 Эта лекция вводит к оптимальному контролу. Во-первых, я покажу, что оптимальное контрол опять о том, что есть. У нас сетя. У нас есть, как обычно, контрольный систем. Продолжение к вам. И сейчас я надеюсь, что нужно продолжить. Так что я надеюсь, что этот человек продолжает с респектом у Q. Он смеется с респектом Q, когда Q фиксирует с респектом у Q. У Q теперь в другом манифолте. И это у Q F U Q это продолжение. Это очень defined, это продолжение на клоуже. Во-вторых, базовый результат, который я буду объяснить в этой лекции о оптимальном контроле это максимальный принцип, и это очень важный для Q F U. Это странно, но это очень важный для Q F U. Но это очень важный для Q F U. Если мы надеюсь, что это продолжение на клоуже, на клоуже у Q F U. И потом Q F U это очень важное. Но другие результаты, в частности, existence of optimal control нуждают больше правительства для более специальных сетей и это я фиксировал. Но сначала, я расскажу о оптимальном контроле. Я уверен, что мы уже знали, в первые лекции, мы изучили проблему для получения Q F U от Q F U. И еще один ремарк. Сейчас, если я бы хотел найти что-то оптимальное, то мой способ контрол-фаншен должен быть клоуже. Как вы попробовали найти что-то оптимальное? Если вы уже знали, что вы можете пойти Q F U от Q F U, вы попробовали сделать это лучше и использовать сегменты. И вы хотели эту минимальную сегмент, чтобы конверть где-то. Так что, ваша цель оптимального функции не может быть пистолетов. Потому что, когда вы возьмете сегменты, это будет больше и больше пистолетов, и в этой лиме, вы потеряете сегменты. Так что, вы нуждаете какие-то полные места. И в этом случае, это удобно использовать контрол. Фаншен, спецфон контрол-фаншен это U of dot, это негативная функция, T в У of T, это Линфинец. Так что, это в негативном объеме. В сегменте, в котором это можно быть в негативном объеме, это всегда в негативном объеме. Это в негативном объеме. Это в негативном объеме, Линфинец. Так что, это в негативном объеме. Но, как мы говорим о оптимальном контроле, мы нуждаем что-то, чтобы оптимализовать. Мы считаем, что есть какие-то функции, которые называются C, G, с параметром T, которые мы хотим оптимализовать. Это будет просто в негативном объеме. Так что это генерализация классической калкивосоверии. Т Ф Q T U of T. И мы пытаемся оптимализовать эту функцию на всех тракторах в нашей системе. Так что это зависит от тракторской контроли. В более простых непрерывных проблемах классических проблемах мы не имеем нетривестного динамича и тракторской трактора. Тогда у нас F U equals U и U只是 скорость. И здесь у нас Ку-дот. Ватик, у нас есть ку-дот, когда ку-дот выстроится. И мы имеем коз, который зависит от... Мы имеем смыслы ков. Велоситеты, контроли велоситеты в таком случае. И коз зависит от ку-кю-дот. Это ваше, в которое он генерализирует классический проблем. Так что мы минимумим этот человек. Мы все отраджекторы, которые собирают ку-зеру в кю-дот. Отраджекторы, которые собирают ку-зеру в кю-дот. Так что мы нужны хотя бы 1, чтобы найти минимум. И давайте сейчас объяснить, когда мы минимумим 2 версии в этом проблеме, когда мы фиксируем ку-дот, мы хотим получить ку-дот именно в кю-доте. Или мы не можем, в чем время мы делаем это. Так что это только 2 версии. В версии... Это 3-й раз, кстати, как я объяснял. Просто можно быть сочетанным в постоянном времени. И в общем-то... Мы можем формулировать много разных версий для оптимального контроля. Мы можем взять еще один кост. И еще один кост. Мы можем фиксировать конкуренты. Возвращаем, что конкуренты перейдут к субменифу, здесь перейдут к субменифу, или просто отраджекторы. Все эти проблемы, с тем, что сеттинг очень генеральный, у нас есть орбитарные динамики. Все эти проблемы, более-менее, могут быть reducing в 1, 2, и еще в определенные детали. Фиксируемый и свободный раз. Конечно, всегда есть какие-то детали. Это не полностью эквивалент, но метод изучения это то же самое, и редакция. Эта реформуляция тоже работает. И в общем-то, если вы хотели увидеть эту проблему, как я сейчас объясняю, то это о том, что сеттинг очень генеральный. Для меня, у нас есть тут слишком много символов. У нас есть динамик, у нас есть лагранжен, функция, и я хотел бы унифицировать. Сейчас я объясняю вам, почему я хотел бы, чтобы выяснить одну проблему и другую проблему. Я хотел бы посмотреть это более геометрично, и не использовать слишком много символов. Я хотел бы представить, это не проблема, чтобы представить эту проблему как в корпорации в динамике. Что мы делаем, давайте представить эту проблему. Мы можем выяснить эту проблему в следующий раз. Мы считаем F equals... Мы считаем First of all, instead of m, we consider the m that is r multiplied by m. Я хотел бы унифицировать эту проблему. И потом я возьму F u of q is a phi of q u F u of q. И я пишу систему, и скажу q bar q bar здесь и я пишу систему. x, x belong to m. q. Сет of controls is the same as before. And I consider the dynamics x equal x0, q. Because x0 is the number and q is here. And I can write write the system x0 dot equals phi of q u and q dot equals F u of q. And altogether, it can be written as x dot equal F u of q. So a new system in the little bit greater manifold and if I resolve this system then solution x of t x of t is just So how to resolve? So this guy does not depend on X. First you resolve this, if you plug in here control, now plug in here control, we get trajectory which is a solution of this system. We also consider consider x of 0 as initial condition. As initial condition we take 0 and q0. So the original system was in the manifold term. Any manifold term is a plane. And this is the extra coordinate. So this is point q0 and extra coordinate start in the 0. To resolve this system we resolve this one, get a trajectory and then plug in trajectory here control is here and integrate because here is just integral. So we obtain Jt Jt of u if q0 is fixed it depends only on u, Jt of u. And here we obtain q q of t u of t dot. So that is what is the solution and absolutely obvious fact is as follows. Then we consider attainable set for time t for this system, for the extended system we consider attainable set a a of 0 of t that is that is set of all x of t x of t u of dot like that that is a set of all x of t u of dot for all L-infinite controls and I claim claim is absolutely obvious claim if u is optimal if this cost attains minimum and control u ut is optimal x dt for fixed time optimal for the problem with fixed time t1 then x t1 belongs to the border of a x of 0 of t1. So that is a totally totally trivial look for main properties so not all boundary belongs to the optimal but somehow we modify costs but the main question is to characterize those which arrive to bone geometrically what I would like to say geometrically to be optimal means to stay on the boundary of attainable set of certain dynamical system control system related cooked from our original and the cost it is always like that this is the main meaning of optimality why it is like that why it is like that but it should be clear let me draw again this picture this picture of m and r and let us draw attainable set here we have q0 here we have q1 we start here ok so our trajectory of original system arrive to q1 if we don't leave the trajectory of extended system arrive to this line all coordinates but first one is q1 don't to a1 let us consider our attainable set it is some set we start here then we arrive somewhere it is some set here attainable set 81 we assume that we can come to q1 with different with different cost of x of 0 and so we are in q1 if and only if for extended system we are on this line and we are optimal if and only if we arrive in the lowest point lowest point of the attainable set so this is automatically in the board of course not all board also maximum of the board is on the board there are other parts of the board but main point but what is what is the problem huge number of trajectories which arrive controls which give which give you your point here such that we can move we can any control we can move and the principle properties if we are interested in optimal control we are interested only on those we have to characterize them first of all this way we characterize attainable set because we know the dynamics of attainable set and minimize any kind of cost so this is the main message you know I started from problem this comes from q0, q1 but now I formulated it as a problem which should give you all possible solutions for all possible q1 if you for all q1 for any q1 its acid calculus is always like that in constrained optimization instead of study study system with precise conditions we consider family and characterize all possible solutions for all family somehow you divide your problem in two parts first from huge infinite dimensional terrible number of admissible controls you select those quite restrictive family which can be optimal and then working with this which can be optimal those which are extreme which go to which lead to the boundary and then among this finite dimensional quite restrictive family extreme control you solve boundary value problem try to characterize those of them those of them which connect points you wish and also the natural application of control theory also in engineering there are a lot of applications from engineering first it was engineering subject then it was developed a purely mathematical subject which can be applied to other parts of pure mathematics but again the basic idea is like that if you have some dynamics and cost also cost can be incorporated in dynamics then you at the beginning dynamics and cost not on the boundary condition and looking at this studying those guys who arrived to the boundary of attainable set you give rules of optimal behavior independently of boundary condition any this is for applications you can very naturally formulate it so if you want to be to be optimal you have to take your control very particular family family depends on the dynamic only and then in this particular family you can visit so main idea of mathematical theory engineers of course they have their condition they have to solve this but it essentially remains for them with their tools but mathematical question is to give this rules of optimal behavior independently on the boundary condition and we have such corresponding tools but first what we have to discuss this is one more one more one more remark maybe how to reduce what I say concerns concerns fixed time problem but actually any time problem can be very easily reduced to the fixed time again when you take reduction one problem here we added to reduce optimal control to reduce the problem minimizing integral cost to the characterization of the boundary we added one variable in the state space m if you would like to reduce problem with free time to the fixed time we add one control parameter there are two examples of simple manipulations and reduce one problem to another what control parameter we use because the advantage of this setting is quite general dynamic you may put here a lot of different problems in one setting one one how to reduce free time to fixed time very simple x dot equals f u of x of t you consider such a system and then reshcale time you simply make change time you take a time reshcaling and you consider some time substitution of time phi not phi already being a which stands monotonic function r to r a dot call it say b and consider problem like this so you go along the same trajectory more or less because u is there the same trajectory was different velocities it's one more freedom if you have two time you have free time it means that you you don't have other trajectory but you can you can change velocity so you write system like that x dot equals v f u of x where and now your control control now control is v u v scale belongs to r plus multiplied by u and we have a problem and now we can take fix time reshcale time we can always choose fix so and this is just just two examples but there are many other formulations that each time you reduce essentially the problem to characterizing characterizing guys which go to the boundary ok so and let us try to understand how to solve this problem how to characterize this extremal control that is the most fast or most slow that goes to the boundary and first of all of course we have enough of them to characterize the boundary and to solve optimal control problem only if boundary is really attained only if attainable set is closed ok if attainable set is closed so you see we have then the minimum exist minimum if not unlikely of course you can go deeper deeper say which kind of the boundary is here which is not but at least for the purpose of the course these details are not important at all we would like to discuss principles ok when we can guarantee that the guys close first of all it's not a good idea bad idea to assume that you is compact of course that control is really it seems to be very restrictive because classical problem u is not compact u is free and still we often have regular problem with existence theorem but if you think about this this is one so we assume now that u is compact for existence as I already mentioned we need more than the equation which characterizes optimal control if they exist optimal control characterizes determined by some equation it satisfies some vc, protragon maximum principle even if u is any set but we don't know if it exists if it exists it must satisfy this conditions but to be existed but u compact is certainly first of all how we reduce very briefly variational problem one more reduction I don't explain otherwise I don't go far but just give a hint how in principle you can do it you know in calculus of variation to guarantee the existence you need that the Lagrangian grows fast enough when you go to infinity it grows fast enough when you go to infinity and otherwise it doesn't exist but then what is the idea what is the idea you take your cost you take your cost as a new time your cost as a new time so you start from the problem say q dot we'll see you in calculus of variation but you can generalize it that maybe some dynamics which simply that you can be free and dynamics should grow f u should grow less than not so fast divided in 5 q u as a new we take new times if we take new times say s is a integral of 0 dt phi of q u dt is then dq over ds is going to be to be like that is going to be like that and this if this guy has super linear growth when you go to infinity then the right hand side go to 0 when you go to infinity and super linear growth is a natural conjectural with total guarantee existence and then you can simply compactify u instead of u u was here some rn and you simply compactify it add infinity and get compact guy as sn and your system is well defined if you see on the conditions which people put for existence the exactly condition that guarantee u that it's still sufficient for us to have a well posed question including infinity when you compactify it you take u this one more reduction and you appear in the certain vr with a compact u it's also possible but compact u is not sufficient that is very clear should be clear also from the previous lecture it should be clear from the previous lecture but it's not sufficient because you know that if we what does it mean compact u it could be just two points for example two points are compact we have one dynamics u1 u2 and as I explained last lecture we can and if you consider the system q dot say alpha f u1 of q plus minus I can write even v to see that it's closer to control 1 over v of t then we can always treating v as a one more control we can always approximate we go like that like that like that and we can always approximate it by admissible trajectory but the trajectory itself is not admissible this trajectory satisfies v smaller than 1 I take a convexification any trajectory whose velocity belongs to convexification can be uniformly approximated by admissible but it is not admissible and of course for simplest possible problems with constant vector fields if vector fields are constant so these guys are indeed indeed broken lines particularly if it is piecewise constant what we are minimizing is lens then of course minimally straight line and it is not admissible but if you convexify we have it so absolutely if you would like to have general result absolutely unavoidably to moreover it is very natural instead of fu to take a convexification of fu and convexification may be realized again by adding if you want by adding extra parameters because if we have if we have a set in a rank here in the tangent space to a dimensional manifold then any point in the convexification can be obtained as a convex combination of no more than n plus 1 points so you add this coefficients of convexification of linear combination adding new controls you take your system with a convex and convexification is the first step to do because indeed it is like to go from rational number to real numbers we can approximate anyway we can uniformly approximate any trajectory with convexified velocities and it's when we would like to have something really close it's like approximate real number by rational sequence and the new point so if something close it is the first step to do it's not restrictive condition it's a net of your problem you have hidden convexity here if you you have a chance to have existence if you are convex and if you are not convex maybe you have existence but anyway you can approximate orbitary well your optimal solution that you obtain from the convex problem so main conditions now this theorem is a philipov-serium but what I would like to say this philipov-serium rather simple by the way but it should be yes, since it's in Russia it is philipov and from in sixties proved even in sixties that contain if you think about that use some reduction all basic existence theorems in calculus of variation it says there is a fall and if we have a dynamic q dot equals fu of q q belongs to u this is compact compact and fu of q is convex if your manifold is not compact you need some this I perhaps omitted some gross condition of f simply to guarantee that solution will not blow up I just say that in particular if this guy grows in a rank this guy should grow not faster than linear functions when q goes to infinity or it's not necessary a rank you can formulate it also if m is a complete you need some estimate on infinity and in principle your system does not even depend on you you have dynamics I did not discuss it just to economy time sometime but in fact the flow if you have a vector field in general you do not have flow if you are not on the compact manifold because trajectory is defined only for small time and may blow up if you take control from you it's easy to write but it's not not related very much to what we are studying so if we we use just controls from compact set from this compact set and go that time then we stay in the compact we start from q0 take control from compact set that assumption we always mean or you may simply assume that the manifold is compact that is also not so terrible assumption because this compact can be extremely big if you are interested in what happens in some domain you do not care what is about what dynamic is about your trajectory also you can multiply right hand side by some cut off function because you are not and then everything will be fine no growth at all out of compact because you are interested only in trajectory that stay in some compact that go out you do not care essentially our series about that because of that I do not discuss much this completeness of right hand side of right hand side of properties ok so convex and some conditions some growth condition on f completeness plus completeness this is compact this is and also f u of t are complete vector fields in a random complete vector fields in a random for this is sufficient that f u of q smaller than c 1 plus a growth linearly that's everyone should know in particular this is this is one of condition and it's analog in the completeness manifold is also easy to write ok so this theorem says that in this case then a t a q0 of t is closed is compact simply compact is bounded because we assume that blow up is not possible and it's closed main main main statement is closed that is not it's rather simple I do not want to because already one half of my half of my lecture passed and I do not want to discuss a lot but maybe I say something how you how you prove it first of all since you consider admissible trajectories actually we prove even more not only attainable set for us it's most important that attainable set it's compact for the existence it's important for us but in fact also also the set of admissible trajectories on given segment are closed and compact incisirutopology incisirutopology first of all since we assume that velocities are bounded we do not have blow the trajectories are uniformly bounded if you consider u of t of q of t since trajectories are uniformly bounded right hand side is continuous then we start from point q zero then we see that since velocity is bounded all trajectories are Lipschitzon with the same Lipschitz constant so it is heavy donut not convexity convexity of the right hand side they are all Lipschitzon with the same Lipschitz constant so they are pre-compact in the incisirutopology so for us as a theorem there is if you have u one of dot some sequence of now what we we would like to show that attainable set is attainable set is closed okay so we have a sequence of trajectories such that end point converge you have these guides q of u n of dot u n say i omit this dot at u n and we know that q of t one u n converge to some point q hat say okay we would like to show that to find control such that such that that to show that u hat such that then this is a question they exist u hat such that q is q of t one how to do it we cannot guarantee that it's a problem for the moment I never used convexity of right hand side we can guarantee that exist limiting sub sequence of trajectories so we can go to the sub sequence we may assume that q of t u n uniformly converge to some trajectory to some to some curve to some say gamma t is the same leapshitz constant that's all we know from Oskola Art Salasseri but this of course do not converge velocities the trajectories uniformly converge but the velocities do not converge more of u n do not converge in general but the trick is and at this point we have to indeed in general since velocities do not converge that in general this trajectory is not admissible but if you think about that you see that convexity save save us and if right hand side is convex then we find another control not limit of this one but another control then velocity of limiting will be always in the convexification of this without details I just it follows from the you know that we have a limiting guy which is leapshitz it's differentiable everywhere what remains is to find to find velocity of this limiting curve and to show that it belongs to the convexification of velocities of admissible velocities it's rather simple we take this gamma of t plus epsilon we take first we take q n at t plus epsilon minus q n of t we remember what does it mean velocity and you know that this satisfies our differential equation so we rewrite this integral equation it will be one of epsilon from t till t plus epsilon f q n of dot of q n of dot of t of tau d tau this is just because there are solutions to before limit there are solutions to our differential equation and look this guy converges moreover this is uniformly converged uniformly converged to gamma so in this way we obtain this difference when we go to limit here and here this is not this is not but we take anyway one over epsilon and integral it's average of this guy it is average so it belongs to convexification and in the limit we obtain that gamma of t plus epsilon minus gamma of t divided by epsilon belongs automatically belongs because each time it belongs to convexification and this converges to gamma or this convexification are compact so everything converges uniformly and it's belongs to one over epsilon t plus epsilon and here we have a cone f u of gamma k of tau d tau for small epsilon again gamma then we go then we send epsilon to zero this goes to the derivative of gamma and here we get just some element of convexification of f u at point gamma of t because here we have a union of convexification or better to write here not like that it is more maybe more clear to take to be sure a little bit bigger space t tau t smaller than t plus epsilon f u f u of gamma tau this is for sure because all of them belong to it so this difference for sure stay because when we take integrals we do not leave this we do not leave this space and then when epsilon so you see that velocity belongs to convexification there is still boring detail that velocity so if velocity belongs to convexification and we assume that the set of admissible velocities is convex then for any we can find some u such that gamma dot is f of from the convexification from the convexity condition we get that gamma dot f gamma dot of t is f of some u r of t of gamma of t that we obtained and and still there is some boring details we need some force to show that this u can be selected measurable it's not automatic but it can be done it can be done u always can be selected measurable because control it's not some function on t but it's still measurable but it also can be done no problem and that's it that's all so if we are natural condition which I add immediately natural condition for existence this is natural conditions that u is compact and velocity are convex but for now I would like to try to characterize to characterize control which go to the boundary I can do it now more quietly but there are natural conditions when they exist but when I characterize it I do not care I just assume let this guy goes to the boundary and try to see which condition it's a bit spicy and what conditions are called panthragon maximum principle again I do not give many details but I try to explain actually what is going on maybe I even formulate formulate formulate maybe first some hints and then I formulate even formulate the problem I do so the geometric picture because geometric picture is important here so we have q0 we have attainable set we have trajectory which goes to the boundary of attainable set okay aq0 and trajectory goes to the boundary okay first thing to do is for the moment what I do now it's totally realistic because in the panthragon maximum principle appears some extra parameters which at first glance can be look quite artificial and first I just explain the meaning of this parameter without rigorous mathematics what means essentially this parameter and then I formulate the result when you already understand that parameter somehow natural and then I try to explain I believe that I will have time why it is correct how to prove it so assume that everything is perfect absolutely perfect that's attainable set maybe quite complicated but to explain meaning of this extra parameter I assume that it is absolutely perfect it is a smooth manifold with a border and then if it is smooth manifold with a border then there are tangent plane tangent hyper full dimension if it is not full dimensional then our condition is going to be empty because it is not full dimensional then any control lead to any point in the boundary but we know we already studied this question we have natural condition one thing is full dimension this we already know so we assume that it is full dimension because it is to degenerate problem and actually normally unfortunately in these four lectures I cannot explain everything but if that full dimension guaranteed by breakage generating condition but if we do not have breakage generating condition it means essentially that we have some variables touch that your problem simply does not depend on this variable we need to reduce if the algebra this algebraic object is low dimensional then we can reduce our problem to the low submanifold weight is full dimensional it means that that essentially means that all m is sliced by submanifold we have some filiation such that attainable set belong to the leaf of the filiation and then we simply reduce things to the leaf also this part of this area so again like convexification, breakage generation condition is somehow is not really restriction it simply give you that all variables your problem depends on all variables otherwise you exclude also it does not depend on so fine but assume that it is body like that and then there is there is a normal there is a hyper plane that that assume that this is velocity of our trajectory velocity gamma dot and it looks like that so we arrive on the boundary there is a normal and this in order to characterize those which go to the boundary it is actually most natural to characterize to characterize these covectors which we can also normalize so we consider let us lambda is a belongs to t star of gamma of t m it is a linear form lambda is a linear form of t m to r linear form on the tangent on the tangent space and such that it is it is annihilate the boundary the tangent space to the boundary and we can also normalize it such a way says that that lambda gamma of t dot is a constant is a constant is a same one say one now look since we deal with evolutionary problem that if if gamma of t I claim that if gamma of t belongs to the a to the boundary of a q zero of t then gamma of s belongs to the boundary of a q zero of s for any s smaller than t for any s smaller than t why it is like that why it is like that it is also very easy easy to see here it is like that because of the fact which we always always use in the controllability in the study of controllability because if you have admissible trajectory it's it's ruled by some control and this control give us the whole flow not one trajectory but the whole flow look assume assume on the counter I would like to explain this property assume that on the contrary there is some s that that gamma of s belongs to the interior of a q zero of s and this gamma gamma dot is f u of d is a solution of this OD it's particular value just we studied now one trajectory but it's give you you can call it you gamma if you have admissible trajectory you have some control but with this control now consider the flow p to t that send any q of t in the q of t invert you or say for us it could be s maybe better invert you of the equation q dot equals f u gamma of t q ok so what we do now to obtain obtainable set we fix the initial position and apply all possible controls but now since we have one particular control we fix this control and apply it to all possible initial conditions we always play this game this game also in controllability theory we did like that so this is just a defymorphism p to t is invertible defymorphism mm mm and moreover p to t minus one minus one is just p t p to t s it's defined also not necessary when t is bigger than s when you just take a fixed od e time varying od e you can go back forward whatever you want so it's invertible if t is it's invertible you can go back and forward you have a unique solution and then what you do assume that you are in the now take your q zero your attainable set a lot of trajectories that cover that is our gamma and this is a q zero of s ok we are inside this is gamma of s gamma of s then what we can do I claim that if we apply p s t and trajectories of p s t are all admissible because all of them respond to the same control gamma if we apply p s of t to a a q zero of s then it is obviously contained in the a t a a q zero of t what we do here we use all possible controls until the moment test until the moment test we use all possible control to get our our attainable set and then apply special control just this is a u gamma this is q of t u gamma and then apply the flow so we take this set at moment test and apply flow but flow is a defamorphism so what we obtain will be subset of this guy those which obtained with very particular control so before s it was arbitrary control and then just fixed and since defamorphism is invertible so if we were in the interior here we will be in the interior here so if we at the end point at the border we were in the border all the time and you do not need it is almost set theory all you need is correctness of Cauchy problem you do not need special estimations so it is absolutely universal property due to evolutionary nature of our it is just evolutionary nature of our problem just to see why this property are going on so when we go ahead we have more and more possibilities new and new possibilities appear all we reached is somehow we already more and more controls set of controls set of controls control function on segment s can be treated as a subset of control function on segment t so our possibility is increasing it does not mean of course that attainable set is increasing because it may happen that it is moving but it is increasing in this sense which I explained so that's what I would like to and now we know that for any s we have a lambda of s this we call lambda i'm sorry this we call lambda t and for any s for any s smaller than t we defined lambda s that in the same way in the same way this is as that is normal since we always on the boundary on the border we may define lambda s then such that this is a a0 of s such that this is gamma of s this is a point gamma of s such that it is a normal vector to to the attainable to the boundary of attainable set again of course what I write here it's all always only if boundary is fine but the general result we will not depend on these details the point is the message is that this lambda so we have a lambda of s lambda s which belong to t star gamma of s m we have this attainable set moving but we always on the boundary so there is a vector in the nice case if attainable set is nice it is like that so we have a lift of our trajectory to cotangent bundle and and conditions characterize this lift maybe those who who remember Huygens principle it is essentially wave propagation is essentially version of the universal Huygens principle so principle it's natural to characterize such a in mechanics it would be momentum this covector would be momentum in mechanics because mechanics is also ruled by by the variational principles but this action principle and but this has so general nature for optimal it can be defined for optimal control problem in extremely general setting without this condition on the smoothness of the boundary so we have for any good trajectory extreme trajectory there is such an ellipse like a draw picture so this is going to be m this is t star if this is q0 this is t star q0 and this is all t star so we have our trajectory here unfortunately since cotangent bundle is even dimensional it's hard to draw pictures so you can draw it only 4 dimensional picture you can draw and I need to use one dimensional there is not a good idea because no many trajectories one dimensional but you can imagine that it is n dimensional this is n dimensional so we have our trajectories gamma of s t and we have over it we have this is fiber cotangent spaces this is a lift we have a lift in the cotangent bundle and the message is that this lift survive well all possible degenerate situation and now I formulate the main result panthragon maximum principle and this is a good characterization of panthragon maximum principle for this geometric version of optimal control problem then we just characterize trajectories which lead go to the boundary of attainable set theorem theorem so this guigan's principle can be applied to very general situation so it is panthragon maximum principle so pmp panthragon maximum principle so write the following I even rewrite this this is f u of q this is our gomiltonian system I'm sorry this control system but very soon will appear also gomiltonian system and and then we write the following function on cotangent bundle elementary function simply lift this vector fields to cotangent bundle so so we write hemiltonian of your system is h u of pq this is a this is a just p f u of q nothing else and as we know I do it in coordinates but you perhaps know that that if you have a cotangent bundle or any function you can associate hemiltonian vector fields and and then then the theorem then the theorem the theorem theorem p of p if this is notation and now the theorem if q of t q of dot belongs to the boundary of the attainable set then there exists a solution of the hemiltonian system then there exists lambda s that is lift belongs to the t star of gamma that's what I thought gamma s of m lift to cotangent bundle and the meaning of this lambda you already know such that the theorem our conditions are only only I recall that we have a continuity with respect to u of the right hand side smoothness with respect to q how many derivative you see but you see not too much I think I don't care and u is any subset of some finite any not necessary close but if you if it is not closed if you do not have convexity we do not reach all points of the boundary but if by chance we reached one by some trajectory it must satisfy this condition that's the point and now conditions such that this lambda is now look this lambda actually let us write this one is do not introduce all intrinsic notations so it's good idea to think about lambda as a point in the cotangent bundle because we we would like now to write a dynamical system in the cotangent bundle but actually in coordinates you can split it this lambda s is p s gamma of s so if we have coordinates so we have a we have a point here and we have a point on the fiber momentum then then there exists lambda s that is p s gamma of s such that it satisfies Hamiltonian system so p dot equals minus dhu over dq and gamma dot is dh over dp u of s if our time is now s and t is just the last time last moment of p s gamma of s and this is dh over dp of p s gamma of s so if we have such a Hamilton it satisfies such a Hamiltonian system so for the moment it is just nothing of course we can have any control trajectory, we can write Hamiltonian system and any solution of this Hamiltonian system we need a condition which eliminates u this is not one Hamiltonian system but Hamiltonian system depends on the orbitary function u of s we need a condition and this is the so it satisfies this Hamiltonian system and also also we have this Hamiltonian let us introduce one more Hamiltonian sometimes people call it master Hamiltonian just h which depends only on pq only on pq here lambda is here lambda is pq lambda is pq you always think in two ways because intrinsically you cannot separate without coordinate p of q but lambda is defined also intrinsically covector is defined intrinsically Hamiltonian h of pq is defined like this so you take p and apply to f of u but p is actually u of q but it can be also written intrinsically and covector lambda apply to tangent vector u of q covector lambda belongs to cotangent space this is the space of the lambda and of lambda and if you can see this each lambda knows its q because it is cotangent bundle is a fiber bundle we take any point it belongs to some fiber and this fiber is q intrinsically it is like that you do not separate you have fibers but you do not have horizontal guys horizontal guys appear only in coordinates so lambda is pq this is like that so this can be written also in more intrinsic ways lambda dot equals h u u of s of lambda of lambda s vh v0 is a Hamiltonian vector field associated to this Hamiltonian function h this system can be written other way and now condition which which allow us to eliminate to eliminate u so condition is like that that h u h u of s of ps gamma of s if you want it is lambda s just this p is equals to the maximum sorry, what is written here h u h u is like that sorry, I did not write it h of pq is the maximum maximum for all u of h u of pq master Hamiltonian is the maximum the maximum so here we have a family and on the parameter u to eliminate this family we take a maximum so the condition is that this if we are extremal if our trajectory goes to the corresponding trajectory goes to the boundary of attainable set then there exists solution of this Hamiltonian system such that along this solution our Hamiltonian attains this maximum with respect to u always maximum with respect to u so this is something that is always true always valid and in such general situation now I give an exercise simple exercise of course in our setting this maximum is not is not something non-smooth certainly non-smooth it's maybe non-smooth function but in particular if our u if u is segment in particular let us we start with the most simple problem when we have just two dynamics first step is to convexify them this is a point q u1 of q this is fu2 of q so we may always assume minimal possible situation and when dynamics is a segment if we take now maximum and what you would like to what is written here is maximum you take some covector p covector p some linear form you simply maximize linear form p psi in this situation h of p this velocity start point q q is the maximum maximum of p psi where psi belongs to our segment we take maximum with respect to all admissible velocities of psi where psi belongs to this segment they data belongs to data clearly it's continuous of course but of course it is not smooth because when you p is transversal to the segment that it is just a linear function because the maximum is attained then maximum is attained on one and the same p but when it becomes but then you switch and smoothness and your function is not differentiable because your jump when velocity jump if p is like that then maximum is attained here if p is like that linear form is constant on data but then it switched to other but if p is like that then it switch here so you have a kind of derivative with jumps so in general it is like that in many interesting situation it is like that somehow it is like that always and that for us is most interesting situation when this attainable set when set of admissible velocities has a smaller, it's some convex set if it has a smaller dimension what most interesting for us it's when it has rather big dimension if it's smaller dimension than ambient space then you do not have smoothness you always you always jump because for some p are going to be orthogonal to this attainable to this set of admissible velocities and pass p which is orthogonal you always jump you have discontinuity of various kinds so for optimal control problems we have to deal with this situation but this guy this is always true anyway these conditions are always true anyway but if we are likey if we particularly in regular variational problems we are indeed likey this maximum is smooth if attainable if set of admissible velocities strictly convex body then maximum is smooth full dimension then maximum is smooth then maximum is smooth then assume that this is smooth exercise if c2 if h of ps of pq is a c2 function then the pair p of s gamma of s satisfies pmp satisfies pmp if and only if it is a it already shows essentially what this condition if and only if it satisfies Hamiltonian system with h control here in this situation we can first eliminate control simply maximizing Hamiltonian and then write Hamiltonian system and all solutions are here gamma dot equals dh of dp of p gamma so this is a simple situation this is our optimal low for good behavior for good behavior very clear recipe we write Hamiltonian which depends on you eliminate you simply maximizing maximizing the scalar product and all all solutions are are here so it looks like that it looks like that and somehow p instead of control now we have p but p is well controlled we have this p but p is well controlled and somehow if you count parameter this is the maximum we can do indeed look in our cotangent bundle this is now m you assume that this guy has the same dimension t star m fibers have the same dimension we have some point q0 q0 and we have attainable set at any point this is full dimensional at any point of attainable set we have some trajectory if trajectory goes to the boundary it means that they exist but when H capital master Hamiltonian is smooth it means that that the derivative is fine is Lipschitz C1 this determines the flow if we have q0 and also p0 then we have well defined trajectory unique well defined trajectory of p0 and this p0 p0 we have n-dimensional family of trajectory starting from q0 but in fact if you think about it little bit that that since Hamiltonian is homogenous then Hamiltonian is homogenous because this is homogenous with respect to p this is even linear linear with respect to p if you maximise linear function you obtain some con by the way it is always con automatically convex it is a function of p but it is not homogenous just some convex functions you see that this admissible is low dimension that is never smooth but in full dimension it is smooth so my time is over I am just finishing few words to say since it is homogenous then reshkilling of p does not change if you reshkail p you reshkail homogenous of good degree 1 you just reshkail the whole trajectory ok then p0 by constant you simply multiply trajectory you obtain it is simply multiplication of your original trajectory by the same constant so this this the q is so in fact the q of t up to multiplier is determined by p up to scalar multiplier ok so we have in fact really we have only n-1 dimensional set of admissible parameters of initial conditions and we have n-1 dimensional border of obtainable set and due to existence theorem to any if you are convex of course now assume that we if we are not convex at the very beginning that Hamiltonian again not smooth because when we take convexification we have some flat pieces in Hamiltonian not smooth some p can be orthogonal to the whole phase but if we are convex we know that there is optimal trajectory going to any point on the boundary attainable set is closed and we have n-1 we need n-1 dimensional family of trajectories to cover the border to characterize attainable set optimal guys and we have this theorem give us already n-1 dimensional family so minimal possible the law of optimal behavior so that's in the smooth case but the main point the advantage of this result why it is more powerful more interesting than classical things when you also can find these rational mechanics and then calculus of variation then I many times saw that it as it formulated it works even if maximum is not smooth if maximum is not smooth this is not defined it is not so clear how to interpret solutions and not even if you can invent a way to interpret solutions of somehow right hand side of discontinuous right hand side but not at all all solutions are interesting for optimization problems but this result always true so the good idea it was not to eliminate you from the very beginning but first right ODE where you is presented and additional so the whole system to study and essentially when you study it even here everything is well defined because here we have we have a smooth right hand side it's not smooth only with respect to TIBA that for us is not a problem that we always can do it's well posed Kashi problem and the whole system of condition is such differential equation Hamiltonian system and and identity and point wise identity so this is valid for all S for all S from zero T so you should first right ODE and then use this condition to eliminate and of course there is a well developed theory how to do it how to characterize different kind of of of trajectories which satisfies that also can be brain but in principle it can be normal situation is just to finish unfortunately I do not I do not have time to explain the proof the complete proof is bit technical I have an idea to have a simple explanation but perhaps I need one more lecture for that but at least I showed the meaning of this covector so in general of course this guy is not smooth but we consider first order approximations of attainable set locally we are in local first order approximations and first order approximations due to hidden convexity which you perhaps already understand is a kind of convex convex cone and we take this covector which defines the plane of support for this convex cone that what we do and if convex cone is full dimensional then also because it is first approximation it is full dimensional higher order terms do not destroy it is full dimensional then in first approximation you are inside you cover the whole neighborhood and the higher order terms do not destroy this fact it is some elaboration of not exactly implicit function theorem but technically not much more complicated then implicit function theorem so you characterize it just normal covector to first order approximation and it is important the first order approximation is convex for any problem because of that hidden convexity because of relaxation of hidden convexity essentially that's how it works and when we apply it the situation is like that so I already draw this picture with a segment or it could be something more complicated of lower dimension so in fact of course your maximum is not smooth but it is convex smooth with respect to Q and it is smooth on some domains on some subdomains in smooth and then you have a conos angles and things like that so you have object to study in fact in the domains when it is smooth you are governed by such a Hamiltonian system you have different smooth Hamiltonian flow that links non-smooth subsurface surfaces of non-smoothness and what you have to study and this is a sufficient material to understand many things what you have to study how good trajectories which satisfied globally for all team for dragging maximum principle must be must be glued in this surfaces of non-smoothness some of them belong to the surface of non-smoothness for the most interesting maybe just go along the non-smoothness surfaces but they also can be studied so this is essentially what you do in optimal control ok thank you for attention I finish my presentation