 не для всех лекций, но для основной части этой лекции. Я не оценю, как для любого уюта смузной вектории, но я не оценю на природе уюта, и это просто индекс. Это просто семейный параметр. Если в таком генеральном состоянии, я рекомендовал, что контролл-фаншансы — уфт, уфт, уфт, уфт, уфт. Это варьи уфт. Для генерального состояния мы можем использовать только письминсменный констант. Письминсменный констант — это всегда сочетатель. Для нас контролл-фаншанс будет письминсменным константом. Это очень важно. Письминсменный констант, письминсменный констант, не фиксированы. Вы можете выбрать любое, что вы хотите. В принципе, если вы имеете, скажем, кондиционную зависимость на вас, то вы можете опроксимировать, читать любые функции. Да-да, это не... Это неплохо, из-за того, что у вас линия-фаншансов. На самом деле, это странно. Это фаншансы, но очень хорошо здесь. Если вы изучаете контроллабильность, то, что я буду делать, то это достаточно. Результаты для этих функций будут одинаковыми. Если вы считаете, что есть хорошие зависимости от вас и большие функции, это очень большой технологий. Это будет просто та же история. И, конечно же, для оптимального контроля, perhaps в следующей неделе будет деликателема для оптимального контроля. Вы не можете делать что-то на этом состоянии, просто потому, если вы пытаетесь минимизировать какие-то стоeedы, минимизировать секвенцию, и ваши письминсменные функции Вы не можете найти оптимальный контрол в этой классе, обычно. Поэтому это мы будем needing. Исцентрально-континентно-депендентная ассампция. Но для контролабилития просто феймеров объекта фильтров. И это худ, худ-синх феймеров объекта фильтров. Мы полагим это здесь. И мы получим троектор. Мы начнем с первых точек. Q0 equals Q0. И мы получим, что мы называем, endpoint map, который дает Q of dot into Q of t U of dot. Которое, которое соответствует каши-проблем. Солюция, уникальная soluция каши-проблем. Как это сделано? У вас есть несколько динамичных систем. Вы пойдете однажды на одну систему, потом выключите к другим, выключите к третьему, и вы выберите свои выключения, что-то, что вы хотите. В любом сегменте. Так что это статистик. И мы считаем, что это оптимальные сети. Оптимальные сети. А, скажем, Q0. Может быть, я не был очень precise in formulation of my series. Q of t is simply, simply point. Мы начнем с Q0. И мы получим, что они будут быть в этой сети, в письмах, с траекториями. И мы получим, что, используем много, мы можем выбрать очень сфистикательный контроль, в письмах. И мы получим все эти письма, и кринерсериум, который я формулировал, действительно, был для того, для того, что я сказал. Поэтому, когда мы не фиксируем exact time, это та же история, но Q of t, Q of dot, 0, my formulae, I recall it now. И, конечно, очень similar result can be stated also for t fixed, but condition is a little bit stronger. Later, I discussed it, maybe it's exercise. Conditions for fixed, it's almost the same. Но это должно быть странно, если бы вы хотели... Итак, теория. Сначала Ли... Ли... Я defined... Ли-Квизиру... ...Оф-Ф-Ю... ...От этой семьи. Это просто... ...Ли-Ня-Хал... ...Оф-Фол-Ли-Брэкетс... ...Оф-Фол-Ли-Брэкетс... ...Спэн... ...Ф-У-1... ...Ф-У-К-Оф-Квизиру... ...Со-Ок... ...Всё. И что я сказал в этой ситуации... ...Сериум говорит... ...Сериум. Для любого позитивного Т... ...Я сомневаюсь, что... ...То Ли-Квизиру-Ф-Ю... ...Это субспортом в танженном спейсе. Веллифт сфэкта-филта... Веллифт сфэкта-филта... ...Веллифт сфэкта-филта-филта... ...Станжент-фектор... ...И это субспортом субспорта в Генере... ...Суммается, что... ...Нес... ...Сиум... ...Ли-Квизиру-Ф-Ю... Люди говорят, что наш системы ракетируют, а потом, а потом, интерьер, я могу выхожу так вот, а, а, q0, smaller than t, это содержится в кложе интерьера. В частности, интерьер не сомневается. В кложе интерьера или в кложе интерьера. Но не сомневается и любые моменты можно вывести из интерьера. Окей. Так, и я начну показать. Это, в моем мнении, это кринерсериум, это кринерсериум, я думаю, это простая и элегантная обстоятельность. Окей. Так, я начну. Во-первых, это сэффективно показать, что мы сэффективны в следующей ситуации. Основная вещь, Основная вещь, Основная вещь, Основная вещь, это так. Так что q0 содержится в клозе. Это эффективно показать, что initial position содержит в клозе в клозе в a2z t и это, кажется, то, что это ни у ole t, ни т. п ни positive t, а то, что очень маленькие. Дай, сейчас, почему это так, Ассюм, что мы знаем, что это Q0, и мы говорили, что струкция может быть здесь, но всегда полная дименция. У меня есть маленькие улыбки, а это просто кложа. Кюзировка не должна быть. В моей пикче Кюзировка не должна быть в интерьере. Кюзировка должна быть в кложе в интерьере. Все истории о кложе. Семь сезон, этот человек, это эквивалент, чтобы сказать, что это так же. Потому что в кложе есть то же самое. Потому что у наших траекторий без своей смыслы. В кложе это так же. Это очень маленькая траектория. Это очень маленькая траектория. Это очень маленькая траектория. Это очень маленькая траектория. Теперь, Assume that 4 times smaller than t, we can reach point Q. Q belongs to A of smaller than t Q0. We can reach it. There is a trajectory. I don't know how it goes. Certainly it goes inside this set. Because it's admissible. There is some trajectory, which comes here. Look. What is the crucial point about control system? You could think about control system of a priori. If you would like to generalize and forget all unnecessary structures. You can think about differential equation. So you consider, you have some restriction. Velocity at any point belongs to some subset. You cannot go too far in this setting. It's very important that this is a family of smooth vector fields. Why? And we always do like that. In controllability, if to have a working efficient theory, we need the following fact. We need the following fact. That if control is fixed, as soon as control is fixed, we don't have only a trajectory. Yesterday, formal calculation, I also used it. But we have the whole flow. We have the flow. We have a family of defymorphism, which sends any initial point somewhere. It is quite good, smooth with respect to initial data. It is in our setting a smooth with respect to t and smooth with respect to initial data. It translates everything ahead. And that's it. What we do? If we have one trajectory, which comes here, then we have the whole flow that comes here. And we do the following. We have a very small time. We know that we can reach this guy by some sophisticated switching. We can have this neighborhood, including Q0 for a small time. And then we apply the flow for prescribed time. This belongs to some tau, where tau is smaller than t. And we go a little bit bigger time, because first we apply the small epsilon time, and then we get the whole neighborhood. And then we apply the flow. And flow is just a change of variables. It's transform. And the result will be, OK, maybe write it more precisely. So assume that Q is a Q1. Say this Q1 is a Q of tau, U tilde of tau, of particular control. And we consider the flow, Pt, that sends Q of 0 into Q of t, inverting of differential equation, inverting of this differential equation. And then I claim that then P, this is defamorphism. If M wants to defam, I need a chalk. Then I claim that Pt of A epsilon of A Q0 of epsilon, small epsilon, if you want, is contained automatically in A Q0 of tau plus epsilon. Это автоматически, because I can change this switching as I want. First time epsilon I do to realize this guy. And then transform all this set by prescribed control by U tilde. We can also write Pt, if you want, because control is U tilde. So that is автоматически. And that's all, because now our Q1 is the closure. Clearly the closure belongs to the closure of this set. So the real question is about very small time. И это правда, что мы можем найти интерьерпоинт, интерьерпоинт в оттеннице. Орбитрия очень близкая к Q0. И здесь, очень простая сегмента, которую я пытался объяснить. Это один из этих сегментов, но когда вы пишите все сегменты, вам нужно очень много символов, но сегменты очень простые. Я пытался сделать это по словам, для того, чтобы сделать это очень claramente. Это действительно очень clarifiably. Вот мы делаем это. Это как раз индукция. Как вы видите, мы покажем больше. Мы покажем больше, чем этот факт. Что-то более важное. И это удобно сейчас, чтобы definить, что это действительно легало, чтобы definить сolution of time invariant equations. So, that's an autonomous equation, Q dot fu, where u is fixed, is constant of q, flow generated by this equation. I denote by EETF, its balloon is the defiomorphism, its mm. Like in the case when f is a linear vector field center, and it is exactly matrix exponential, nothing else, but it has good properties. It's a good idea to define it like that, to denote it like that. That's one parametric subgroup. And here EETF sends a Q of s into QT process to show that it is indeed one parametric subgroup. But it's general notation for any vector field, but in our case we have index here. So it's convenient to do like that. So, let us start our induction. First step. There exists a vector field, so we assume the bracket generating condition. First step. They exist at u, fu, such that fu. One say, they exist. u1 in u, such that fu1, fu1, q0 is not 0. If all of them are 0, then all brackets are also 0. We do not have bracket generating condition. We also say, because we generalize it to higher dimension, that if all of them are 0, then all of them are tangent 0 dimensional. Submanifold, that's a point q0. And we know that if vector fields are tangent, some submanifold, then also bracket tangent. We generalize it, it is just a starting point. So there is at least something, it's not equilibrium for all. Okay, fine. This trajectory, e, e, say, t, fu1. This trajectory, small piece of trajectory, orbitary small piece of trajectory, in the blackboard, it is small, I drew it like this, but it is supposed to be small, e, t, fu1. Then it is one dimensional submanifold. It is a smooth one dimensional submanifold. I claim that there exists a point. For us, it's important to go overhead. We take it for positive t, positive and small t. Positive, this is q0, positive and small t. It's one dimensional submanifold. So if our original manifold was one dimensional, we are done. It has interior point. If not, then we know that there is a point, there is some vector field in our family that does not tangent to the submanifold everywhere, even for small piece of it. There is a point q, this is say t1, such that, and the vector field fu2, such that this is a point e t of q0, t1 of q0, such that this vector field is linearly independent. If this vector field is linearly independent, let us cook two dimensional submanifold, considering the map, say how to go, s1, s2, e s2, f2, f2, e s1, f u1 of q0. Where we take, we take, look, what we have here. If we compute derivative of this map with respect to, with respect to s1, at this point, where s2 is 0. We have another field here. The derivative of this map with respect to, call it phi, phi2, say, d phi over d s1, clearly at point t1 0, t1 at this point, is of course f u1, evaluated at this point. And d over ds2 of phi at the same point is, of course, f u2. We call this point q1, for example, of q1, of q1. So, they are linearly independent. These two fields are linearly, these two derivatives are linearly independent. So, if we consider image of phi on the small neighborhood of this point, not the small neighborhood of 0, but on the small neighborhood of point t1 0, this is the immersion, because it has non-degenerative derivative. This guy is linearly independent. It's immersion. So, we get a two-dimensional manifold near our point. So, it's orbitally close, because we could start from orbitally small segment here, orbitally to q0. So, we have a two-dimensional, piece of two-dimensional submanifold at our point. Absolutely, absolutely. That's a picture that often happens. No, starting point, no, because we always go ahead, we do not assume any symmetricity, we cannot go back. Very soon I explained that if we can also go back, then we have complete controllability. But if we go only ahead, not in general, of course, there could be, yes. No one field is zero, and all could be lead in the same, more or less in similar direction, close, but, and then you cannot agree it on. This I also explained. So, we have this two-dimensional piece of manifold, and again, if our original manifold was two-dimensional, we are done. If it is not two-dimensional, we continue the same story. There is a third field, there is some point in this two-dimensional sub-manifold, and third field, which is linearly independent, we build, we cook three-dimensional sub-manifold and continue. So, it looks very, it works, it works, it is a result. But now, think about it little bit more detailed. That's important what I say now. Because actually starting from this third step, situation little bit different. So, I not really cheating, but it's good to see that apparent contradiction, which is not contradiction. Then, assume, assume that that it is so generally that that you that you cannot have just one element, you capital, but two elements is fine. For generic couple of vector fields has bracket generator property. If you take any couple of vector fields and perturbed them a little bit in C infinity topology, it becomes bracket generator. And, okay, if we have just two vector fields, then FU3 must be one of these two. So, if you take FU3 but but at this point the trick is and it looks like a contradiction. Our argument is totally correct. But at this point at FU3, of course, linearly dependent of if you have just two vector fields, one of them. But the point is that tangent space to this sub manifold is not linear half of F1F2. At the starting point it is like that but if you differentiate if you differentiate this for S tangent here we built this map of F1S2 and tangent space to the sub manifold is just linear half span of d phi over dS1 d phi over dS2. Okay? But the point is that differential of this map with respect to S1 is not FU1 if you compute this differential when S2 freezes U2 0 nothing here and differentiate with respect to S1 then it is S1 otherwise the differential of d phi phi over dS2 is of course FU2 but differential of d phi over dS1 is E S2 F2 star applied to F1, okay? and corresponding point. This is not F1 anymore but this is a tangent to our sub manifold and we can find field which is transversal at some point and built d phi 3 S1 S2 S3 F3 FU3 S2 FU2 S1 FU1 and according to our construction to our choice this will be immersion so derivative with respect to S1 S2, S3 are linearly independent Even in the case of just two elements set here we have again U1 but derivative with respect to S1 is not F1 That's a picture and this is the end of the proof we may continue and so on and look that actually we have more and this is important more than the fact the fact that we have non-empty interior because to prove that we have non-empty interior we used actually implicit function we used we built a defymorphism that sends some family of controls some family of controls by the way just we need only n pieces we know that with n pieces of n pieces of controls we obtain the map we see that what we proved what we proved we have proved let us analyze it let us analyze it we proved that we found family of controls U S of dot this belongs to s1 sn n is the dimension of our manifold it's also sometimes useful that number of pieces is just the dimension of the manifold and such that the map the endpoint map F such that the endpoint map better to say the map which sent s1 sn into into at the point sum of si U S of dot call this phi such that this map has a regular point in the positive or tanned orbitary close to zero it has a regular points orbitary close to zero that's very important somehow endpoint map any point at least in this setting almost any point in the image of the endpoint map has a regular preimage it's much more than to say that map is open at some point it has a regular preimage by the way already here you see that this family we constructed for our map it is perfect if we compute solution we have a smooth map but from the point of view of the classical functional analysis it is not so good because when you move switching points it is not so nice if you work in the space of functions of controller space of functions some one spaces some you do not have smoothness here it's not smooth that's the end of the proof now corollary corollary I assume that we can go forward and back f of u equals minus f u we can go in both directions forward and back then and we have a break a generating condition break a generating condition so li q for all point let me do it now for all points because I would like to have a global corollary q f u is t q m then I claim that a call it a infinity maybe we do it locally first we do it locally just take one point and then global global result is obvious corollary then q zero is contained in the interior what people say that we have a local controllability it could be many photos very big dimension it could be interior I did not finish it I write of smaller than epsilon q zero it could be just two vector fields as I mentioned and many photo-forbit really dimension if we are break a generating actually it means that we can go immediately in any direction but for that it is essential then we will try to weaken this property that we are symmetric and this is quite easy and moreover again actually we have more and this is important that we have more actually q zero has has regular of the endpoint map q zero is normally any interesting system q zero is of course belongs to the critical level of the endpoint map because it is critical if we if we then I show but it has regular preimage arbitrary close close to zero and then this is the main idea of all controllability result actually in the preimage we can choose control arbitrary close time to arbitrary close to zero always constant control yes such that we come in our points and we have perturbed switching then differential of this map which just perturbed switching is full non-degenerate, it's full dimensional here we need two end switch again we can estimate number of switching here we have two end switching and it is again extremely simple again we need that as soon as we have control we have also not one trajectory but defymorphism this is extremely simple let me show it so indeed indeed ok ok so we may since we are symmetric let us consider consider system reverse time if we go back we are still admissible if we reverse time we can consider system q dot equals minus f of u it is the same as to go in the opposite time so in the opposite time ok we have a q zero and we know due to bracket generating family from bracket generating property that we have good points here regular points here interior and that achieve regular point of the endpoint map but ok but now do it for system that we can do but if our system is bracket generating even without symmetry property without symmetry property this is this is this is also bracket generating of course so if we go backward if we go backward we obtain we obtain some a priori some some good points again ok some good points again and then what we do we obtain some good points no this moment backward is not so important I am very sorry go forward we obtain interior points take one of the interior points and some particular control u which goes to this good interior points regular points and this q is say phi we constructed this phi of s1 some sn and tilde very particular so that is that is that is exponential of some sn tilde f u n s1 tilde f u1 of q0 denoted this defiomorphism forgetting q0 denoted p tilde everywhere tilde this is defiomorphism this is p tilde this is defiomorphism m2 and consider composition of the map first you go and since now for the moment we did not use that the set of of vector fields is symmetric but since it is symmetric then also p tilde minus 1 can be realized by p tilde minus 1 is minus s1 tilde f u1 minus s sn tilde f u n so it is also can be realized but moving along admissible controls due to symmetry because we know that this vector fields are presented also presented so look and then what we obtain we obtain that q0 belongs to the interior and has regular preimage belongs to the interior of the map of the then we obtain obtain we consider the map p star minus 1 phi or s1 sn where s1 sn belongs to our neighborhood in the first order one sn composition of this map so first we go to some point and changing s2 s1 sn then we freeze them put them tilde one of those we freeze them and apply and apply go backward because it is now allowed we consider such a map such a map s1 sn s1 sn are variables here and fixed here but p as applied is worked p is a is a difammorphism this is difammorphism just difammorphism change of variables and it transforms neighborhood neighborhood around this point q tilde into neighborhood neighborhood of q0 if we consider this map for s1 sn equal s tilde it is just identity we don't move for s1 sn equal s1 s tilde this we come to point q0 and if we compute differential of this map at the point of s1 of s1 sn this differential is not degenerate because it is chain rule it is differential of phi product with differential of difammorphism which does not depend of s1 sn this difammorphism does not depend of sn so we got neighborhood of q0 moreover we get it we get it in a good way as a regular image ok so now it is to globalize this result if this property is satisfied at any point then and manifold of course is connected we certainly can jump from one connected component to another then we we can go whenever we want from any point we have manifold q0, q1 on the manifold and we can as usually we connect we connect on some curves maybe not admissible it is not admissible curve just connected property and then at any point we can find neighborhood where we reach any neighborhood this means that we may find a way maybe very sophisticated way to step by step to go from q0, q1 time may be long now this corollary is for short time we have small neighborhood but corollary is at that certain if if we are symmetric lq fu is the whole tqm for any q and fu equals minus fu then say infinity q0 for arbitrary time is the whole manifold for any q0 we have complete controllability people say that is complete controllability now if we analyze the proof you also have and this is important for applications to mixing this is really important we have more I always insisted that we have this regular way to reach whatever we want if we analyze this the proof we have moreover not only for symmetric situation but for any situation so that is sufficient condition to be completely controllable but let me formulate proposition where proposition assume that fu is break it generating and assume that and assume that we have controllability aq0 of infinity particularly in the symmetric situation it is like that then I claim that if we perturb little bit in c0 topology that may be surprising we still keep controllability in general we do not keep break it generating problem we still keep controllability because it is like that if you realize your a regular value of a function when you perturb little bit vector fields then in f0 new fields is again smooth but close only in f0 then your map is close in f0 and it is cover the neighborhood small perturbation application of browser still cover the neighborhood it may happen that it have critical points because we approximate only in f0 but it is cover the neighborhood so this is a c0 c0 stable property if we have controllability sometimes we say that we have solid controllability small perturbation does not destroy do not destroy in c0 do not destroy in az remark that assume that another very nice property of controllability it happens that in this setting in finite dimension but symmetric still still rather restrictive so I would like symmetric at least one sufficient condition for controllability we have but symmetric rather restrictive still our agreement gives something for any controllable system and I will try to give conditions for non-symmetric system that is still globally controllable but assume that we have global controllability another proposition or color is like that assume that we have breakage generating and proposition again is breakage generating and all points generating and also we have approximate controllability closure of iq0 infinity for orbit really time is the whole breakage generating plus approximate controllability I claim I claim that then we have exact controllability again in this very stable stable manner then we have this equivalence that what never happens in infinite dimension if we would like to use finite dimensional control and our manifold is infinite dimensional then there are no chance to get exact controllability but we can by approximation by finite dimensional objects often it is absolutely non-trivial with some analytic technique reach approximate controllability but in finite dimension it is important that it is just equivalent of course if we have breakage generating breakage generating is not before explaining one is true one is true I also say that approximate controllability breakage generating property is not of course necessary approximate controllability you may have if you have just one vector field and minus this vector field if you take a torus and directional winding of the torus just one vector fields you don't need even minus directional winding of the torus you approximately for big time but as soon as we have a couple of vector fields and breakage generating property it cannot be just approximate but automatically exact in a stable way but the arguments essentially the same very similar to what we did in symmetric situation so we okay we would like to get q0 we would like to get q1 from q0 there is q0 then we have q1 let us consider system starting at q0 with opposite time now I do not do not assume symmetry but still breakage generating of course because breakage generating is linear property consider the system q dot equal minus f u minus f u so we have some trajectory which lead we do not have some trajectory we just wish we have q0 q1 go guy with opposite time we do it in q1 with opposite time we do it in q1 guy with opposite time okay so here is our assumption our assumption that this proposition our assumption that closure of attainable set from q0 is the whole manifold okay in this point we have a very good map we have interior the whole neighborhood of very good good points okay that was our first in particular since attainable set attainable set from q0 here with every where dense we certainly can reach one of this point one of this very good point in the interior of attainable set for reverse system for reverse system here we take reverse system at q1 and since we have breakage generating property we have very good points on the reverse system and fine and that's all since it is reversed when we go from here with opposite time we we just move this neighborhood in a regular way but for controllability you don't need even that so since it is in this point is again we have open neighborhood of points reachable by reverse time when I go backward since attainable set from q0 with every where dense we can catch one of this point with positive time from q0 then go from positive time change time again and we come to q1 because that was for reverse system so we have this and if you look about it in more details maybe add one more piece we have this property but this we already proved in general so we have complete controllability means that we have complete controllability so that what does breakage generating property in finite dimension but now I would like to work with non-symmetric situation because it is important in natural system some dynamics some drift which does not depend now wheel may control by control we just perturbed somehow so what how to manage this situation first first observation that we can always the idea is like that so we have our we have our vector fields and we will try to add more vector fields such that closure of attainable set does not change related compatible with this one and if at the end we arrive to the symmetric situation then we have controllability this is a basic idea and first first observation which is a corollary of very general relaxation простигия if we have a few if we consider convictification of a few if it is controllable then also a few is controllable something like that if we have three fields three fields say breakage generating like that it is not symmetric it is not symmetric but convictification is symmetric and we are still controllable so this is this is actually follows from the general general relaxation procedure that I would like to explain in the way that you can apply not only to controllability because it's important and quite universal property of the endpoint map I explained it like that but to see why it happens it's not so hard so if you have two vector fields it's quite natural if you have two vector fields so we go along one vector fields we have a flow like that along other vector fields we have flow like that and now if we switch if we switch very fast from one to another we start from Q0 switch very fast from one to another then we approximate convicts combination convicts combination of trajectory just switch from one typical relaxation procedure so it is a clear but at that point it is good to formulate mathematically important general result because if you do it well you see that it is maybe applied also for PDEs it somehow gives you Gromov's Gromov's Convex integration let us do it for that I come back to to endpoint map analysis of the endpoint map so we wrote we can write now ODE flow now we deal with now we consider some very general not even control system but just our world I even do like that XT of Q where XT is time varying vector field and we have a map that can be can be written in the operator form as linear as linear equation so if we take p yes just this guy appears when we plug in control but result I would like to give general result I would like to explain deals just with non-autonomous as when we consider expansion of the endpoint map what is control? we have just some universal universal map that associate to any non-time varying vector field a flow that send time varying vector field into the flow and control endpoint map is just specification is for any control function for any control function it's just composition of two maps first for control function for control function we associate an anti varying map and this first step is trivial it's a point wide if you have u for control system if you have q dot equals fu of q we have actually first step is take u of dot into f u of dot u of t even it is point wise very simple map and then we take this time varying vector field and apply universal map main properties information and this universal map we control just specification to particular family of time varying vector fields so if we now consider pt the map which send send q of 0 into q of t this is a defiomorphism of this equation then we can write this equation like that equation in operator form and then pt we can treat ptx we can treat as a variable operator for scalar function if we have a now function just real function and m pt applied to m for us is a pt linear operator on the space of function vector fields is also linear operator on the space of function that we know directional derivative this is change of variable then we can write our system like that p dot we did it, but you can check again xt it's cache problem on the system of the group left invariant system of the group of defiomorphism if you want p0 is just identity p0 is identity we do not move so let us rewrite it as integral as integral equation we can rewrite it now as pt equals s operators since the linear operator I can rewrite sums whatever you want I get other operators from 0 to t p to x0 so this is the equation integration by paths defiomorphisms like that are quite legal in this setting behind c infinite defiomorphisms vector fields infinite defiomorphisms topology is a usual topology of uniform convergence of derivatives up to certain order on complex subsets it's a free shell manifold space if all operators have space so what I would like to show then this map we prove that this map that send we consider the map which send x dot p dot we prove that it is smooth smooth map if we consider topology for non-variance vector field we consider topology like that norm norm of x not hard to deal with topology norm of x dot it's just integral norm with respect to t integral norm with respect to t in usual k ck norm ck norms with respect to x x dot is l1 with respect to g but actually lp lp is fine here l1 with respect to 1 let us work of course of finite segment t from 0 to 1 otherwise we need more norms more complex also compact segments so x xk xk actually we in fact prove we used it for control system specification but all calculations for general non-autonomous vector fields we proved that this map is smooth if we use such a topology of free space and then we found the Taylor expansion more or less explicitly as operator now very important relaxation relaxation relaxation property this phi phi is continuous is much weaker topology in topology with norms like that topology defined by the norms with norms defined by the norms so we do not take a norm and then integrate with respect to t and then take the norm it's continuous in this topology and this is very important because in such a topology we may that approximation affect in such a topology if you if you jump from one value to another very fast then in average approach just combination it's not ODE anymore it's not differential equation that average approach it's a weak topology somehow if you switch very fast average approach linear combination okay convex combination I'm sorry convex combination but what is the what is the interesting and why I say that it is more can be applied also for PDE and gives actually actually convex integration if somebody heard about that because it approach not only a long trajectory but the whole flow I say that continuous is map which send vector field to flow so we switched very fast with respect to t and we approximate in this way if we do it in all points simultaneously we approximate we approximate the flow in CK all derivative in any compact approximate orbitary well so that's so it's a level of flows so we have two kinds of variables here time and initial condition for your ODE and you approximate uniformly with respect to initial condition convex combination with all derivative with respect to initial condition only uniformly without of course any derivative with respect to t but uniformly with all derivatives in the what converge what converge is this guy not values of derivatives but average converge and trajectory converge uniformly derivative not of course it is something like that trajectory converge uniformly to convex combination but uniformly with all derivative with respect to initial data now let us prove it and then just two words how тупие идеи and of course it implies immediately that property that I thought any trajectory getting by convex combination can be uniformly approximated by original by original admissible trajectories so if we have if we have a controllability for convex combination then we have approximate controllability for original and we know that approximate controllability is equivalent to exact controllability so we can so we can consider always convex combination and in operator setting the proof is so obvious of this rather strong property let us prove it of course some work is needed but simple functional analysis gives you that integration by path is legal integration by path is legal I don't think we have to do here is integration by path let us what I am going to prove now I would like to prove that if this converge to zero lemma if this weak norm if we have a sequence which depend of n if this star converge to zero as n goes to infinity then pt uniformly converge to identity so that that will be continuity with respect to our norms at identity at zero for vector fields so I am going to prove it but I will not continue I will not prove continuity at any point because it is reduced situation do you remember maybe in the previous lectures I showed how to reduce by changing of variables reduce study of endpoint map near orbitary x near orbitary point to zero by changing of variables so it is next step but it is very simple so actually since this guy is left invariant in the theory language if it is left invariant it is sufficient to prove analytic properties of it at zero at identity and then just translate translation is fine translation is continuous and no possible topology just done translate that what we did actually in the previous lecture by hand we translated it so it is enough this and this is integration by part you write ptn we integrate by part p integrate this and differentiate this minus minus and here we have here we have since we integrate this we have integral from zero to tau x theta no no no minus integral from zero to t pto xto because when I differentiate pt I get ptxt it is integration by part first integrated part and then we differentiate pt we get this one we get this one and integral of xto ok just a meaning this one minus this one this is integrated part this one minus ok it is just integration by part of this guy first we consider integral of p and integral of x this minus derivative of p and integral of x and now look and then this integral goes to zero in ck norm this integral goes to zero in ck norm and this diffeomorphism is limited if vector fields are limited so we have this sequence is bounded because if vector fields are bounded then also pn are bounded and this goes to zero that's it ok in the most general nonlinear system finite dimensional this is valid and then you can translate it also any point so it's indeed like that and we can always take convexification so we start with some family of vector fields can be discrete we take convexification but break it generate break it generate and it can be discrete then we take convexification look if convexification not necessary symmetric but at least contains some neighborhood of zero symmetric we go back and forward then we are completely controllable but if not because main application I like that we have a system q dot main applications are q dot equals fu of q where fu is f0 plus sum of ui fi so we have a nominal dynamic somehow and ui are just real numbers normally for ui we expect that they take both both signs otherwise we simply change f0 so we have a kind of and then they can be bounded so we have q0 we have this f0 we have some piece of low dimensional but at least one dimensional of admissible controls and that f0 it's already convex it's already convex so with convexification you can achieve anything but still very often it is controllable if not for small and normally if f0 is strong s is written if f0 is not 0 at a point then then I don't have much time I have to do to do for nothing but something I say I say no f0 if f0 is not 0 then at least for small time for sure all velocities are contained in the acute cone and we cannot cover the neighborhood for sure if we draw the wall we do not intersect this wall for small time because all velocities will go to one side of the wall but в принципе there are two very interesting possibilities to study local controllability in the case when in the equilibrium of f0 very interesting and deep theory exist when we try to check somehow test problem check controllability test also for fine optimal control theory because optimal control next lecture I explained it is very similar to controllability somehow to local controllability and when we have an equilibrium assume that f0, fq0 is 0 but just equilibrium because this is important dynamic it may be it some dynamic which taken from mathematical mathematical physics so it could be Navier-Stokes equation some reduction of Navier-Stokes equation and whatever you some finite dimension or dynamic that can be approximated also infinite dimension or PDS and it has equilibrium but the isolated this equilibrium isolated everywhere 0 so at equilibrium we have symmetricity but symmetricity in one point is not enough not at all when we at q0 admissible velocities is like that but then they start to be like that then they start to be like that very interesting and is fantastic effects is a study to characterization of generating but not symmetric actually and when we still can go in any direction from here that's equivalent to reach equilibrium for any direction some kind of local stability property of control can we stabilize our system using this perturbation the equilibrium may be not stable but you have some way small c even to control it but only but the dimension of control is small is much smaller than that it may be just one it can be just one plus uf u smaller than c that's very interesting problem when it's controllable you can somehow segregate break so break generating gives you only full dimensional dimensional attainable set but it starts to be important by what kind of break it reaches full dimensional some break it's a good, some bad it is a deep story it was well developed 90s essentially 80s 90s Exhausted somehow but there are still interesting open questions almost forgotten I hope that the technique developed that it's a test problem a kind of sport problem to characterize say if you have a two vector fields like that equilibrium to characterize in terms of break it when it is locally controllable it is test problem but technique developed has much more general meaning of course but another story if it is not like that if we are not in the equilibrium that we do not have a chance to get local controllability but still quite often may get global controllability now assume that this f0 and f0 you know somehow not unique in this setting because you can always we have a fine a fine space of a piece of a fine space a fine ball of a fine box of admissible velocity so of course f0 we can take any guy inside in principle we can take any guy inside but assume that f0 is conservative q dot equals f0 q is conservative in the following sense the manifold can be compact or non-compact k such that exponential of tf of k of k equals to k just invariant con equals to k and it preserve volume if the system preserve volume so divergence of f is 0 in this sense I mean conservative then then vector field minus f0 is compatible then vector field so my time is over maybe very very it is not a big deal to explain proof it is the same argument but maybe I just explained by words so then then in this case if the system is controllable divergence then the system fu equals f0 plus uifi is completely controllable somehow what you do you prove that you can go approximate trajectories of minus f0 points of trajectories of minus f0 of course for arbitrary time for arbitrary time obtainable set infinite time is the whole space so this is corollary of our first Kriner idea Kriner result about nonzero interior and upon correct for such a system as we know trajectories inside k if you take such a system then take any neighborhood here any neighborhood o and apply et f0 then they exist always e touch that this is which guy moves and it comes back any neighborhood intersect with itself this is a famous one of the deep results which 3 will prove essentially because if volume is preserved, volume of compact is finite and you do cannot cover everything by by nonintersecting by infinite number nonintersecting domains you use also group properties but I do not have time to repeat even if it takes 3 minutes to prove the spawn query recurrence property and if you have this property again we use mainly the same agreement as before the same agreement of before we would like to show that vector field minus f0 is compatible with trajectory this is et f0 of q starting from q and we would like to show that we can approximated this by go back with trajectories like that go back using going in positive time but using also control again we consider we consider if we are breaking generating there is open set it is important that we can reach some open set near q0 near q0 and we can reach open set near any so it is like that so we can reach trajectory of f0 come back intersect any neighborhood of such a point we take a point with reverse time with negative time and we know that that if we take a small neighborhood of it and apply just vector field f0 then we appear we appear again in the same neighborhood but now what we do what we do we consider so this is q0 this is e-t 0 say f0 of q0 it is reverse time but then with our system again with our system with reverse time we can find attainable set some neighborhood that is attained and if this neighborhood starts to travel by f0 in positive time then it is intersect itself at some moment so we take interior point of attainable set from this one and then apply just f0 but in positive this neighborhood is attained that we apply f0 just for positive time and then neighborhood is intersect itself in future so we appear here and it is orbitery close to this point everything is done orbitery close to this point exactly in negative direction minus f0 is available and convictification is symmetric if we put here plus minus plus minus f0 convictification is already here so that's the agreement since we have full dimension we can also do that but of course it is like upon career result not theoretically nice but not really applicable because you know that neighborhood is very small and we should wait too long to get this controllability still it is important because it is a kind of existence here if we have control if we have I am finishing if we have a fine system this conservative drift this called drift, this controllable path this conservative drift breaking generating then it is completely controllable for very big time, fine it means that we can really get any point from any other and then we can start optimal control but it is important existence because if there exist some path connecting two guys then they exist optimal also time optimal and this is a great achievement because time optimal satisfies usually in calculus of variation are critical points of something satisfy some differential equation and we already know so among that not so long guys there is one for sure one we resolve much more restrictive problem that connect Q0 with Q1 this existence give existence of optimal so thank you very much sorry for long