands because at any time t this is a vector in rm. Integrate this from minus infinity to infinity. Integrate this with respect to time. This should be a finite value. This finite value itself of course we will very soon define as the L2 norm of the signal f. But if this norm is finite, if this after integrating if you get a finite value then that f you will pick and put include into this set L2. So, what is L2? The set of all such signals. It turns out this will be a vector space. What is vector space about it? You take any two functions f here. You add them that will also continue to be in L2. If you take any function f and you multiplied by a scalar it will also continue to be in this. And also the 0 vector is there. For this set the 0 vector is a 0 signal. The signal with which is equal to 0 for all time t. So, this is what makes L2 r to rm into a vector space. Of course, you may not be interested in r. You may be interested in only functions that take values from 0 to infinity to rm. These are the function that we are considering here. The domain could be different. This one is a co-domain, the vector space in which it takes its values at any time t. But t itself need not be varying from minus infinity to plus infinity, but it could be varying from only 0 to infinity. This is defined as set of all f from this domain to rm such that integral. Now, we will take not from minus infinity, but only from 0 to infinity of this f of t dt. If this is less than infinity then we will go ahead and put that function f inside this. So, notice that this 2 here is different from this 2 here. This 2 here refers to f of t as a vector in rm and there you take the Euclidean norm. But sorry, there is one important thing I missed. What is this 2 then? This 2 refers to that you take the norm and you take the square here. Also here I should have been putting a 2 here. So, this 2 up here refers to this subscript below this L. Let me see if I miss the 2 in the previous slide. Here also I have only half written this. This is slide number 4. Here also we need a 2. This 2 here refers to this. So, let me just quickly speak about and these 2 do not have to be related. More precisely, we can speak of L2 space from r to rm f. Take all those functions f from r to rm such that integral from minus infinity to infinity of f of t. This norm we can also take as infinity norm but still you have to take the second power dt as an infinity. I hope this is clear that if you are taking L2 space here, then the power should also be 2 here in the superscript. After taking the infinity norm of the vector f of t at any time t, you take the second power and then you integrate from minus infinity to plus infinity and that should be a finite value. That should not be infinity. The reason that people often keep writing this is because the set that you get here is eventually the same. The set of all functions that you take will eventually be the same whether you take infinity below or you take 2 below or p equal to 1 or any other p greater than or equal to 1. Whatever norm you take in rm, that norm does not decide what functions come into this set L2. What certainly decides is the fact that you took the second power and still it is finite. Clearly, well it is not obvious but it is important to know that Lrrm is not equal to L2 r to rm nor is that equal to… We will quickly see some examples. L infinity we are yet to define but this 1, 2 or infinity refers to what you put in the power after taking the norm of f of t. That norm of f of t we have indicated here that is a norm in rm. These sets are not the same. The reason that people often skip the specific which norm we took in rm is because there is some very important result saying that all these norms in finite dimension vector spaces are equivalent. Because of that it turns out to not matter which norm you take here. Of course, the actual 2 norm of L2 norm of the function f does depend on which norm you take but as long as you do it consistently it will not matter much in all our arguments. Let some function f be inside this L2 r to rm. This already means that it is square integrable. This condition that we have written here means that it is square integrable. Whether it is square integrable or not that particular statement does not depend on whether you take infinity here or you take 1 here or 2 here. It only depends. Square refers to the power 2 up here. So, you take f here then this f we will define as L2 norm. L2 norm is defined as integral from minus infinity to infinity f of t to dt and we have taken already the power square here. We are going to take square root here. So, notice that I have skipped writing this here. Whether I take 2 or infinity it does not it does matter what the L2 norm of f is but it will we have to only do it consistently and for this course why do not we just stick to the 2 norm in rm. One can also speak about the L infinity norm. So, for that we have to define what is L infinity r to rm that is defined as the set of all f from r to rm in which for the purpose of this argument why do not we say continuous, f is continuous. Continuous is not required but then I will tell you why it does make a difference eventually max. Max overall t of f of t, max of this is finite. This is what is this is set of all the functions whose maximum value is finite but then this max is taken over what? t varying over what range? t inside r. So, notice that this r is not a finite set. It is not a bounded set. It is not to be precise a compact set because of that this max may not exist. We are a little concerned that over a bounded set over a compact set to be precise the maximum value always exist but over r there are these strange problems. So, we need to correct this max to sup. So, I will tell you what the difference is. So, suppose here is a function f of t where sort of goes on increasing and is saturating and it saturates seems to saturate to 1 but for no value of t does it become equal to 1. f of t always strictly less than 1 for all t but at t tending to infinity reaches 1. So, we want to know what is the lowest value which is above all values of f of t. You take all values of absolute value of f of t and look at the smallest value above all of them that is called as a sup. That is not strictly above but greater than or equal to. So, in this case it is equal to 1 that sup is equal to 1 but sup is always strictly greater than f of t. It is never equal to f of t for any value of t why because f of t is strictly less than 1 for all t. So, here in this example the max is never attained the maximum values the maximum does not exist why because is 0.99 the maximum value you know it goes and exceeds 0.99 eventually. So, there is no maximum value but the supremum exists the supremum is equal to 1. So, this is the important subtle difference between sup and max and over compact sets over a set over which you are looking for the maximum if that set is closed and bounded then the 2 turn out to be equal then the sup is equal to max that time the maximum is attained to be precise then the supremum is attained. So, now important statement is now coming back to why we assume f is continuous now we want to come back to whether L infinity whether the L infinity norm turns out to be the limit as p tends to infinity of Lp norm. There it turns out that if it is not continuous then here is an example that look at this function say which is equal to 1 eventually, but only for 1 value only for t equal to 3 it takes some different this is equal to 1 here but for t equal to 1 it is equal to 5. Of course clearly the graph is not to scale but f of 3 equal to 5 otherwise f of t is less than or equal to 1 for all t not equal to 5. Look at this graph there is a small hole here only for t equal to 3 it turns out to take value equal to 5 but for all other values of t it is less than or equal to 1 only for one value of t equal to 3 it takes value 5. So, one can ask below this how much area is covered under this point area is 0 thickness of this area is equal to 1 thickness is 0 and height is 5. So, height into bread so area is equal to 0 because the breadth is equal to 0 thickness is equal to 0. So, you might say how much area is covered under this point under this point area covered is 0 for the area to be non-zero you need at least you need to be you need equal to you need f to be equal to 5 for at least some width at only one point if it is equal to a large value that cannot change the L infinity norm that is indeed it turns out to be the case if you take the power p and you let p tend to infinity. So, let me tell a slightly more correct definition not slightly more this is indeed a very correct definition but this is when f need not be continuous. So, L infinity consists of the set of all points like this and at only one point it cannot matter it cannot become infinity anyway. So, here notice that here we are not requiring it to be continuous super all t in R of f of t norm as long as the supremum is less than infinity as long as the supremum is less than infinity you will take all those f and put it inside the set L infinity what I have written here is less than infinity. So, this is what constitutes the set L infinity but if you have a f inside L infinity what is its norm till now for every p it turned out to just be its Lp norm but for L infinity it will turn out to be slightly different we need that essential supremum which I will come to in the next slide. For any f in Lp for p greater than or equal to 1 but not equal to infinity what is its Lp norm it is defined as integral from minus infinity to infinity why minus infinity to plus infinity because we wrote the domain of f is equal to R of the norm of f of t the pth power for any time t f of t is a vector in Rm which norm to take in Rm that we are supposed to write in the subscript we have decided to not write why because we are going to stick to the same norm for all our arguments you can take the two norm there you do not you are not forced to take the pth norm in Rm for that purpose. We will integrate only when p is less than infinity after taking the pth power but now that we are defining this as the Lp norm sorry this is already finite if you are taking the Lp norm this is already finite you will take the pth root of this value for any f in Lp we know already that this integral is finite only because the integral is finite we have decided to put this f inside this set now you take that f and compute this value you know it is a finite value that we will define after taking the 1 by pth power we will take the pth root and that is defined as the Lp norm of the function f but for L infinity norm this we are going to say is supremum over all t in R of f of t f of t is norm f of t is norm which norm to take again it does not matter as I said but we prefer writing this essential supremum what is this essential supremum this full form stands for essential this what I am saying here is what I studied several years ago thanks to my teacher Prasab Bhanawar this essential supremum stands for that this f should have reached this value over at least over at least some small interval it cannot be equal to this value at just one point yeah so this is t f of t and at one point if it is equal to 5 that is t equal to 3 then essential supremum will be equal to 1 and not 5 the essential word ensures that the supremum value there are many values of t close to the supremum value very close arbitrarily close and 1 qualifies for this number but for f of t equal to 5 f of 3 equal to 5 there is only 1 point t equal to 3 that is essential ensures that that for this particular f of t l infinity norm l infinity of this particular function turns out to be equal to 1 so let me take a concrete example so here is a function minus 5 at 1 point is equal to 6 at t equal to 4 yeah so here is a function f which is never equal to minus 5 but it seems to be saturating to minus 5 on t tending to minus infinity it grows like this 40 equal to 4 alone only 40 equal to 4 it suddenly becomes equal to 6 this is where continuity would have helped us but 40 equal to 4 alone it has suddenly jumped to value 6 after that it seems to be saturating to 1 so f of t max after taking absolute value max is equal to 6 sup of f of t also equal to 6 sup and max over what over t tending varying from minus infinity to plus infinity but essential supremum of f of t equal to plus 5 plus 5 yeah we are taking the norm is the essential supremum attained for any value of t no it is converging to minus 5 as t tends to minus infinity it is never equal to minus 5 for no value of t it is equal to minus 5 but still the essential supremum turns out to be equal to 5 because it is for enough values of t yeah to be precise for those who are inclined for a set of measure greater than 0 for this length strictly greater than 0 it has come very close to minus 5 so when you take the norm it will become plus 5 so that is why the essential supremum in need is equal to 5 yeah so f f l infinity norm is equal to plus 5 yeah and this will indeed be equal to the value when you take l p th norm of f of f and let p tend to infinity that is the best part and if f were continuous then the then this 6 would automatically get ruled out for continuous functions for each value is attained over a set of measure non-zero is not not necessarily attained but it comes very close to it over a set of measure non-zero that is what that is how continuity helps okay these this is only about norms of signals now we are to also see we want to think of all our operators all our systems that take input u give output y and that is an operator h yeah so h takes signals from where is the question and gives and gives output in are the input and output in L2 are they in L infinity that is the question we want to answer for that purpose it turns out that we will have to extend our spaces L1 L2 L infinity Lp will all have to be extended yeah so we will quickly see what is the meaning of L2 e R2 Rm this extension is this e for extension is coming because our systems take signals from not L2 necessarily but L2 e and give you signals possibly in L2 e if h is not stable we eventually want to say that even though u is in L2 y might be in L2 e yeah that is the purpose that we are going to define this extended space as soon as we define this L2 e space we will see some examples of what is in L2 what is in L infinity what is in L infinity but not in L2 etc those examples we are going to see now so consider sin t is this yeah this is how this is how sin t looks is this bounded as t tends to infinity is this finite yes many is there a value that is greater than or equal to every value of absolute value of sin t sin t bounded yeah what is the maximum value plus 1 what is the minimum value we are not supposed to see whether sin t is bounded we are supposed to see where absolute value of sin t is bounded yeah luckily that value absolute value is also equal to plus 1 sin t is less than or equal to 1 for all t yeah so that is why sin t is in L infinity what is the range from r to r at any for any value of t sin t gives you a real number yeah of course it may not be giving all real numbers as its output as its range never the less every value sin t gives is a real number so sin t is an example of L infinity is it in L2 that is what we will see now can we integrate can we take sin t to norm second power dt from minus infinity to infinity yeah we have taken sin t this is absolute value of sin t because sin t is actually a real number it is a scalar to know its Euclidean norm is norm is nothing but absolute value yeah so for each period there is some area under this so over this yeah when you integrate from minus infinity to plus infinity this is not less than infinity yeah it becomes unbounded this integral does not exist we will say minus infinity to infinity sin t dt square does not exist of course in layman term we say the integral is equal to infinity that is not correct mathematical language we will say the integral does not exist so we will say sin sin t is not that so we are not allowed to take this function put it into L2 yeah it is not in L2 but now we can ask what if we chop it what if we stop it at some yeah for the purpose of this L2 e we prefer considering L2 0 to infinity only on R plus to R yeah the same signal sin t yeah sin t is also in L infinity from 0 to infinity to R yeah why because over this range also it is bounded in fact it is bounded from minus infinity to plus infinity so over a subset also it will of course be bounded so this it is inside this so we are interested in extending this space actually because we are interested in only future being chopped or future being extended so this sin t which was then L infinity R to R is also in L infinity 0 to infinity to R yeah so this sin t is unfortunately not in L2 0 to infinity to R also it is not in that neither in yeah so we ask the question is sin t question mark is it in L2 R to R we have already answered the question as no what about 0 to infinity yeah answer to this is also no yeah what is this sin t square now we are going to integrate this yeah area under this is all for each period it is some nonzero positive value so when you integrate from 0 to infinity 0 to infinity of sin t square dt this does not exist does not exist yeah so we will say sin t does not belong to L2 0 to infinity yeah of course if you make on this side from infinity instead of infinity if you take a finite value then sin t will belong yeah that brings us to this extension so what we are going to do is we are going to say that our signals sin t etc don't really belong from 0 to plus infinity they are always up to some finite value yeah we never ask as t tends to plus infinity what happens whether the norm whether the total integral is finite or not is not a question we ask for so far so much in the future so what we will do is we are going to say some f is said to be in L2 e 0 to infinity to Rm if f tau we are going to define what is this f tau is in L2 for each tau in R plus each tau in positive value only make sense of course the question arises I have already defined this f tau without telling what is I already define L2 e using f tau without telling what is this tau f tau so given f and we also take a tau tau also inside this range it doesn't make sense to chop at negative values yeah so f tau is a new function that is defined as is equal to f of t for 0 less than equal to t so equal to tau yeah and is equal to 0 for tau greater than sorry for t greater than tau yeah so what we have done we have chopped whatever for all values of t greater than tau we have chopped it to 0 this is a definition of the function f tau f was already a function from 0 to infinity to R somebody gives us a value tau some positive value and we have used that value tau to define a new function f tau f tau is also function from 0 to infinity to R it behaves like f as long as t is less than or equal to tau but t is strictly greater than tau it is just equal to 0 so we will see a graph of such a function so for the purpose of this discussion we are considering f defined only for non-negative values of t this is how f of t looks suppose let me take a different color pen so here is another this is for this blue one is equal to f 3 where we have assumed that we have chopped it at tau equal to 3 yeah we can speak of f 5 as f 5 is this new function then that is in green that gets chopped a little further from here onwards it becomes 0 yeah this is a green one so we can chop it at different different values of tau and once it is chopped it is sent to 0 that chopped version is what we call f tau yeah f tau is also an element of from 0 to infinity function of 0 to infinity to R yeah if f is in 0 to infinity to R this is the meaning of a chopped signal so now we are concerned that there are many signals which are in L infinity but not in L2 yeah there are various types of signals which are not in L2 perhaps their chopped versions are that is our next thing that is what we will see in detail now consider sign of t again this is not in L2 yeah we already saw that it is not in L2 L2 which from this from this to R this is how it is called the domain codomain scientist is not in this why because it has what we call infinite energy over this range but for any for any chop yeah chop chop sign t at any at any tau in that range over whatever range it is defined yeah chop scientist yeah then then sign t chopped yeah we know that that is in L2 yeah scientist is not too bad as soon as you chop it for any value of tau it comes into L2 the chopped one yeah so we will say sin t though it is not in L2 it is in L2 e so how is this an extension sin t was already in L2 but for every chopped version of sin t for every tau you chop it and it is in L2 so why don't we put it in not L2 but L2 e will extend the space L2 to L2 e where for every chop that function is in L2 yeah so this is recall our definition this is how we define so this is our 14th slide so we'll say f is in L2 e from this domain to this codomain if f chopped is in L2 but for each tau yeah it is not okay that you chop it at some carefully designed tau you can always chop it at tau equal to 0 and always come back to L2 it will become a 0 signal no for each tau in R plus you chop and it comes into L2 then you will say that function f was not too bad you will put it into L2 extended yeah in you one should note that what is not obvious is that this is genuinely an genuinely an extension yeah what does it mean to be an extension L2 0 to infinity the extension word is justified because of this property L2 e if without chopping already the function was square integrable of course and for every chop also it will be integral yeah that is why this one is contained inside this and we already have an example to say that the two sets are not equal sin t is an example of what is here but not here and this set is a strictly larger set of this in other words this is a proper subset of this yeah this is how extension is defined let us see some more examples of L infinity extended L1 extended etc so consider f of t equal to t yeah we are interested in f only for positive values of t this is our so-called ramp this is a ramp signal so it is is this f of t is it in L infinity answer is no answer is no why it is not even bounded it goes on becoming very large yeah but you chop f of f to any value of tau yeah for whatever value you chop any f tau looks like this f tau is also again a function of t tau is some value here it goes on increasing till there from there on it is 0 yeah for each tau it is bounded what is the value maximum values tau so for each fixed tau for each finite value of tau it is bounded is it uniformly bounded no the bound depends on tau in that sense it is not uniformly bounded in tau yeah so this f tau is in L infinity thank god for each for each tau in hence we will say f is in L infinity e L infinity extended yeah hence f of t equal to t f is in L infinity extended what is the domain and codomain so from now on anyway this is always going to be our domain yeah and this codomain is often clear from the context so we will just say L infinity e yeah so the function t is in L infinity e but is t in L infinity no yes yeah so it is not bounded what about this this also shows again that L infinity is also a proper subset of L infinity e yeah so we need this extended class of signals all our signals live in this space extended spaces whatever step input step signal impulse ramp sin t i sin of t okay cos cos t sin t cos t are expected to not be very different what about t square yeah so these are all in what check whether they are in L1 L2 L infinity L1 e L2 e L infinity this is an extremely good exercise to try oneself for each of these signals why these are important signals step signal ramp sin t cos t t square impulse is not a function yeah no need to check of this for here because impulse is not yeah till now we had been saying f is a map from as a signal it is a lab egg integrable function all all our f's where functions is impulse of this type impulse no yeah so please don't check for this if somebody says is the impulse in L1 it is not even a function is it in L2 not a function no question of checking whether impulse is in any of these sets because it is not even a function strictly speaking it is what we call a distribution yeah thanks to Schwartz who made this theory very precise impulse is not a function it is a distribution of course impulse is a limit it is a limit of a sequence of functions sequence of functions each of which has area under it is 1 and it is not equal to 0 for a smaller and smaller thinner and thinner interval around 0 yeah it could be the limit of a sequence of functions but itself upon limit upon the limit itself is the impulse distribution not a function so this brings us to whatever operators our h we want to ask is this h stable is the magnification it causes to the input to give you output is a magnification bounded by some value that bound magnification for that we need these L2 these norms it is with respect to these norms that we speak about the bound so h operator yeah takes takes input signal and gives output yeah this input and output live where they live in some such extended space yeah you can say L infinity extended and output is also L infinity extended these extended spaces themselves do not have a two norm they do not have an infinity norm the operator possibly nonlinear yeah of course we will see what all this means possibly nonlinear what this all means in the case of linear system is of course something we will see in much detail so now we are going to ask when when input input in L2 e then of course and of course output by in L2 e yeah this is not too surprising but what is good about what we will define a stable systems is then you in L2 then why in L2 question mark yeah if so then h is what we will call stable stable of course is in a very general sense this is input output stable this is not Lyapunov stable there it was for autonomous systems and here we are speaking of input output stability so this is the question we are going to ask and in a very loose sense we will say if the u was in L2 for every u in L2 if the output is also in L2 then in some sense we will say h is good why because in general u in L2 e goes to y in L2 e if you restrict the input to L2 from L2 e when you restrict it to L2 that does not mean why we will also get very conveniently restricted from L2 e to L2 yeah but if that happens then h is kind of good then h is stable so we will drop today's lecture with the precise definition of stable so h is called finite gain stable finite gain stable that refers to L2 stable let us say yeah please introduce this L2 stable if y tau L2 is less than or equal to some gamma this is gamma u tau L2 plus beta for all tau and u in L2 e yeah what is gamma and beta if there exists gamma and beta 0 such that this inequality is true for every chopping tau and for every u in L2 e for every single u that you give as input the corresponding output is bounded by such a quantity in which the gamma and beta are some some two positive numbers that cannot be changed when you take different different tau and u yeah please notice the sequence of arguments so we will call h as finite gain L2 stable if you can find two numbers gamma and beta both greater than 0 such that this inequality is true for every tau and for every u in L2 e yeah so this is easy to confuse this gamma as y this is gamma while this is y so this is the definition of finite gain L2 stable yeah notice that of course gamma and beta clearly not unique if you have found one gamma and beta you can take a larger gamma and larger beta and still this inequality will be satisfied because each of these quantities are positive this quantity is positive this quantity is positive you multiply them by a larger number and still this inequality will be satisfied so clearly gamma and beta not unique so greater than 0 as soon as such a finite gamma and beta exist you will say this h is finite gain L2 stable so in our next lecture we will see what this means for the case that h is a linear system yeah that is the next thing that we will see in our next lecture okay this ends today's lecture