 to have never been affiliated to the University of Bonn. So that's a formal part. Then that's what my talk is dedicated to control of ensembles of dynamical system. And it's a particular type of nonlinear system in infinite dimensions. So the whole story is about trying to extend the algebraic approach about what you have heard in the lectures in these days and to promote this, to extend this, to the infinite dimensional setting. So let me start. My first slides are perfectly known. I start with something different, not with ensemble control. That's just the history. I mean, how this topic of control and mixing, what was the beginning of this beautiful friendship, so to say. So it started with some question put by Sergey Kuxin, I believe, on the control of nonlinear partial differential equation, in particular on Navier-Stokes equation. And with degenerate control, so we have an infinite dimensional system. And we have a finite number of controls, so we control the modes, so to say. And we wish to study controllability, say approximate controllability of such system. And to see, more importantly, even was to see how the energy is transferred from mode to mode. What is the cascade? And the idea was to look for the algebraic structure into this transfer of energy along the modes. And so that was the idea of what we were used to in finite dimensional situation. And I repeat, the point is that we have a very limited number of controls. We always deal with so-called degenerate control, et cetera. So there were some about 15 years ago, 15. This work started. And in these 15 years, there was a progress. I mean, several results obtained in this direction. The main method applied was the method of the extension, which has been mentioned along these days also yesterday, for example, in context of stabilizability and stabilization. So the idea is that you can get of extended control system. So you can apply the algebraic approach and the extension in order to get extended control system and extended control, which are applied along the lubricates of the original vector field. So that's the idea, how you follow. That's the way to see how you follow the transfer of energy, how you see the cascade structure. So you look for the lubricates of some vector fields. And there was plenty of results obtained in this and some plenty of publications obtained in these 15 years. I repeat, that's just a historical introduction that the true story goes after. So the important thing also mentioned yesterday in the talk by Professor Zouif is that the important is the use of fast oscillating controls. So the controls which are fast oscillating in time. And another important notion also mentioned by several times in these days, a notion of solid controllability. So some kind of controllability which is structurally stable, so to say, stable under perturbances. So there was number of results obtained. So obtained in this direction, there are some controllability results. I will not write any equations. I mean, that's just a historical thing. And different people were involved in this. This is, I believe, not a complete list. In particular, the recent contributions were done by colleagues who are here from PDE. So the initial, so to say, push was done by control. People specialized in nonlinear control and geometric control. And then there was a lot of contribution of Armen Shrikan and Wagner Sessian who resolved plenty of analytic problems. So let me say now some general things before I pass to Ensemblers of Systems, which is a completely different situation, as I say for young people. So let me say there are still plenty of open problems. And there are two type of difficulties. When you seek to apply the algebraic methods to infinite dimensional situations, in particular to nonlinear PDE. The first type of difficulties are analytic. Because you have nonlinear partial differential equations, for example, with something at the right-hand side, which is of the following. Something like something of this type, for example. So you have something which is, so to say, high gain. It has high amplitude, and it has faster oscillations. So you need some nodes, I would say, non-canonical, non-standard, non-classical continuity results for this type of nonlinear differential equation. That is the first difficulty for nonlinear PDE. It was kind of a new situation which has been treated. And the second, but yeah, and certainly plenty of difficulties, standard difficulties with boundary conditions, et cetera, et cetera. So that has been the result. That's the great contribution that has been made by Armin and Vaharn. But the second difficulty, and this difficulty is not yet a result satisfactory, that's the second difficulty of the algebraic type. So I mean, you have to, for example, if you deal with Navier-Stokes equation or no domain, then you have to study algebra of, so you have some spectral geometry of this domain, and you have to study an algebra of eigenfunctions of the Laplacian on this domain. That's essentially what you deal with, at least in 2D case, in two-dimensional Navier-Stokes equation. Then this is completely not studied. So there were some progress. In periodic case on torus, there were some progresses for hemisphere. But as, for example, I can say to young people that the problem on the disk is an open problem. It's an open problem. That the problem on the disk, as far as I know, if my colleagues do not correct me, is open. I mean, there is no controllability result for the two-dimensional disk. And the main difficulties that you have here with the eigenfunctions of the Laplacian in this case, which are defined by means of Bessel function. And you have a very complicated algebra of this Bessel function. And there is plenty, plenty of things to do. So I addressed it mainly to young people. There is a lot of interesting algebra in it, I mean, to study algebra and symplectic geometry and a lot of these things. So that was a historical part, I believe. I'm done with this so I can come to some concrete thing. So we will consider another type of infinite dimensional system. Now forget PD. It was just a historic. And now we pass to another type of infinite dimensional system, which is also rather well-known. And there are quite some many publications on it. I will, in a few slides, I will explain the origin of this. So what is an ensemble of systems? That's essentially a family, a parametrized family of control systems. So T is a parameter here. And so the solution, as far as T enters the right-hand side, the solution depends on theta. So we have a dynamics in infinite dimensional space of the functions of the parameter theta. So theta here is an arbitrary complex subset of Rd. So if theta is trivial, so it is a singleton, we have a one system. So it's a standard control theory. So interesting cases when it is not a singleton, it can be finite or infinite. And so the origin of the problem setting comes probably, I mean, as far as I know, comes from experiments in NMR, the famous Bloch system, when we have some dispersion. So it's all about a system with, how to say, indeterminacy. Yes, it's a system which has some undetermined parameter, parametrized indeterminacy. So I believe the starting model, it all started in the beginning of 2000. And as far as I know, the first publications were due to Haneja and Lee. And it was related to so-called Bloch equation that I just give for information. That's a very rather simple, simply-looking equation on Ethel 3. And this simply-looking equation contains parameters omega and theta, and both are subject to some dispersion. So they are not defined exactly. So they belong to some intervals. And we want to simultaneously control it, for example, from point to point. So what is, what means that you do not know a priori the value of parameter, but you still want to control, to invent some control, which is independent on parameters, and to transfer a system from point to point independently on parameter. So that's the idea. See, approximately, approximately. You cannot do it exactly, but approximately you want to do this. So that was, so to say, the first model of interest and the first attempt to apply also algebraic methods, since we are on a league group. So it was natural to apply the algebraic approach to this. In the case of Haneja and Lee, it was just Campbell-Hausdorff formula or some things, some tools of this kind. Later on, there was a much more extensive work on this done by Karin Boucher, Jean-Michel Coronet-Pierre-Chon on this, more extensive, where the problem was kind of settled, at least, at least. Well, we may say it was settled in this work. So now we come to a general situation. So it was an example, an example of what is interesting in these kind of models. And we come to a general situation. So again, we consider this family of non-linear control system, and they also allow the parameter in the initial condition. So moreover, today I will speak about this second case when the parameter is so we will have, in a few slides, we will have a control system which is not dependent on parameter. And we wish to transform some parameterized family, for example. So to say before, well, let me say so, the main setting today will be we wish to do the following thing. So we have, for example, we have an initial curve, yes. And we have a target curve. And we have one control system of the following type. And our objective is to find the control which transforms this curve, this initial curve, to the neighborhood and one or another metric, if you wish, integral metric or C0 metric. In the neighborhood of this, so we want to find the control such that the respective flow corresponding to this control system and to this control transforms the initial curve into a finite curve which is close in some metric to the target curve. That's the idea. There is an obvious generalization of this. We may speak about control of defiomorphism. So we may just wish to control the flow. We might just consider the flow in the group of defiomorphism. And during the talk, I will provide some results on this. So that's the problem setting I will consider. So for a moment, the general setting can be done when we have also a parameter. The system depends on the parameter. But later on, we concentrate on the second case. So there's standard things. So we assume that we are on some manifold, connected manifold, Riemannian manifold with some Riemannian distance. So what we call ensemble. Ensemble will be called just an image of continuous map from the set of parameters theta to this manifold. Theta, we consider theta to be compact. It's a Lebesgue measure set if you want to define L1 or L2 measure. So and we denote it by E theta. E theta of M is the space of parameters. And it is obviously infinite dimension. And if theta is an infinite set, yes, if it is a finite set, it is finite dimensional. Then in general, it is an infinite dimensional space. Of course, for example, or it can be surfaces, et cetera, et cetera. So we wish to control something like this. Yes, and so what I said, we wish to define a control dynamics. So the control dynamics is, I said it already. But essentially, this is the whole following thing. So if we add here a parameter theta, yes, then the flow generated by this family of systems also depends on the parameter theta. So we have a flow which depends on the control and parameter theta. And it depends on time t. And we apply this flow to an initial ensemble, alpha of theta, which are the initial conditions, say. Oh, gamma of theta is here. There, yes. OK, gamma of theta. Gamma of theta. And then we get a result. So that's our controlled evolution. So for each control, we have a curve in the space of ensembles. For example, curve in the space of curves or surfaces, yeah, kind of. What episode? That's the controlled dynamics in the space of ensembles. And we said controllability problem for the action, for this action. And well, we can speak about exact controllability. So that's when you can transform an ensemble, an initial ensemble into a finite ensemble exactly. So this, as I would say, is available only in the case of finite ensembles. In finite ensembles, we can achieve finite exact controllability. I believe I trust it can be done as theorem so that in infinite case, you cannot do this by the methods like Paul Slamrod non-controllability result. This can be done in the case when the number of parameters in the infinite, you cannot generally cannot do this. We can define the attainable set from the ensemble. So again, we'll be subset of the space of ensembles. And again, well, a binary mark that the ensemble is trivial. So we deal with single tones. The whole story coincides with the respective terminology for single control systems. But as I said, yes? Continuously? Yes, yes. Ensemble is continuous in theta. No, control is theta-independent. That's important. Control is, yes, yes. That's extremely important. Yeah, control, that's the whole point of the thing. No, no, no, it's f of theta is continuous. Yes, f of theta continuous. But control is, control is theta-independent. That's the whole point, yes? So we wish to find a theta-independent control, which kind of transforms the whole curve into a curve. We do not know that theta. Yeah, the whole family. You have a dispersion in parameters, and you wish to control independently of the dispersion. Yes, control is always independent on theta, yes? Yes. Yeah, yeah, but after I mean that, in a sense, it's more interesting situation. Well, it's less general, probably. We will take f independent on theta. We will wish just to control curves and surfaces, so to say. That's the point. And we come to the notion of approximate controllability, which is, I would say, the only one available in the case of infinite or continual, we call it continual ensembles. So we cannot expect the exact controllability. So we introduce an obvious notion of C0 or LP approximate controllability. So in the integral norm, so that's the first definition is a steerability. I would say that we can steer an ensemble to ensemble. Sorry, I forgot. Here, here should be theta also, virgula theta, because we are still in general setting. And that's the definition of approximate controllability in C0 norm. This is the definition of approximate controllability in integral norm of LP. So when I speak about C0 and LP, I speak about C0 of theta. Yes, of theta capital, the set of parameter, and LP of theta capital. So that's our definition. And we will distinguish two cases. So for us, it will be convenient to distinguish two cases. So the cases, so that's a little bit delicate. I mean, we distinguish the cases when our control, well, the most general cases when we have dependence on theta in system and in initial condition. Yes? That's the first case. Or we can consider a particular problem when there is dependence only in theta. So we have a dispersion on parameter. But we wish to steer point to point. So the robust control, so to say, the control, which does not depend on parameter, and steers the whole family of system to the needed situation. That's the first case. About this, we've developed some, say, algebraic approach. It has been published three years ago in a paper mentioned there, kind of general algebraic approach to some problems of this type. But today, I wish to concentrate. It will be convenient because the formulations are different. The formulations are different, and it will be convenient to consider the second case, or what I mentioned in the beginning. So from now on, we will have the system which does not depend on control. The initial condition depends on parameter. And we wish to approximately transfer the initial curve to the neighborhood of the target final curve. So we wish to find this control. So the most general problem setting will be to control on the group of diffeomorphism. So we wish to find the control which transforms a diffeomorph initial, say, identity diffeomorphism to a neighborhood of a given diffeomorphism. OK, then I believe that I would like to emphasize that we have a limited, very limited control. So our control is only a function of t. We do not allow anything else. So it's a very, very, very poor class of control. So that's our problem setting. So I will concentrate today on this second type of controllability. It seems to me it's rather beautiful from geometric viewpoints. So we control surfaces, curves, and such objects by controls which are independent of the parameter on this surface, yes? And so we can define this. So first I define why I would explain it. It's very easy that these ensembles can be given a structure of Banach manifold in an obvious way. So we consider just a continuous function. So imagine that m is rn, for example. So we can consider continuous functions defined on theta rn valued. And for example, in the case of finite ensemble, it's just a Cartesian product of n copies of m. So we can define, since m is a smooth manifold, we can define the tangent space on the space of the ensembles. And that's a formal definition. So that's a coordination of the tangent element of this tangent space. So it's a very nice Banach manifold. So in the case of finite ensemble, that's enough for us. And we get exact controllability results easily in this case. So there is no problem about it. The problem comes in infinite dimensional case because the algebraic approach is not completely adapted to this case. And as we will see, this structure of Banach manifold does not help much. I mean, we have to do it, to study it, how to say, too. We cannot act with standard, general results for Banach manifolds. It's often the case. So it's infinite dimensional systems are not part of a theory of general nonlinear maps in infinite dimensional spaces. We will see it later, and that's important. So yeah, so we did not say so. With the same ways, we have defined the tangent space on the Banach manifold ensemble, we can define vector fields on the ensembles. And moreover, that's important for us that if we have a vector field on M, if we have a vector field on M, this can be lifted to a vector field on the space of ensemble by biobusing that you can act on ensemble differentiated by T. And then you get a vector field. And the vector field, this is very easy. It's easily just calculated along the ensemble. I mean, that's very easy. And then the very easy computation. And also, you can define the brackets of this vector fields, of this lifted vector fields. So the whole story looks simple in the sense of definition, but not in the sense of results, as we will see in a moment. So such properties like bracket generating, et cetera, are not at all simple here. So that's the whole nice story, the whole nice picture, which we will not use very much later for the reasons which are explained. So again, I mean, we define, I repeat, the definition of approximate controllability. So I will concentrate today on C0, approximate controllability. For example, it's more natural here. So we wish we have a control system, and we wish to find a control independent on parameter. And we have initial and target ensembles parameterized by T. And we wish to find the T independent control, which transforms alpha in C0 close in an ensemble, which is C0 close to omega. So that's our problem setting. And OK, we start with elementary case of controllability, which is a finite ensemble. That's an easy case, because here everything can be done by standard Treshevsky show result. Because in finite case, our manifold is just nth degree. So we take n-capital copies of m. We take Cartesian product. So we consider n-pulse of points of m. So they belong to Mn. And here in this case, it is convenient for us to consider injective ensembles. So they are different. We assume them to be different. So that's a case of n-pulse, which are parameterized different in coordinates. And so we wish to consider a control system. And control system, we have a control system on M. This is control system on M. And obviously, this control system can be lifted to the M to the power of N. And we want to put a problem of global controllability of this system. Well, let us consider something. So we wish by simultaneous control. So that's the problem. We have n-points. And we, by a unique control, we wish to transform it in n-points. So for geometric control, people is a very obvious problem, not always obvious for, I have heard from people, say, from a sabbatical physics, many questions. Is it possible to start with different initial condition and transform it to some other final terminal conditions? So for geometric control people, it's a rather obvious thing that it's not very, very different from the case of one system. So let us consider a simple case. I mean, obviously, if we have difficulty in establishing controllability on M, certainly it's even more difficult to establish controllability on the N's power of M. So we consider something simple. So we consider a control linear system of this kind. So essentially, so we have this and that controllability can be treated by standard Rachevsky-Chou theorem. Well, the only point is that here we have very particular vector fields. So their particular field vector fields are anouples of identical of the same vector field, repetition of the same vector field. So our dynamic is very degenerate, yes? So we have a vector field of this kind, FFF of the same vector field repeated, yes? So x1, et cetera, xn are guided by the same system. The initial conditions are different, but the system is the same. So we have a very limited controllability situation, but the question can be answered. So we obviously define and define the brackets in this way. So we take it component-wise and repeat it 10 times. So we can define a control system on this manifold on M to the power of M, and the result is obvious. So we can define a bracket-generated condition or Hermann conditions on the N's fold of M, so to say. So we can evaluate the iteratedly brackets. And yeah, that's important seeing that if we have this property of bracket-generating of N folds, so what are N folds of vector field? N folds of vector fields are these vector fields. So we bracket-generate, bracket them component-wise. Yes, certainly, yes. And if these N folds are bracket-generated on this big manifold, then we have by Rachevsky-Chow's theorem. There is nothing new. By Rachevsky-Chow's theorem, we have a controllability. An important remark is that if N is the number of elements and the ensemble is greater than 1, this property is obviously strictly stronger than the standard bracket-generating property on M. For example, if these vector fields generate nil potential, finite dimension on the algebra, then there is no controllability of sufficiently big ensembles. So obviously, there is an obvious relation between the dimensions. Maybe I wrote it, but it's an obvious relation. So you cannot control big ensembles if the algebra is finite dimension on a particular nil potent, yes? So that's important. This is a proposition. I repeat that from control theoretical viewpoint, there is nothing new here. That's just Rachevsky-Chow's theorem applied to the N fold of the manifold M. We were looking for an interesting result, which is known in control theory, one of the first results of nonlinear control theory. It was a result by Laubri, who was saying that if we have a system of this kind, then the controllability property for this system is generic, that there is an open dense residual set with respect to some metric C of sufficient high order, such that any R opal, or how you say, of these vector fields from this set is controllable. That's a result of Laubri from the 70s. So we wish to get the similar result for the ensembles, for finite ensembles. And we managed to do this. I mean, that's not a repetition, unlike the previous result. That's not a repetition of standard Rachevsky-Chow, because here it is written. I mean, so when we speak about genericity, genericity means here that we allow to perturb this vector field, but not the whole story. We cannot perturb them independently, yes? So that's a different problem, stating that the classical Laubri result, yes? We have much more limited possibility of the perturbation of vector fields. And still we managed to prove this theorem, that in this case for the ensembles, so the set of the bracket generating property for the set of ensembles is generic. It's verified for a generic aeropole of vector fields. So we have our control linear system. So I mean, the proof of Laubri goes by the end. It requires some rectification arguments, or rectification of one vector field. And then at the end, applies to the transversality theorem to get genericity. So in this case, we have to employ something a little bit more delicate, which is John Mazer's multi-jet transversality theorem. Because we have here vector fields computed in different points of the manifold. So we have to involve here multi-jets and to use this result of Mazer. OK, and now we come to more interesting. I mean, the finite ensemble is intuitively clear case. So now we come to continual ensembles and motion planning. I mean, the formulation will be a little bit at the beginning. It will be a little bit particular, I mean. So we talk not exactly about controllability, but more about control along something, or it is also called motion planning. Again, for simplicity, we consider the symmetric control system, control linear of this kind. So what we do? We consider a finite and initial and finite and target ensemble of points. So we consider two ensembles of points. And we consider some dephiotopy which joins these two curves, for example. This dephiotopy is generated by some flow. What we call the dephiotopy is generated by some flow, by some equation of this kind. And y is, say, C1 and y is small and continuous in T. I mean, important that this y capital is not related to control system, obviously, because otherwise everything is trivial, is we can transform the ensemble in ensembles. So y is just a vector field, not having to do anything with control system. So we take just two dephiotopy ensembles. And we wish to, it's written here, yes. Yes, well, in alternative, we can consider something which is, so to speak, approximately dephiotopic. But OK, it's not probably very important. So we can find, well, OK, that's not extremely important. And yes, that's what I said. That's important, that the flow which transforms this initial ensemble into finite ensemble has nothing to do with the control system. So it's absolutely nothing to do, because otherwise the problem is trivial. And so what we wish, we wish now, we have some dephiotopy, and we wish now to find the control. So now we come back to our control system, and we wish to find the control which transforms this to the neighborhood of this. We will do it in such a way that we will remain close to dephiotopy, so to say that I'm talking about them. And then, yeah, so that's the problem setting. And we seek for control which transforms the initial ensemble to the finite ensemble. Certainly, we need something like breakage generating. Yes, breakage generating condition in this situation. And we call this condition, the analog of this condition as bracket approximating condition along the dephiotopy. I will later explain why we haven't reduced the dephiotopy, et cetera. That's a little bit delicate story. So we introduce this condition which is called the bracket approximating condition. And that's the following, that's the following, that's the following conditions. So we need some notation. So we consider the, so we have our system is, our system is, so this is our control system. This is our control system. So we consider the algebra of this control system. So we take the vector fields from which are liberated, liberated, liberated of this. And then we take a subset because we take a subset of this algebra of the vector fields which are Lipschitzon with some constant lambda. So we take something bounded in the Lipschitzon metric, yes? So we have a bounded Lipschitzon constant, yes? On some neighborhood of this, on some neighborhood of the dephiotopy. And now we introduce an assumption which looks a little bit cumbersome, but I will later explain that there is a sense natural. So our assumption of C0, Libre, C0 approximating condition is the following. That if we take the vector field y, which generates our dephiotopy, then this vector field evaluates along the elements of the dephiotopy. Yes, so we have here a violation of this dephiotopy by the curves, by the image of the curves under the dephiotopy. So the evaluation along this dephiotopy differ from the evaluation from, can be approximated by the vector fields from the algebra of f and also with this restriction on the Lipschitzon constant lambda. This looks rather cumbersome and a little bit ugly in comparison with nice breakage generating condition, but in a few minutes I will explain why this is needed. I mean, that's our main analog of breakage generating condition. So Libre, C0, yes, and this is done uniformly in theta for theta belonging to theta and for ht belonging to t. So this is our condition. So we manage to approximate this vector field by elements of the algebra under controlling the Lipschitz constant of these elements. And then, yes, there is another technical assumption. That's the first one is very essential. Second is more technical. So we assume these fields to be C0 smooth in order to calculate a lot of Libreckets. And we assume them to be bounded on the whole manifold. Them, if m is compact, it's all OK, it's not needed, are bounded on the whole m and together with their covariant derivatives of each order. So imagine, for example, some of these vector fields, if we are on non-compact manifold, this vector fields say multiplied by some rapidly decreasing function. And then, yeah, this is related to the second condition, is related to the question of completeness, et cetera. And then we get a theorem, which says that if these two conditions are satisfied for the dephiotopy, then the initial ensemble of the dephiotopy, this alpha of theta, alpha of theta, can be stirred to the neighborhood of the finite ensemble. I mean, I will not give a proof, but anticipating the proof, which you can find in the preprinting archive. I mean, you can imagine that the whole evolution, control the evolution of this ensemble, will go in the neighborhood of this dephiotopy. So it's kind of important. I mean, the result looks a little bit partial. I mean, what are the alternatives? I mean, let me, I believe I do not have much time. Well, so that's something about the proof, which says, let me be quick a little bit, because I want to spend some time with discussing the formulation. Discussing the formulation is more interesting here than the proof. So essentially, there's a standard technique extension. So we consider an extended control system, where these vector fields are the liberators of the initial vector fields. And then we have to approximate the flow generated by the extended system, by the flow of the original system. So for people from geometric control is known. Yeah, and the third result, I forgot to say, the third result, which I would like to mention, the whole story, I will not comment much on it. I mean, the whole story is that we can also make a formulation, and probably this is a nicer formulation for the group of dephiomorphism. Essentially, this result says following the following. If we have a flow generated by extended control system, so this is probably more comprehensible by this control system, where the vector fields X are the brackets of the system, then this dephiomorphism can be approximated by dephiomorphism generated by the original control system. So we have an approximate controllability in the group of dephiomorphism, a little bit reduced approximate controllability, I mean. Because this is extended control system, this is original control system, so there is a relation between the system. But still we have a result on approximate controllability. Okay, but let me in the last, in the last, so this is a coral elephond in the last 10 minutes that I have. There is an example, I mean, an example in R2, a simple example in R2 where I explain how we can verify this approximating condition, but let me pass since I have only, this is a computation of only brackets, so in the whole story reduces to the density of the space of hermit, of the system of hermit polynomials in this example. But let me discuss the formulation because, I mean, the first formulation we were not, I mean, why this formulation is a little bit peculiar, so why we need to involve dephiotherapy, et cetera. So let me discuss something. I mean, so we have, this is a standard Rachevsky-Chau series in finite dimension. There is no need, everybody knows this from the lectures of Professor Agarachev, et cetera, so this is standard Rachevsky-Chau series in finite dimension. There was, there are results which generalize the Rachevsky-Chau series with bracket, bracket generating condition to infinite dimension, to Banach-Bailey-Folz. The first of such results was in 1980, so some two Ukrainians, two Ukrainian, so Soviet mathematicians, published this result in 1980 or 1991, the generalization of Rachevsky-Chau to the infinite dimensional case. I mean, the generalization is rather obvious. They take the, they act on a Banach-Bailey-Folz and we have a Banach-Bailey-Folz. They take the brackets of the vector fields and they say that if the closure of the value of the elements of the algebra at each point coincide with the whole tangent space of the Banach-Bailey-Folz, then the system is globally approximately controllable, obviously, because we take a closure, then we get only approximate controllability. Yes, so that's a natural result. It seems very, very, very nice. Unfortunately, this result does not work in reasonable situations. I mean, it definitely does not work in PDE because PDE involved unbounded operators and in this case, they ask this everything be smooth, differentiable, continuously differential, so it will never work in PDE, does not, to PDE does not. For formal PDE, but what you call approximately, what you call approximately for a formal case. Okay, but I mean, that surprise, I mean, that it does not work for PDE, it was clear. But surprisingly, it does not even work in our case, which is very natural, I mean, just computation of vector field along curves. For two reasons. I mean, the first reason is practical, so to say, practical, because, I mean, look, what they ask, what this theory of ask, they ask verification of this condition at each point of a huge Banach-Bailey-Folz. So in our case, such kind of condition will require verification along any curve, say curve, if we are one dimension, any curve in the space, I mean, which is absolutely, I mean, unreasonable to do. That's the one reason for the less important reason for taking the Diffie-Otopy, et cetera, because here when we verify our condition along the Diffie-Otopy is a condition which should be verified along one parametric family of curves, in this case of curves, yes? So it's much more economic, but it's a less important reason. The second important reason that this condition is extremely unstable structurally, so it will be violated, it does not hold, it's not stable even locally. It is not stable even locally, I mean, yeah, here I explained that we can even for single system, we can find a version, the longer Diffie-Otopy for single system, but it's not important. I mean, that's the main reason that this condition, that condition on this general infinite dimensional Rachevsky-Chow theory, in our example, Bailey-Folz often fails rather even locally. That's a simple, it's a simple example. If we have a continual ensemble, obviously not for finite ensembles, everything is okay. For infinite ensembles, for example, we can take, if we have a curve, so we can reparameterize. So in any neighborhood in C0 of this ensemble, we can find an ensemble which is constant on some set of parameters or some non-trivial set of parameters, so we can reparameterize it that at some point, we stay at a point, yeah, it's constant. So we reparameterize in such way that we have a trivial piece, trivial piece, probably reduced to a point. And it is obvious, it's easy to check that in this case the bracket generating condition just will not, in their sense, will not hold. It will not be bracket generating in the sense of the theorem I have just demonstrated. We have a constant piece. It's easy to check that you cannot approximate any vector field. It's a degeneracy. That's only one type of degeneracy, but you can invent many types of such degeneracies. When you get this density, I mean the K condition, the K condition of this theorem is density of this thing. And if you have a degenerate piece, you will never be dense there, and there are other types of degeneracies. So I mean the key point, the key point of our reasoning that you may get approximating condition on the dephiotopy and not have it nearby, just nearby, they're approximately close to this dephiotopy. So that's why we need some such kind of, to be a little bit heavy in the formulations to involve this dephiotopy, et cetera. We are not able here to formulate such general result like in finite dimensional case because this condition is just very, very, very unstable, so to say. I believe it's better, yeah, what else? Yeah, I mean here there is an example which I probably will not do it. I mean there is an example. If you remember there was another condition unbounded as of Lipschitz constant that here I explain why this condition is important even in the case of one system. I mean there is a rather easy example but you can find it in a preprint. So the preprint is in archive in this here, Agarachov-Sarachev, and you can see there this example. I mean it's an easy example which explains that even for one system you can invent an example when you need some bounds for Lipschitz constant. Okay, I believe I finish here, thank you.