 Так что, я начну мою курсу, как профессор Куксин, я тоже даю 4 лекции на контроль серии. Это будет интердакторный курс и геометрический контроль серии. Это не очень формально, я не буду говорить про очень новые вещи. И я считаю, что этот курс... Дело в том, чтобы объяснить какие-то геометрические идеи с минимальной аналитической техникой, потому что эти идеи могут быть использованы. Они уже были использованы и я считаю, что часть этой идеи уже будет использована для миксирования, для эволюционации. ПДА будет использована по стакастике. И я считаю, что тоже что-то еще будет использовано в будущем. Итак, я начну просто с дефинитива. Что это контроль-систем? Контроль-систем — это простая форма. Если у вас есть эволюция, или дефинитивная эволюция, я буду работать с аудией, но как профессор Куксин, это был син. Курс, в принципе, вы можете думать об эволюции пидей, как аудией в более сложном месте. Геометрия — это значение. Если вы считаете, что все аудией в новом диаметре, то здесь может быть и та же геометрия, как пидей, но больше анализов не нужны. Мои форты могут регулярить. Мои форты могут регулярить. Это очень хорошая миксация. Но геометрия, в принципе, та же. Я уже показал, что это очень важная миксация пидей. Это работает. Можно быть оплотен. Но я делаю с аудией, поэтому я считаю, что систия. Я стараюсь с очень геометрией, чтобы я был немного меньше геометрией. Геометр-контроль-систем — это так. У вас есть только динамичный систем. Экс-биллун 무슨? У меня есть геометр-контроль-систем. Мой геометр-контроль — это очень �гיבие, но если вы не очень уверены. Некоторые из вас не очень уверены. Геометр-контроль — это просто domain in run. ФФ, давайте мы снижем. ФФс — это динамичные системы. Если вы хотите, и контрол-система, это формально, это фамилия динамичного системы, которая зависит от некоторых контрол-параметра. Для меня для вас будет открытый, но большая часть контрол-система, или просто вектор-спейс, потому что это используется для миксирования. Но в общем-то, это может быть тоже нон-линия, и вы можете быть в более компликатом специи, в ближайшем, даже более компликатом. Так, это просто фамилия динамичного системы, параметрирована с некоторыми параметрами. Мы надеялись, что это все с респектом. Но, в общем-то, в моем курсе я считаю, что это просто фамилия динамичного системы. Специальный случай, FU of X, это F0 of X. Мой случай для нашей курсы и аплодисменты. И FU — это просто FU of RK. Так, что у нас здесь? У нас есть какие-то динамичные системы, и это респект с другими динамичными. FU of X, это параметры респекта. Так, это почему мы делаем специальный серий, мы пишем много книг, если это просто... Это полностью включено в такой ситуации в теории динамичных системах, потому что у вас есть X, и у FU does not move, и это все, окей? Это параметры. Но это не... Мы в контроле серии, что мы делаем, мы контролируем U as control. Так что это параметры, которые мы можем выбрать, как мы хотим, и менять наш выбор в каждый момент времени. Это такой момент, окей? Для нас это параметры контроля, и функции контроля. Я уже думал, что я не буду дать много деталей аналитических деталей. Вы можете найти это в книге. Я не знаю, как это произойдет, в сайте или нет, в сайте, где все. Присоединяется. Провод в книге Агора Чев, Сачков. В контроле серии от геометрии. Геометрия. Это шпрингер 2004. И есть еще хорошая коллекция лекций. Здесь, в ICTP, если я понимаю, что это totally free, может быть, у них даже принятые копии. Я не знаю. В школе мы организовались здесь несколько лет назад. Это массематическая контрол-сери. Два вольера. Есть много курсов. Контрол-сери. ИСТП, лекция-сери. Это самое простое, чтобы найти здесь. Лекция-сери. Так, это ресурсы. Сейчас, окей. Контрол-фаншен. Это просто функция на сегментах НРПЛАС. Т. Ти. Ти. Титаль. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. Ти. если мы не depend linearly. На ухе мэп не очень хорошо оборудовывается и не очень хорошо на литературе, но в этом случае ухе-то это не очень хорошо оборудовывается и в инфинити это всегда возможно, без проблем ОЛ2, ОЛПС, ОТРАКУДБИ. Теперь, что мы делаем? Так что, это, действительно, невозможно, в случае стакастик-пертурбационных контроллсерий, в классике контроллсерий, УФТ — это то, что мы totally free to choose as we want. And the main object to study is actually the map. We can define the map, FT, family of maps that depend on parameter T, that we can, in order to... So now, what we do? If we select, we have our dynamics, dynamic is something that is done by physical law, or I don't know by say... We have a dynamics, we have a dynamical system that depends on parameters, and we can control some parameters. That's the idea. We have a family of dynamical systems, some parameters, like a car, or like a human body also. We have a body that actually, whose position depends on many, many parameters and some of them we can control and move and change these parameters whenever we want, in certain limits that are indicated here. And so we simply plug in this U of T, we get the map, control U of T, instead of parameter U, we are writing U of T now. And then we have under this setting, we have a well-posed evolution system. So Koshi problems in finite dimension is well-posed. At least for small T, but that details which I will not create. Of course, solutions may blow up in some time, but this I will not care about. It's easy to control anyway. Just do not to spend time for that. At least for any initial condition, X of zero, we have one defined solution. And let me in order to minimize number of variables to simplify formulas just to fix. So in order to get a solution, we need to choose U of T and control function and the initial condition. I fixed initial condition. I just fixed initial condition. This is going to be fixed from the very beginning. And then I have the map, which I call family of map, which I call F T, that send U of dot into X of T U of dot. If I have U of dot, I have unique solution starting from here F zero is just of U of dot is just X zero. So I assume that initial condition is fixed. We have a unique solution starting from X zero. And we take it and evaluate it moment. That's a map of T. It's actually the main object to study. There is a quite rich map, as I think you will see soon. What are properties we are interested first of all. Of course, we are interested where we can arrive. We start from position X zero. We can start without any position. But for the moment I fixed it. And we select control, we move somehow. And the question is our control capacities, where we can arrive. So what we are interested in, this is called controllability problem. To fix, to understand somehow the image of this map. Where we can arrive. Let me also denote by U. This is going to be space of controls. This will be script U. U is the space of this function U of dot. Admissible U of dot. As well as in your capital. And fine. So that's what we try to study. And the controllability, controllability problem. Is it true? System is controllable. That F of U script is the whole manifold. Can we arrive everywhere or not? That's the first question. For time T. Controllability for time T. Or for infinite time. Then just T is free. Let me explain immediately, before specification, that even in so general setting, relation of this problem, it's actually a problem that is relation with mixing. So here everything is deterministic. No stochastic thing. And as I thought, we simply study, select any control that we need. But assume that we don't know how to select our control. And we have some probability distribution. Some probability distribution on U. And just apply with certain probability, according to certain velocity distribution. Now I start to be close to the setting of Professor Kuxin. We have a time segment T. We divide it in small pieces. Or we may do more advanced things, consider kind of divided limiting case, kind of Brownian motion. A stochastic process with continuous time. But it's totally sufficient to divide it in pieces. And we use just piecewise constant control. Even piecewise constant control. Or some particular classes of controls. For piecewise constant control that I distributed with respect to probability measure on U. Some probability measure on U. We apply it here, arrive to some point. Then independently apply on the second segment, independently on third segment, independently on third segment. And we are interested. And then as a result, at time T, we will get not a point, but probability distribution. We have such a random work, if you want according to prescribed dynamics. We obtain a probability distribution at the end. And we are interested on the properties. And mostly we are interested. Does it has a limit for T equals to infinity? That limit that does not depend on X0. Unique limit for T equals to X. It usually does not depend on X. So I claim that it's a look. There could be, and I think in Professor Cookson lectures, at least in this papers that he mentioned, he used more sophisticated things. So they were not piecewise constants. So the probability distribution was defined on the space of U restricted to small segment, to the given segment. So it's richer things. It's not piecewise constant, but important that you take a certain distribution of control in the space of controls on given segment, and then you repeat it, but totally independently. Crucial words here independently. And what will happen at the end of the day or at the end of everything, for T infinity? I claim that the mixing property is extremely close to the just controllability properties. If you do not take naive, not actually mathematical problem, that question is not a mathematical question. So it's at the level of set. Normally if we use some real tool to check this, you have more. Because if you create, give a way to select control, which arrives at any point, it is not just you prove that it is. I explained this later in my course. You have really more. So what I mean? We have a kind of qualified. I say, I claim that a qualified controllability, controllability with some stability, some solid properties is essentially equivalent to mixing at least in finite dimension. In finite dimension yes. Because in infinite dimension we normally do not have We never have this. In infinite dimension we may expect only the closure of that. And it is a little bit other story. And you need much more analysis. In order to study the properties of my measure, you have to project it on finite dimensions. It's more delicate. And you cannot do it in the universal way. You need a certain class. It works for a certain class, but important class of PDEs. But in finite dimension it's more or less. What I mean is the qualified or solid controllability. Assume that we start from x0, and at time t we arrive. And there is a... And we have a... And as soon as we have ux... So this is x of t, of u t, of dot. We arrived at some point using specific controls. We start from x0. This is a trajectory. And at time moment t we arrived here. Maybe I drove it in another part of the box. Blackbot. We have a huge blackbot. So let us use it. x0, we arrive to x1, x of t, u t, of dot. And assume that it is not... We have a bigger property, greater property. We assume that the map is... Our map is open in the point u tilde. Ft of some neighborhood of u tilde in the space of control functions. Kavas, some other neighborhoods. Say like that of x1. Assume that it is open. And moreover, this size of this. And assume that it is like that. So essentially, normally not all neighborhoods is necessary. Since we have a finite dimensional manifold. Here I start to use a finite dimensional. We have a finite dimensional manifold of x. We can normally find a family. It's maybe not necessarily the complete neighborhood by some finite dimensional family. Family of controls near u tilde. Some perturbation of this control is efficient. But assume we have this and moreover somehow we may control the size. This guy may be much smaller than this one, but still with some minimal control. Absolutely minimal control. And how it can be controlled? Very often, and I explain it in the next lecture, maybe probably today afternoon, very often when I say qualified controllability, it means that if I perturb a little bit f, then we still have this property. Which maybe neighborhood becomes smaller, but we still have this property. So I do not only go to the, can go to the point. First of all, I go to the neighborhood. I have a sufficiently rich. Somehow this is the only way you prove controllability, you automatically prove. There are no other way. You use analysis to prove controllability. So using analysis, you automatically prove this property. But at least it's finite dimensional. We can have more. It is a stable. That property is stable. So if we perturb a little bit uf, uf is a finite dimensional family. So you have u tilde, and you have some f tilde of dot, and you have some finite dimensional family. This is, you can take it as a neighborhood of ut. Not completed, it's too much normally. And if you perturb also this guy's little bit and f little bit, you still cover neighborhood. So this is what is called solid controllability. Now look, now look. Take a time, say 1. Now look what happens. Assume that we have it. It means that now switch to the stochastic situation. In stochastic situation, this property means that we cover small neighborhood with a nonzero probability. If you have a distribution, such that all controls are presented somehow in this distribution, okay? Then we, we, we, we, it's maybe very small, extremely small. Exponentially small size, has a smaller size than this. But still with nonzero probability we can cover and enable it. It means exactly that. You can, you can understand it. Because if you, you, you, you can use a family of full probability. So, and if it is like that, switching is almost free in finite dimension. Look, the independence comes in the game. For time one we do it, okay? And I'm sorry because of course now I see that when we do stochastic view point it is not convenient to, to, to use a, use a fixed once forever initial condition, okay? So, so this effect actually depends also on X0, okay? Let us write like that. They play different rule. Like X0 and this play different rule, okay? And then what we do? Then what we do? Look. So, we have time one now, two. Yes, but I say about initial condition. Initial condition I cannot feel, because I have a dynamics. I have a semi-groups. When I go time one, then I start from the, then I continue going. I keep going, but start from another point. I start from the point where I arrived, okay? So, this property has to be somehow homogenous. Homogenous with respect to X0, okay? That property, this property of X0. But it is, it is, because indeed that what I say we have something bigger, because if we take little bit X0, then it means that F change just little bit, okay? And if we have a forgiven X0, we have a qualified controllability, then we have a for any, for any point, for any X0, okay? At least when we start from compact, we can cover by neighborhood and things like that, okay? We have qualified, solid or qualified controllability, which I mentioned implies that it works for any X0, okay? From a compact. So, look, we, now we understand that for any reasonable distribution of controls, we arrive to a point, if we really solve controllability problem, not just to claim something abstract non-checkable, set theoretic, that the map is surjective. If we really prove it, we got this, then, then, then what happens? So, we arrive to a point with some very small probability, starting from X0, starting from X0, we obtain X1, the small neighborhood of this, of X1 with very small probability, with probability mu distribution here is mu of that neighborhood, this one is epsilon, very small. So, very small probability, but still, it is separated from zero. Probability to arrive to such a neighborhood. But then we continue, we repeat our game, starting from here, and let us look what happens. So, if we arrive here with probability epsilon, it means that we do not arrive, mu of f2 arrive, arrive in O of X0. And probability, and probability mu of the whole guy minus O is one minus epsilon. It is a little bit smaller than epsilon. But then we repeat our game, but we do it independently. We do it independently, it means that for time two, we look the same neighborhood. For time one, do not come here, it is minus epsilon, it is probability minus epsilon. Then do not come here, for time one and for time two, it gives you, it is a time one. Time one, time two, time equals to one, time equals to two, that is a minus epsilon in square, just independence. We do not arrive for time one, and neither for time two, it is a product. Then we continue for time equal, and it is minus epsilon in power n. So, what we see, that eventually this probability, if we keep going long, long time, but our idea to do it up to infinity, probability not to come here is goes to zero. Absolutely, earlier, later, we come to this neighborhood, and if you have a controllability property, it means that we come to any neighborhood. Earlier, later, we come to any neighborhood, and with some positive probability. And this is essentially, almost all you need for mixing. Because look, we wait for the moment when we come here, but then still, if it is very long time, still it is, we have infinite time ahead. So may we repeat, repeat it, and at the end it is quite classical in the mark of processes, mark of chain, that everything is mixing. Due to that, first of all, we arrived everywhere, that I already showed, but since we arrived everywhere, still in finite time, we repeat this guy, which travels everywhere, infinite number of times, at the end everything is mixed, this is a basic property of mark of chain, at least in dimensional, in finite dimension, it's okay. And infinite dimension, as I mentioned, the main problem is that we still simply do not have this. We do not have complete controllability, we do not have enough resources. So that is, I would like probably exactly to start to explain why mixing is so close to control. And at the end of my course I will not use absolutely stochastic language and mixing, this remains to Professor Kuxin. And I will just study this controllability problem, controllability problem, but not only actually, in particular, in finite dimension this normal controllability is enough. In infinite dimension we have some intermediate step, that finite dimension of projections, because only, not solution itself, but only its finite dimension of projections may have this property. And then, thing starts to be more delicate, because when dimension grows, estimates become worse. And it starts to be very important not just to come here with some, but to come here in the best possible way. So here optimal control problem comes again and I will discuss also optimal control. In this language, how I can explain optimal control, optimal control questions. Two basic questions of control theory, controllability and optimal control. Contrary control used for applications, used in a lot of, for many things, but let me explain how it appears here, why it is getting to be essential here, here also in the, in the, in the, not without, without any technique actually, look. So, as I already explained, the main thing is to understand that, that we come with some controls, close. As soon as we arrived at some point, with some controls close to, to this point, we will cover a neighborhood, but still, but then, when it is only fine, if we actually need a family of that, this growing and growing dimension, that property simply, it's not enough, it does not work. It starts to be not applicable. And what we need? We need to cover it by, when I told, I did not, for me it was all the same, which kind of control I used to cover the neighborhood, okay. And for applications to PDE, for me it's important that it is controlled not big enough. It's well described family, and that is not so far from UT. Not so far from UT. It starts to be very important for estimator, okay. And for that optimal control, optimal control is useful. Essentially, what gives optimal control? Assume that we can get point x1 from x2. Point b from point a is in the school, in the high school. And then, and we have a cost to say if our control is good or not. And optimal control try to minimize simply the cost. Cost is maybe integral cost. In this situation, so we come, utility centers here, utility centers here, some utility say plus v, centers in the point nearby. Natural cost here in particular is integral of v square is just l2. You can build p on what we can really check. On a particular LP. And we try to minimize guy which sent us nearby. And there is a good, we can characterize for, as usually, it's a kind of generalization of calculus of variations. So you can characterize streamals, so we have one point, another point, and we try to find a corresponding trajectory such that that is as cheap as possible. Perturbation as cheap as possible. We can characterize it by some differential equations. So there is some calculus which extends classical calculus of variation. And this is actually optimal control series. I also discussed about it, what can be done. So this is what we plan to with. And now let me start. So as you see, as you see, the main here is the endpoint mapping. So which is quite rich. You see that it is not, it depends on the family of dynamics or even dynamical systems quite rich subject, but this one also. Let us consider, and I immediately, and at the beginning to explain some basic things and some maybe nice geometry, I restrict myself to the not even a fine, but linear case. So I consider the situation when f u of x, so we do not have dynamics with u control. When control is zero, we just stay. f u of x is just some of u i f i of x. And here the theory starts, and that so let's we we start at point x zero and we immediately would like to see where we can go, which direction we should go. Interesting things also in control theory it's what control theory is about when dimension of of m is case much smaller just smaller, much smaller than dimension of m. If dimension if k is a dimension of m and f i, f i let us assume linearly independent, locally at least just for simplicity, that may be dependent at some point. And if dimensions are equal, of course we can go in any direction controllability is obvious because u i f i give you combination of f i give you all kind of velocities, you go in any direction. Everything is fantastically good and interesting cases are when it is smaller and when applications to p d e interesting cases are when case finite but what I say dimension of m much bigger in p d e is infinite it's still we can manage at least finite dimension of projection let us look where we go we can go just write this equation we use analysis let us use small control f i of x put here epsilon and one of f i of epsilon u is x 0 plus with small control we can write a principle term we can write a principle term and it will be it's very clear if control is principle term here is just average of f u of x 0 d t plus of f so we can write like this so this is x of t without any any remain the term but if you would like to develop to extend it it's just a writing od what is here it's a written od but still we already have epsilon here and x is epsilon close since velocity velocity has size epsilon close to original point that we can write it put here 0 of epsilon square as it was obvious expected in the first approximation we just go in the direction of linear hull of this guys what I would like to mention here that there are another way to do things in particular maybe there are two parameters here time and size of controls we may have with a not small control but for small time and obtain essentially the same if we write u epsilon u epsilon of t equals u of t epsilon and this is maybe not small then we get changing variables in od e we get more or less the same the same just reshkailing time reshkailing time I omitted this calculations we have the same it could be epsilon u that could be u u epsilon of dot and for time epsilon since we start to use the effect that we can select our controls we take u then reshkaile the same u the same values of u of t one after other but go faster go faster not in the first approximation in general we have then f epsilon I am sorry just the same just the same to go time one with guy u or to go time epsilon with guy one just reshkaile fine approximation we do not have much we have something trivial but there is something here now I give first introduction for that part of this lecture then I start to be more precise I will explain how things appear but now I would like to have example of result simple result but to demonstrate theory works then I will develop it in more detail in more details so what we go is average velocity in direction of average velocity but if average velocity is zero let us consider u such that now assume that of t dt is zero so if average is not zero just we move in the direction of average velocity it's not so much but clearly we may expect much better as a direction if in first approximation simply don't move now change little bit notation starts to be more more geometric average of u this is linearly depend on u this is linearly depend on u it is yes so if average of u is zero then also average of f is zero because f is computed in one point we go in the direction of average and the initial point what we do in average of u the idea is to to see some laws where we go in terms of u so let me consider the curve gamma of t just primitive of u just primitive so that property means that gamma is a closed curve not closed curve in the space of controls moreover now I would like to specify more because my idea now to show not to drive lectures to show something geometrically interesting rather simple but geometrically interesting immediately so gamma is not closed so we go if f then we go in the direction f of gamma of one because integral can be taken here you can write this is gamma of one simply and if gamma of one is zero if the curve is closed gamma of zero and then where we go I show it's a simple exercise that you can do even by yourself then where we go then I claim that it follows that f epsilon of u epsilon is we have to compute second form that is quite easy here quite easy here and this is a epsilon square epsilon square this gamma let me now specify assume that k is 2 I would like to study just k is 2 but what I say at this level there is a nice generalization but I will not much more notation of what I say what I am going to say for any k assume that k is k equals 2 then what we have so this is a plane curve this is a plane curve and this is a domain curve it covers domain curve what I have here is a and it is a very simple exercise perhaps omit we get area of gamma multiplied by f1 f2 of 0 plus of epsilon curve so we go slow but we go another direction this is a I should explain perhaps so this is a free bracket so f1 f2 in coordinates f2 in coordinates just df2 df1 minus df1 over dx f2 of x, computed at point x0 computed at point x0 so this is what is called bracket commutator of vector fields so we have something new if field do not commute may recall it will be very useful for me also more formal definition of commutator why it is commutator and again I leave with you a sexy size we may think f is a vector field so at any point we attach some velocities to go we normally since in all text book and analysis it's immediately explained that we can think about it as first order differential operator when we apply linear differential operator this non-linear field is actually linear differential operator which are x like that if we have a smooth function say cinfinite of m then fx on a it's just definition fx of a fa the other function is directional derivative of a in the direction of f of x is a dxa f of x that we know then since it is linear operator we can we can we can we can iterate it we can write f1 f2a just a composition operator that is second order operator that is second order differential operator we can also write f2f1 this is another second order clear that when we apply f2 we get the first derivative of a then for the result we apply f1 for result we have second derivative here we have also second derivative but commutator that is everyone knows but it's not occasional this is a peos commutator and you see that nothing else could be appear here because since it is a we have some expansion of some we have epsilon of epsilon x a cure it's a cure in m its expansion of the cure starts from x0 second term is is a tangent vector must be tangent vector and it appears some combination of derivative of f1f2 it's only one of this you can expect it you can expect it so then f1f2 it is actually if we treat these guys as separators it is f1 it is indeed commutator f1f2-f2 f1 it is indeed commutator so now let me ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... И потом, может быть, я скажу сразу, что это не может быть саунд, потому что это может быть различные саунды, конечно. Это саунды. Если мы обмениваем ориентацию, то саунды меняют. Здесь и здесь, и здесь, и здесь. Не обмениваешься, а просто обмениваешься. В результате, это должно быть то же самое. Это саунд. Этот пацан воритый и воритый. Я не помню, что это, может быть, саундово. Но это не важно так же. Это саунд. Так что, в особенностях, если у тебя курица выглядит так, и ты идешь в эту сторону, начиная с 0, саундово. Это дает тебе плюс, и дает ориентацию в opposite way, и дает тебе минус. У тебя это важно, потому что мы не хотим остаться с саундами. Мы хотим идти вперед. Мы хотим идти вперед. Так что это саундово. У нас уже есть что-то делать. Если мы хотим идти, то всё. Мы берем абсолютно любую курицу, так как эта саундово есть позитив. И ты идешь здесь, если саундово идёт в opposite way. И ты идёшь в эту сторону, и эта курица может быть totally independent. Если вегтофилдсом умеет, то мы не берем любую курицу. И даже если ты хип, то вы всё равно не берёте курицу. И это важно для тебя, потому что это не очень тяжелый интерсайс. Если вегтофилдсом умеет, то это очевидно. Тогда это не важно, где ты идешь вперед, где ты идешь в следующий раз. Если вегтофилдсом умеет, потому что ты можешь его менять. Это означает, что результат... Многие Dinge могут быть объяснены перед calculацией. Результат зависит только, когда ты идешь в 1, в 1, и как долго ты идешь в 2. И не из-за сфистикации. Ты можешь выиграть любое, но вегтофилдсом умеет, и ничего не изменит. Результат зависит только от оборудования. Потому что всё это дает тебе. Если они умеют. И потом... Мы не можем это продолжить, потому что... Для контролации серии, это просто тривелый случай. Если они не умеют, то это невозможно. Если они не умеют, то это невозможно. И мы, конечно же, у них есть. Так что... Некоторые люди дают тебе... Некоторые, как это, дают тебе мгновение, но коэффициент это другое. Коэффициент зависит... зависит от оборудования, от оборудования, от оборудования. И... Окей, конечно. kasura, мы хотим максиминière система. Илиuned. Это оборудование. Я думаю, что мы хотим установить эту сторону. Мы хотим его adherence к тур Joan Erichon. Мы хотим Pierre Chamotект wisee. Так как кюфа-шота. Мы получаем много кюфов. Кюфы-шота. Кюфы-шота. Мы получаем, если мы считаем салюты изопериметрической проблемой. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Кюфа-шота. Для того, чтобы минимизировать интеграцию u of t, u of t minimizes its absolute value here. Для того, чтобы получить given area. Квивалентно, мы можем фиксировать ленту, мы можем использовать... У нас есть какие-то лимиды в качестве контроля, мы можем взять большой ленту и попробовать как можно и в направлении. Так что мы максимумим гамма с фиксированным лентом. Это та же проблема. Это периметрический проблема. И вы знаете очень хорошо эту soluцию. Солюция — это цикл. Если гамма с фиксированным лентом, это дает вам лучший коэффициент. И вы можете... У меня есть стандартная вещь, чтобы реализовать, что тратить сквей. Сначала один, а потом второй, а потом один, а потом тратить. Это лучше в любом смысле. Лучше, чем с фиксированным лентом. Во-первых, с фиксированным лентом. Возможно, один тратилет. Не именно один тратилет, но ворс коэффициент. И также, твой контроль — это нон-смой, но это не будет. Так что вы должны использовать цикл. И это очень просто. Больше интереснее, математически — в следующем случае. В следующем случае. Я надеюсь, что... Как это сделать? Итак, мы уже знаем, как идти в F1, F2, и даже иметь эффектный контроль. Если мы идем вперед, мы идем вперед, и мы можем повторять, повторить. И мы знаем, что делать. Приметив гамма, контроль, просто ротировать. Ротировать, ротировать, ротировать. Это контроль. Если F3 dimension, это F1, F2, F2. И если... Это F1, F2 — экстрадирекция. И когда мы... В контролье, мы просто идем вокруг цикл. Ротировать. Но в реальном случае, в контролье, в реальном случае. Но в реальном случае, в реальном случае, мы идем здесь, и идем так быстро, как можно. Мы получаем спираль, как это. Мы идем слой, надеваем спираль. Это очень простая пикция. Но уже на этом уровне, вы знаете, что действительно у меня есть часть этого курса. Так что, это важно, что эта law of efficient movement is universal. It does not depend on your vector fields. You find in the control space the best possible law. And then just plug in your system. So, somehow, good control theory is done without control system. It's not yet done, by the way. It's maybe just a dream for the moment. Without control system. We simply plug in any control system, but the laws of good behavior are always the same. If we do it with small controls, of course, we need some small parameter to do it. Or you do it for short time, and then repeating, repeating, repeating. But now I would like to discuss also next case, again with k equals 2. s k equals 2. Next case. So, assume that now that area that area of gamma is zero. So, gamma is something like that. Balanceria. Plus, and here it's not really balanced. The picture. Plus and minus. Balanceria. Then f. Then we can write some other expression of gamma. That is not area. It's a kind of momentum. I will not give details. Later, later. In fact, I can do it. In fact, I can do it. What we get here. What we obtain here. Without geometry so nice in geometric interpretation is here I do it. f epsilon of epsilon. Then now, of course, starting point. Starting point. And plus again, you know, epsilon cube. Again. Since now principle term is a cubic that principle term is just tangent to Yokeov. Yokeov, again, it starts from epsilon cube. But if you reshcale it changes parameters. Take epsilon cube as a new parameter, then it will have derivative. And derivative is a tangent vector. You can take epsilon cube as a new parameter. Derivative is a tangent. It must be tangent vector. It must be valley on some vector fields. And this vector field is cooked from f1 of 2. No other way. Again. Expansion is universal. f is just symbol. And here you can write it. Let me write. You want. Then you can interpret it also. Geometrics that I will do. But what I write now is a kind of expression which you can continue. You can write for any. It will be similar for any order. And here you have you have you have this is t1 this is t2 so you integrate over the simplex. And here we get control. So I can write it in terms of gamma. f of gamma 2 gamma u of 2 gamma dot of 2 say because the result depends only on gamma. Result depends only on the gamma. Let me even write. OK. But you see the result does not depend on the parametrization. That's what I would like to say. If different controls are just the parametrization of the same geometry is here also. The parametrization of the same curve that result will be the same. Depends only on gamma parametrization. Gamma of t2 f gamma dot of t2 gamma dot of t1 f gamma dot of t t2 d t1 dt So t2 is integrated until t2 t1 is integrated until t1 Essentially such a such an iterated integral that also first of all since we have a triple integral here and derivatives here then indeed it does not depend on parametrization of gamma and again we at the end what we obtain. At the end we obtain a point in the plane where we draw the plane so we started in the situation the span of f1 f2 f1 f2 and now let us draw the plane we have second order brackets and we have two second order brackets actually we have f1 f1 f2 only two because brackets are anti-symmetric we have only two f2 f2 f1 so here argument must be different because we are anti-symmetric and the first order bracket is just one but here we have two and we appear when we apply our control indicated by gamma we go in the direction of this plane we consider this plane we have arbitrary dimension manifold arbitrary dimension manifold and in this manifold we have a kind of plane I would like to show that it is not square on the plane but it is some plane which is not that one first order bracket we know how to go and we would like to go to next next direction that depends on the second derivative it is the iterated bracket so here say we have two guys f1 f1 f2 and another one f2 f2 f1 full orientation is controlled we will not give details because otherwise it will take too time and hard to follow I would like to explain just basic idea and result so we have a plane of controls commutative plane if you want u1 u2 and we have a linear map call it AFC that send u1 u2 into u1 v1 it is other plane because it is not v1 it is not control really v1 f1 f2 plus v2 f2 at this level we have a more direction to move in the first bracket v2 f2 f1 again just one direction here we have the whole plane and to go in these directions we apply gamma like that with balanced area the question is again the same try to find universe low which gives you better coefficient which with limited length of the curve give you best coefficient you have a nice variational problem universal problem it does not depend on f and it is not so classical it is not a parametric problem anymore it is more on linear more involved and it can be solved and you immediately try to continue and it often happens in mathematics if you go to the next it starts to be too complicated but at least at this level in this level you have a very nice solution so you understand that it should be something like that 8 shape because if you have a more complicated guide we cannot be best it 8 shape it indeed 8 shape because we can rigorously solve these problems in detail shape moreover you see 8 shapes kind of gauntel guys are always done with the fact that we have two dimensional results depends on the orientation you can rotate it if you rotate it it is parametric family but they are absolutely equivalent one getting from the rotate it is not like circle it is rotation and variant due to the fact that it is rotation and variant you can just one result and here and what is the best the best solution is 8 shape earlier elastic the unique best solution is 8 shape your earlier elastic earlier elastics are found by earlier equilibrium of elastic road it is in all text books on elastic theory one of the very few maybe unique one non trivial example which can be computed up to the end but they appear in other very often appears in optimal control problems essentially you see why as soon as you need second order bracket you need something like that maybe a little bit sorted but something like that and they have a nice geometry characterization actually in order to not describe equation for elastic they have very nice geolastic gamma is elastic curve gamma in the plane elastic curve that result does not depend on parameterization this property of the curve if and only if if and only if curvature curvature is a plane curve of gamma at point x is a fine function of coordinates curvature is a fine function of coordinates gamma is gamma is say V1 V2 and the curvature of gamma is alpha plus a1 gamma 1 or better I cannot need to write this one curvature is alpha plus a gamma curvature of gamma at any point at any point of gamma where a is a fixed vector constant vector and this is constant so you see you see what happens for first order bracket just if there are no bracket if we would like to go to the direction of linear halve 1f2 of course we go the best way to go along straight line shortest way straight line and the solution is curve with zero curvature if you go to the second first order bracket the best way then we need close curve and the best way to take curve with constant curvature of course they are all special in such formulation they are all special cases of elastic but degenerate most degenerate if you go to third order bracket then you already you use only gamma with a balanced area and among them the universal independently of f the best one is 8 shape elastic elastic there are a lot of them but only two periodic circle of course and this one then and with inflection point looks like that and among them there is exactly elastic here you find a circle when you do not move it always repeats as one of possibilities and here you have 8 indeed if it does not move there are two types of elastic they are related also to pendulum equation and these are two they are related to pendulum so another characterization of elastic is that if you parameterize your curve by the lens so your velocity is always in the circle has absolute value 1 then the curve is elastic if and only if this angle is satisfied pendulum equation standard pendulum equation another characterization classical characterization this is with particular parameterization and non-linear this is total mathematical pendulum theta dot this equation where theta this is a gamma dot and this is angle theta and we have for pendulum equation we have two kinds of behavior it may oscillate when velocity is small or it will rotate when it oscillates according to the definition this is the velocity change direction so this is second derivative change sign and this is second derivative change sign and this family and this family corresponds to rotation like that with big wheel that is what concerns elastic my time is almost over maybe maybe I have 5 minutes maybe I start it was preliminary как я уже говорил но конечно нужно продолжить взять высокие брыки и рассматривать керфы так как керфы это полинома керфы но потом и это полинома и это и это и это это это это это это это это это это это это это это были Это уже для рецепта ж Wouldn Если вы считаете, что кей больше, то это не кей, но я только интересен в первом рейке. Вы получите вырезанную soluцию, очень приятно. Это аналитическая и интригонометрическая функция. Это тоже комплектаризация. Также спешевые кейфы, которые очень-очень вырезаны. Они изменили чебушев. Они тоже имеют другие extreme properties. Как линия? Один один. Один один, но в любом дайменте. В любом количестве контролов. Нет-нет-нет. В следующем дайменте, в первом дайменте, даже в двух дайменте, вы можете найти ваше время для очень специальных рейтей, чтобы получать хорошие кейфы. Потому что потом вы должны быть более балансовыми. В частности, что хорошо сделано, это рейтей, который вы получите, используя хайпер-элиптические кейфы. Так что я не хочу вырезать их. У них есть хайпер-элиптические кейфы, которые имеют более что-то подобное. Так что вы получите много балансов, и с хайпер-элиптическими кейфами, вы получите высокий рейтей. Но очень специальный. У них есть кейфы, у них есть плюс, минус, плюс, минус. Так что вы можете убить 3-3-4-3-4-3-4-3-4-4-4-4-4-4-4-4-4-4. Но у них будет только очень-очень специальный рейтей. Так что для потеки и лосистема нон-комьютатив, это называется чейн-система, когда очень-очень специальные рейтей. Есть много симметрии, а нон-зиру, это работает тоже. Но универсальная пикча начинается быть очень-очень специальным рейтей. Может быть, я перестану здесь, потому что это естественно. И в следующем лекции я буду старт регулировать ревеллобменты этого конкурса, и Introduce some other tools в стадии контроля медипров.