WEBVTT 00:00:00.000 --> 00:00:19.040 So today we have the pleasure to welcome to our seminar Professor Weiwei Wu. 00:00:19.040 --> 00:00:25.400 So she is the Associate Professor of Mathematics at the University of Georgia in the US. 00:00:25.400 --> 00:00:37.400 Before joining the University of Georgia, Dr. Wu held a tenure-track position in mathematics at Oklahoma State University from August 2016 to July 2019. 00:00:37.400 --> 00:00:52.400 She also held a postdoctoral fellowship at the Institute for Mathematics and its Applications, University of Minnesota, on the program Control Theory and its Applications, from September 2015 to August 2016. 00:00:52.400 --> 00:01:04.400 She is also a non-tenure-track assistant professor position in mathematics at the University of Southern California from August 2012 to May 2015. 00:01:04.400 --> 00:01:11.400 She received her doctorate in Applied Mathematics at Virginia Tech University in May 2012. 00:01:11.400 --> 00:01:24.400 Dr. Wu's current research interests include mathematical control theory of partial differential equations, control and estimation of flow transport systems, and computational methods for optimal control design. 00:01:24.400 --> 00:01:35.400 She has been awarded various grants from the NSF and the DOD. She was also awarded the Wimbledon Return Fellowship for Experience Researches in 2004. 00:01:35.400 --> 00:01:39.400 So, Professor Weiwei, the floor is yours. 00:01:40.400 --> 00:01:43.400 Thanks, Andreas, for your kind introduction. 00:01:43.400 --> 00:01:54.400 Today I'm going to give an introduction, actually, on control design for mixing and compressing flows, the topic I have been working on in the past few years. 00:01:54.400 --> 00:02:00.400 So, this project has been supported by NSF and the Aero Force. 00:02:00.400 --> 00:02:15.400 So, of course, first of all, I'm going to explain what's so-called mixing and how we're going to achieve good mixing and how we construct our control for possible optimal mixing. 00:02:15.400 --> 00:02:18.400 So, let's start. 00:02:18.400 --> 00:02:25.400 All right. 00:02:25.400 --> 00:02:32.400 So, first, so this is my outline for today's talk. So, I think I have two hours, right? 00:02:32.400 --> 00:02:40.400 I'll try to keep it within 15 minutes. So, I will start with mixing phenomena. I'll explain what's so-called good mixing. 00:02:40.400 --> 00:02:47.400 And then we mainly will focus on objective mechanisms for mixing. 00:02:47.400 --> 00:02:55.400 And then, of course, here, mathematically, we have to be rigorously talking about what we mean by good mixing, what would be a mathematical measure to quantify mixing. 00:02:55.400 --> 00:03:01.400 And then in the end, I'm going to focus on the control design, both open and closed control designs. 00:03:01.400 --> 00:03:10.400 So, mixing. 00:03:10.400 --> 00:03:21.400 So, basically mixing is to disperse one material or field in another medium. It occurs many natural phenomena and industrial applications. 00:03:21.400 --> 00:03:29.400 Actually, we say this every day in daily life, like mixing in painting, mixing in baking. I guess everyone has this experience about this phenomenon. 00:03:29.400 --> 00:03:44.460 And from large scales, such as spreading the pollutant in the atmosphere and mixing of temperature salt and then the 00:03:44.520 --> 00:04:06.520 And then controllable and fast mixing is corrective in practice, actually, for microfluidic devices, especially with light on chip devices. 00:04:06.580 --> 00:04:18.580 So, basically mixing, as you've already sort of had a basic idea about the application I'm showing. So, mixing means a basic process where both are sterile. 00:04:18.580 --> 00:04:28.580 So, here, sterile is especially referred to as the objective mechanism. Later on, I'm going to come back to this more on this term. 00:04:28.640 --> 00:04:41.640 And then the diffusion occurs simultaneously. So, now, what do we mean by sterile? Sterile means the advection of materials, material blobs subject to mixing without diffusion action. 00:04:41.640 --> 00:04:50.640 So, then let's recall what's called diffusion. Diffusion is the average effect of small scale random particle motion. 00:04:50.700 --> 00:04:58.700 So, in this case, we learned this in high school physics. And then based on this cartoon, actually, let's say, if you look at the figure A, 00:04:58.700 --> 00:05:08.700 so, say we have two fluids in contact with each other. And then the exchange molecules across an interface, next to have black and white, 00:05:08.760 --> 00:05:17.760 which stands for different fluids. So, the molecules crossing the interface between these different fluids actively divide the molecule's random action. 00:05:17.760 --> 00:05:25.760 So, figure A somehow shows before starting the exchange, the black and white blobs start to exchange. 00:05:25.760 --> 00:05:31.760 And then here, figure B shows its entire near state during the exchange. And then, as you can imagine, 00:05:31.820 --> 00:05:45.820 like, although this random motion occurs everywhere in the two fluids, but it does not really cause much difference for the same fluids which are far away from the interface. 00:05:45.820 --> 00:05:53.820 However, in the region near the interface, the molecules on both sides have different properties than random motion. 00:05:53.880 --> 00:06:01.880 The random molecule motion results in permission. If you look at the figure B, right, and then from one side to another side. 00:06:01.880 --> 00:06:07.880 So, such permission is so-called diffusion. And then if you recall the so-called the fake losses, 00:06:07.880 --> 00:06:15.880 the flux of one species through the interface is proportional to the gradient of the concentration of those two fluids. 00:06:15.880 --> 00:06:19.880 And this proportional constant is then defined as the diffusivity constant. 00:06:19.940 --> 00:06:31.940 So, the diffusion mixing can be made effective if there are sufficiently small blobs of one fluid immersed in the region of the other. 00:06:31.940 --> 00:06:39.940 And the length scale of these blobs reaches the diffusion length scales, then both blobs diffusive to the other fluids without including mixing. 00:06:39.940 --> 00:06:46.940 If you want diffusive mixing to occur. So, the diffusive scale of the blob itself is quite important. 00:06:47.000 --> 00:06:51.000 Otherwise, the diffusive mixing is inefficient. 00:06:51.000 --> 00:06:59.000 Then in summary, wouldn't mixing basically, especially here I consider like low diffusivity materials occur in two stages. 00:06:59.000 --> 00:07:04.000 The first sterium in the first stage, and then diffusion in the second stage. 00:07:04.000 --> 00:07:11.000 So, we'd like to sort of chew the property of the vector structure such as chaotic stretching and folding, 00:07:11.060 --> 00:07:18.060 which lead to fluid blobs or fine phalanenins, and then the diffusion is able to take in. 00:07:20.060 --> 00:07:24.060 But here in this part, we're mainly focused on advection dominant test. 00:07:24.060 --> 00:07:29.060 So, in this case, I'm going to assume molecular diffusivity is negligible. 00:07:29.060 --> 00:07:33.060 So, only focus on advection mechanism for mixing. 00:07:33.120 --> 00:07:45.120 And it is known actually, it's possible to utilize the purely advective mechanism to create an additional fluid velocity to obtain complicated mixing or chaotic mixing. 00:07:45.120 --> 00:07:48.120 Even the fluids itself is very regular. 00:07:48.120 --> 00:07:57.120 So, in smooth fluid fields, one is able to achieve quite nice mixing if we introduce appropriate advective mechanisms. 00:07:57.180 --> 00:08:08.180 And especially time variation in the velocity may generate chaotic transport in which it can be actually either or positive approaches. 00:08:08.180 --> 00:08:11.180 So, when we introduce this time variation. 00:08:11.180 --> 00:08:16.180 Now, what do we mean by active or positive approaches for mixing? 00:08:16.180 --> 00:08:20.180 Let's see, I believe these figures are quite familiar with this. 00:08:20.180 --> 00:08:24.180 Think about when you mix coffee with milk. 00:08:24.240 --> 00:08:29.240 So, basically the active approach means supply and energy to the system. 00:08:29.240 --> 00:08:35.240 So, serum, so you can use a spoon to stir the coffee with milk. 00:08:35.240 --> 00:08:41.240 Then serum, the fluid, the back and forth can generate flat weighting velocity with respect to the flow of the barriers. 00:08:41.240 --> 00:08:45.240 Therefore, engender transport across them to achieve better mixing. 00:08:45.240 --> 00:08:50.240 Of course, you can buy something like this, it's more automatic. 00:08:50.240 --> 00:08:53.240 But I normally trust myself better. 00:08:53.300 --> 00:08:56.300 Well, so that is about the active method. 00:08:56.300 --> 00:09:04.300 In contrast, the positive approach does not supply energy, but uses positive mechanisms to add velocity agitation. 00:09:04.300 --> 00:09:14.300 For example, one can have this bends or baffles or curves in the channel to possibly generate anomalous velocities. 00:09:14.300 --> 00:09:19.300 Even the flow in this curved channel, the velocity is somehow steady. 00:09:19.360 --> 00:09:26.360 But when the flow passes through such baffles, it's able to cross the flow barriers and generate transport. 00:09:26.360 --> 00:09:31.360 So in this case, the so-called the passive mechanism for mixing. 00:09:31.360 --> 00:09:42.360 Then after all this sort of introduction, let's see, how can we mathematically come up with a measure to quantify the degree of mixing? 00:09:42.360 --> 00:09:45.360 So basically here mixing, I always like to be well mixed. 00:09:45.420 --> 00:09:50.420 So think about the fluid trajectories given by transport. 00:09:50.420 --> 00:09:54.420 It can be described by the transport equation, actually ODE. 00:09:54.420 --> 00:09:59.420 And in fact, equation one is a Lagrangian specification of the velocity field, 00:09:59.420 --> 00:10:04.420 representing the rate of change of position of each fluid particle given by its velocity. 00:10:04.420 --> 00:10:07.420 So this type of stands for the particle. 00:10:07.420 --> 00:10:10.420 But time t and the particle t are the same. 00:10:10.480 --> 00:10:13.480 So this type of stands for the particle. 00:10:13.480 --> 00:10:19.480 But time t and the position x, the motion of particle is by velocity. 00:10:19.480 --> 00:10:26.480 And solving the two equations, so given the initial condition, the Lagrangian trajectory in the particles in the fluid can be determined. 00:10:26.480 --> 00:10:29.480 So we can follow the particle, right? 00:10:29.480 --> 00:10:32.480 Follow the velocity track to the trajectory of the particle. 00:10:32.480 --> 00:10:39.480 And here we like to consider the case where the flow map is area preserving or volume preserving. 00:10:39.540 --> 00:10:42.540 So here we assume our velocity field is divergence rate. 00:10:42.540 --> 00:10:51.540 So in the case of D equals 2, divergence rate means the particle derivative of v1 with respect to x1 plus its component, 0. 00:10:51.540 --> 00:10:54.540 So that's the divergence rate. 00:10:54.540 --> 00:11:02.540 However, with this presentation, although we know the velocity of each particle, but does not give us information about mixing, 00:11:02.600 --> 00:11:11.600 then maybe one of the ideas is maybe we can sign the particle value, let's say theta, at any time t and the position x. 00:11:11.600 --> 00:11:17.600 So set here as a scalar, then which is conserved move around in the flow. 00:11:17.600 --> 00:11:22.600 Let's see. I think I showed this picture many times in my slides. 00:11:22.600 --> 00:11:31.600 Initially, we have a unit squared containing two fluids, black and white, and I use one stand for black and the negative one stands for white. 00:11:31.660 --> 00:11:34.660 So now I'm sort of signing the particle color. 00:11:34.660 --> 00:11:40.660 So the color one, negative one is giving back to me, referring to the value of theta. 00:11:40.660 --> 00:11:52.660 Now, among all those six figures, you may tell the last one is best mixed in the sense if you run me in for each black dot, 00:11:52.660 --> 00:11:55.660 you see the white dot is evenly distributed, vice versa. 00:11:55.660 --> 00:12:00.660 Then how come we describe what we have observed? 00:12:00.720 --> 00:12:05.720 So the last figure is the most homogeneous one among all the six figures. 00:12:05.720 --> 00:12:09.720 Then let's maybe consider the one-D case. 00:12:09.720 --> 00:12:16.720 Let's say if we have this one-D pair record interval, zero one, I'm going to define a measure of D 00:12:16.720 --> 00:12:25.720 which takes the average of my scalar theta at any point x and with any radius w half, the average. 00:12:25.720 --> 00:12:29.720 And then x, w can take any values from zero one. 00:12:29.780 --> 00:12:36.780 Then we're going to define this mixed norm by averaging D squared over x and w. 00:12:36.780 --> 00:12:44.780 So it's basically like you want to take any point in your domain and with any sort of room you don't alter any area, 00:12:44.780 --> 00:12:47.780 you want to say whether the two is mixed. 00:12:47.780 --> 00:12:55.780 And this metric or measure is precisely equivalent to the negative Sobolev norm, H negative one half of your scalar theta. 00:12:55.840 --> 00:12:59.840 So this is the first time to say this norm. 00:12:59.840 --> 00:13:02.840 I'm going to explain it a little bit more. 00:13:02.840 --> 00:13:08.840 For density field with Fourier expansion, we can always write theta in terms of Fourier expansion here in the domain. 00:13:08.840 --> 00:13:16.840 So here theta pay hats stands for the Fourier coefficients and this k here stands for the wave numbers. 00:13:16.840 --> 00:13:23.840 And then the H from the one half norm is summation is given by this series and the k is the vector. 00:13:23.900 --> 00:13:24.900 So it depends on the dimension. 00:13:24.900 --> 00:13:27.900 So of course in this case, my V is one. 00:13:27.900 --> 00:13:29.900 So k is basically this k. 00:13:29.900 --> 00:13:35.900 And notice that the Fourier, the wave number is in the denominator when you have negative in next here. 00:13:35.900 --> 00:13:40.900 Now let's take a closer look of this Sobolev norm. 00:13:40.900 --> 00:13:48.900 Why this norm somehow can be used to quantify what we just described or observed this homogenization process. 00:13:48.960 --> 00:13:53.960 So let's recall, let's say now I'm going to look at this in the V dimensional torus. 00:13:53.960 --> 00:13:55.960 You can think about this in the theoretical domain. 00:13:55.960 --> 00:13:59.960 With each dimension the length is L. 00:13:59.960 --> 00:14:00.960 Sorry, I put it one. 00:14:00.960 --> 00:14:02.960 So the length is one. 00:14:02.960 --> 00:14:07.960 And I'm going to assume the initial condition as zero means the spatial integral is zero. 00:14:07.960 --> 00:14:15.960 So then if one look at the Fourier expansion of such a scalar, then the zeroes order is zero. 00:14:16.020 --> 00:14:23.020 So then we can somehow define the so-called homogeneous Sobolev norm of index s by the following equation. 00:14:23.020 --> 00:14:26.020 This s for now can be any real number. 00:14:26.020 --> 00:14:39.020 So the homogeneous Sobolev norm, which s is given by the wave number takes norm to the power 2s back and then times the power 2 of the three coefficients. 00:14:39.080 --> 00:14:47.080 Now let's look at a case where if s is positive, then the norm actually must wait on a higher frequency. 00:14:47.080 --> 00:14:55.080 And then that's a function that have a smaller fraction of the mass in the high frequencies will be less oscillatory. 00:14:55.080 --> 00:14:58.080 And then have a smaller F-s norm. 00:14:58.080 --> 00:15:01.080 So you can think about if s is one, that's just as a gradient. 00:15:01.080 --> 00:15:05.080 If you have an energy conserved, let's say if theta is L2 conserved. 00:15:05.140 --> 00:15:10.140 And the gradient theta is bounded, that means certain smoothness. 00:15:10.140 --> 00:15:13.140 So that means oscillation is not that bad. 00:15:13.140 --> 00:15:20.140 Now in contrast, if you look at the case where s is negative, the norm actually puts less weight on higher frequency. 00:15:20.140 --> 00:15:26.140 Then the function that have a larger fraction of the Fourier mass in the high frequency will be very oscillatory. 00:15:26.140 --> 00:15:28.140 So this is actually what we want. 00:15:28.140 --> 00:15:32.140 If you recall the figures they showed earlier, black and white, they really merge very well. 00:15:32.140 --> 00:15:34.140 That means oscillation really bad. 00:15:34.200 --> 00:15:36.200 Between the black and white dots. 00:15:36.200 --> 00:15:43.200 The smaller F-s norm indicates sort of a growth oscillatory scenario. 00:15:45.200 --> 00:16:00.200 Based on the expression about the homogeneous oblique norm and connect to what we have just explained about this homogenization process, 00:16:00.260 --> 00:16:14.260 then it's reasonable to claim that the mixing of theta is equivalent to theta in terms of H-s norm where s is negative converges to zero for every s. 00:16:14.260 --> 00:16:20.260 So basically the mixing of theta is equivalent to the weight convergence of seconds when t goes to infinity. 00:16:22.260 --> 00:16:26.260 So basically here when I write the H-s norm converges to zero. 00:16:26.320 --> 00:16:35.320 So in terms of H-s norm where s is negative, there's a strong convergence in negative surface norm but weight convergence in L2. 00:16:37.320 --> 00:16:47.320 So in fact for any s smaller than zero, the quantity can be used as a measure of how mixed distribution is at time t. 00:16:47.320 --> 00:16:53.320 In other words, any negative surface norm which quantifies the weight convergence can be used as a mixed norm. 00:16:53.380 --> 00:16:58.380 And in fact there's a more detailed study I listed in the references here. 00:16:58.380 --> 00:17:04.380 And in particular in my following discussions I always consider s to be negative one. 00:17:04.380 --> 00:17:12.380 It would be computational convenience also the H to the negative one over L2 stands for certain scale. 00:17:12.380 --> 00:17:16.380 So I'm going to say with H-s equals negative one in that discussion. 00:17:16.440 --> 00:17:22.440 So the previous discussion is based on homogeneous sublet norm in the theoretical domain. 00:17:22.440 --> 00:17:24.440 Then we extend it to the general domain. 00:17:24.440 --> 00:17:29.440 Because later on when we apply my control design I would like always work on general boundary domain. 00:17:29.440 --> 00:17:39.440 So in the general boundary domain I'm going to consider the dual norm H1 prime of H1 space for quantifying mixing. 00:17:39.500 --> 00:17:47.500 So in that space H1 prime norm basically we can take the duality of f with any case of function v in H1. 00:17:47.500 --> 00:17:51.500 So we take this norm as the measure to quantify mixing. 00:17:52.500 --> 00:17:54.500 Any questions so far? 00:17:54.500 --> 00:17:56.500 Is everything clear? 00:17:56.500 --> 00:17:58.500 Right. 00:17:58.500 --> 00:18:02.500 Then I'm going to back to more mathematics. 00:18:02.560 --> 00:18:09.560 So now remember all the definition and then this idea is based on scalar field or somehow assigned to the particles. 00:18:09.560 --> 00:18:21.560 Then we call the total rate of change of the function theta as a fluid of parcels moving through a flow field can be defined or can be described by the Eulerian specification of velocity. 00:18:21.560 --> 00:18:23.560 Which is given by the transport equations. 00:18:23.560 --> 00:18:29.560 So basically if you apply the Periclorestics the Eulerian specification of velocity. 00:18:29.620 --> 00:18:37.620 So if you apply the Periclorestics the Eulerian representation of transport equation will give us the ODE we showed earlier. 00:18:37.620 --> 00:18:45.620 From now on I'm going to consider my domain is open bounded and connected and then the domain boundary is sufficiently regular. 00:18:45.620 --> 00:18:54.620 Then here again theta now stands for say in general mass concentration density distribution or the color of your tracer field. 00:18:54.680 --> 00:19:02.680 Now velocity here I'm going to assume it's incompressible and there is no penetration boundary condition. 00:19:02.680 --> 00:19:07.680 That is divergence base free and normal component velocity restriction of boundary is zero. 00:19:07.680 --> 00:19:09.680 So no fluids going out or enter. 00:19:09.680 --> 00:19:11.680 So the volume is conserved. 00:19:11.680 --> 00:19:15.680 And in fact one can use LLP norm. 00:19:15.680 --> 00:19:21.680 Well you can do the LLP norm estimate LLP norm of your density field always equals the initial LLP norm. 00:19:21.740 --> 00:19:25.740 This P can take from one to positive infinity. 00:19:25.740 --> 00:19:27.740 So basically we have mass conserved. 00:19:29.740 --> 00:19:35.740 So here I'm going to present some known results and open question in mathematical fluid. 00:19:35.740 --> 00:19:40.740 What's the year sort of focus in this problem or what do you care about? 00:19:40.740 --> 00:19:43.740 So if we look at the transport equations. 00:19:43.800 --> 00:19:51.800 One of the problems here people care about is what exactly is the relation between the velocity and the period. 00:19:51.800 --> 00:19:59.800 So the velocity field is actually we like to control or manipulate for cheating mixing. 00:19:59.800 --> 00:20:02.800 So of course the first question is what's the relation? 00:20:02.800 --> 00:20:12.800 So in the paper by Aberti, Krupa and Mazzucato in 2016 they said okay let's say assume the initial condition just essentially becomes zero. 00:20:12.860 --> 00:20:18.860 So the initial condition is essentially bounded and then spatial value the mean zero. 00:20:18.860 --> 00:20:22.860 They actually also assume a similar structure. 00:20:22.860 --> 00:20:30.860 So they figured that they exist let's say the last thing WSP so sub-relative space. 00:20:30.860 --> 00:20:40.860 The spatial derivative is of order S and then this P centriple after take S order derivative is still LP integrable and uniform in time. 00:20:40.920 --> 00:20:53.920 So P is between one and infinity also contains the bounds such that they show that if S is smaller than one for example if the velocity in L2 in space perfect mixing in finite time is possible. 00:20:53.920 --> 00:20:59.920 In the sense there exists a finite time such that when T approaches this T star the mixed norm converts to zero. 00:20:59.980 --> 00:21:07.980 So if they improve the regular velocity let's say if S is one perfect mixing is no longer possible. 00:21:07.980 --> 00:21:11.980 They can show the polynomial because the mixing order is polynomial. 00:21:11.980 --> 00:21:21.980 And then sorry especially and also Oh Jean-Denis Lee said in a material in 2023 a very sensitive paper still on archive. 00:21:22.040 --> 00:21:31.040 The construct then of alternating piecewise linear shear flow which actually is Lipschitz in space and time periodic for exponential mixing. 00:21:31.040 --> 00:21:37.040 So this is one example the construct demonstrate exponential decay is possible for S equals one. 00:21:37.040 --> 00:21:39.040 So Lipschitz is W1 infinity right. 00:21:39.040 --> 00:21:49.040 Now the question is if one increases the regular velocity field let's say if S is strictly greater than one. 00:21:49.100 --> 00:21:59.100 They showed that they can get polynomial decay and the back then it says was unknown whether the mixed norm decay is exponential in time for some S greater than one. 00:21:59.100 --> 00:22:05.100 However later on Oh Jean-Denis Lee showed actually the answer is affirmative. 00:22:05.100 --> 00:22:11.100 If S is greater than one smaller than two in particular has an upper bound of one plus four five plus. 00:22:11.160 --> 00:22:21.160 And P is in this interval that actually showed that should construct the velocity explicitly to show exponential decay is in fact impossible. 00:22:23.160 --> 00:22:32.160 So then later I'm going to come back to those discussions when we incorporate the control times. 00:22:32.220 --> 00:22:44.220 But now I'd like to understand when I first say actually mixing problem back in 2015 Charles Doren is one of the early people looking at mixing using mixed norm from University of Michigan. 00:22:44.220 --> 00:22:46.220 He gave a talk at the USC. 00:22:46.220 --> 00:22:55.220 So normally when they look at this problem they always assume velocity satisfies the constraint and what would be the best mixing or optimal mixing. 00:22:55.280 --> 00:23:03.280 So I was thinking maybe we can incorporate the dynamics associated velocity field because in real life velocity is always governed by some flow dynamics. 00:23:05.280 --> 00:23:11.280 So then my question is how can we optimize mixing by active control through dynamics. 00:23:11.280 --> 00:23:23.280 So as I explained earlier mixing can occur in turbulence flows as well as nominal flows with Reynolds number actually very small and even one even less than one. 00:23:23.340 --> 00:23:31.340 So I start with the set up and then we have transport equation where velocity is described by the Stokes flow. 00:23:31.340 --> 00:23:43.340 So here if you look at this equation Stokes flow this rule stands for the viscosity P stands for the pressure and then we again assume the flow is incompressible. 00:23:43.340 --> 00:23:47.340 And also later on I consider the mixing by the number Stokes flow. 00:23:47.400 --> 00:23:57.400 So numbers of flow here we have this nonlinear advection curve without breaking the V in this equation. 00:23:57.400 --> 00:24:01.400 And then the right hand side if you notice we have F theta. 00:24:01.400 --> 00:24:09.400 So F theta stands for the local forces such as such as the buoyancy for example theta times the unit vector in the direction of buoyancy. 00:24:09.400 --> 00:24:11.400 Then that shows the buoyancy. 00:24:11.460 --> 00:24:25.460 Now I considered the actually the most challenging case is active well if you have these local forces acting on velocity we call the scalar theory the active scalar. 00:24:25.460 --> 00:24:28.460 Active scalar is rather challenging. 00:24:28.460 --> 00:24:34.460 In fact if one I don't know anyone work on Boussinesq equations. 00:24:34.520 --> 00:24:40.520 So if you look at transport equation capital is a bit long the number Stokes is a Boussinesq equation with zero diffusivity. 00:24:40.520 --> 00:24:49.520 And earlier Mafeet in his earlier paper he posed the open problem was asking with a gradient of theta. 00:24:49.520 --> 00:24:55.520 So gradient of the scalar flow blows up as diffusivity converts to zero. 00:24:55.520 --> 00:25:03.520 And then in the end this answer was addressed in Thomas Hull's early paper 2005 they said okay there's no blow up. 00:25:03.580 --> 00:25:05.580 For the gradient of theta. 00:25:05.580 --> 00:25:15.580 And later on when I was opposed to the USC we further showed gradient theta actually has no blow up for initial counting theta in H1 and the velocity in H2. 00:25:15.580 --> 00:25:27.580 So the reason is later on when we apply our control designs very important to understand the well-posed of the governing system itself before we do anything. 00:25:27.640 --> 00:25:33.640 So for example let's say if my how can we how can we introduce our control. 00:25:33.640 --> 00:25:39.640 So let's look at first look at the flow governed by the Stokes equation. 00:25:39.640 --> 00:25:45.640 So keeping in mind here I said velocity has to be divergent spray and normal component is zero. 00:25:45.640 --> 00:25:48.640 So no fluids enters through the boundary. 00:25:48.640 --> 00:25:50.640 Now how I'm able to introduce a control. 00:25:50.700 --> 00:25:58.700 Of course one can think about stirring the fluids inside of your container or vessel that not release to internal control or distributed control. 00:25:58.700 --> 00:26:08.700 But when I work on this it's motivated by the observation that moving walls accelerate mixing compared to the fixed walls with no safe boundary condition. 00:26:08.700 --> 00:26:16.700 The reason is I think everyone may have the experience when you mix in flour with water. 00:26:16.760 --> 00:26:24.760 If you just stir the fluids in your container and then the boundary might have thick sort of layers the boundary layers the mixing couldn't cross. 00:26:24.760 --> 00:26:30.760 So that's what I mean if you have a fixed container with no safe boundary you may have like boundary layer phenomenon. 00:26:30.760 --> 00:26:34.760 The mixing only occurs in the center area. 00:26:34.760 --> 00:26:42.760 Now if you move the wall or stir the fluids along sometimes we see the fluid is moving in the middle of the container. 00:26:42.820 --> 00:26:47.820 Now if you move the wall stir the fluids along sometimes we mix the flour as you may want. 00:26:47.820 --> 00:26:53.820 Okay I'm going to stir along the along the thick boundary so that the mixing can cross the boundary layers. 00:26:53.820 --> 00:27:05.820 Then if you sort of rotate the wall that's going to create a closed orbit near the boundary of your domain so that you have much better mixing region. 00:27:05.880 --> 00:27:07.880 So that's the idea. 00:27:07.880 --> 00:27:19.880 So therefore I consider in the tangential direction I'm going to assume the fluid is able to slide along the boundary and against a certain friction. 00:27:19.880 --> 00:27:22.880 So there's a friction term which is against the velocity. 00:27:22.880 --> 00:27:30.880 So this D this D based stands for the deformation tensor which is the average good invading plus good invading transpose. 00:27:30.880 --> 00:27:34.880 And the N tau again is normal and the tangential vectors. 00:27:34.940 --> 00:27:41.940 So now I'm going to assume tangentially I'm going to stir my fluids and this G stands for my control input. 00:27:41.940 --> 00:27:47.940 I'm going to introduce the control along the boundary and then so that I can control my velocity there. 00:27:47.940 --> 00:27:51.940 And therefore via this friction term I can steer the behavior of my data. 00:27:51.940 --> 00:27:53.940 So that was my design. 00:27:53.940 --> 00:28:01.940 So as I said boundary control you can introduce control via the boundary and then you can just simply stir the fluids in the interior that's introduced. 00:28:02.000 --> 00:28:04.000 That leads to an internal control. 00:28:04.000 --> 00:28:12.000 Then we can always formulate this abstract Cauchy problem right when we have a control system. 00:28:12.000 --> 00:28:19.000 So again I have a transport equation and then the second equation is abstract ODE form for my velocity field. 00:28:19.000 --> 00:28:23.000 And then here okay let me explain a few terms here. 00:28:23.060 --> 00:28:33.060 So this A actually is the Lue is the viscosity times the Ray projector act on my Stokes operator act on my Laplacian operator. 00:28:33.060 --> 00:28:37.060 So this is the Stokes operator associated with a proper boundary condition. 00:28:37.060 --> 00:28:41.060 Like I said we're going to say the normal component is zero. 00:28:41.060 --> 00:28:48.060 So the Ray projector essentially is matched the L2 functions into the Varensevray subspace and with this boundary condition. 00:28:48.120 --> 00:28:56.120 And then if we consider say never Stokes then this MV would be the negative Ray projector act on this Laplacian term. 00:28:56.120 --> 00:29:00.120 And then if this term is zero then we have a Stokes equation. 00:29:00.120 --> 00:29:02.120 Then this U stands for control input. 00:29:02.120 --> 00:29:07.120 B is the control input operator depends on how you would like to introduce a control to the system. 00:29:07.120 --> 00:29:15.120 So I'm going to work on this abstract formulation stands for let's say the linear control for Stokes or Laplacian equations. 00:29:15.180 --> 00:29:17.180 And then my other action term going to control data. 00:29:17.180 --> 00:29:21.180 So that's sort of the basic formulation for the problem going to consider. 00:29:21.180 --> 00:29:31.180 And then the natural sort of as a first step I would like to say okay how can we formulate the optimal control problem for achieving optimal mixing. 00:29:31.180 --> 00:29:42.180 Let's say we can consider for certain given type of T like to minimize the mixed norm of T equals capital T as well as the mixed norm of the time interval from capital T. 00:29:42.240 --> 00:29:49.240 Where is the say minimum cost of my control input. 00:29:49.240 --> 00:29:55.240 And then of course it's quite subtle to determine what should be the suitable country input space for you. 00:29:55.240 --> 00:30:02.240 Because when we choose this the best possible set of control one need to first of all make sure my control is physically meaningful. 00:30:02.240 --> 00:30:08.240 And also need to make sure the problem P subject to my PD concern is solvable right. 00:30:08.300 --> 00:30:17.300 And then very often often we need to use the first order of non condition to solve the optimal control. 00:30:17.300 --> 00:30:25.300 So we may want you have certain smoothness so that the solution is allowed the Gatot different ability with respect to my control input. 00:30:25.300 --> 00:30:31.300 So it's true that this is quite important and sometimes can be a little bit challenging. 00:30:31.360 --> 00:30:38.360 Especially for we have like the active scalar field the two way carbon makes the problem rather challenging. 00:30:38.360 --> 00:30:47.360 Now how can we rewrite this mix norm in a more triteric representation. 00:30:47.360 --> 00:30:58.360 Now it's one crime so in order to deal with this norm I'm going to introduce the ETA which is the higher regularity counterpart of data by solving an elliptic problem. 00:30:58.420 --> 00:31:05.420 So I'm going to solve theta sorry ETA so this elliptical problem with moment boundary condition. 00:31:05.420 --> 00:31:11.420 So here I add identity just to make sure the law of law has no zero eigenvalue in this spectrum. 00:31:11.420 --> 00:31:20.420 So now that if theta plus i is invertible then I can define its square root and I write this as lambda. 00:31:20.420 --> 00:31:27.420 Then the mix norm is one prime is written as lambda inverse act on theta and then it has L2 norm. 00:31:27.480 --> 00:31:34.480 Or the H1 norm of the eta so I can rewrite this term in terms of my eta. 00:31:36.480 --> 00:31:44.480 Now I'm going to back again talking about the challenge we have encountered in analysis and computation for solving problem P. 00:31:44.480 --> 00:31:52.480 So first of all the non-linearity calculation essentially leads to nonlinear control and non-converse migration problem. 00:31:52.540 --> 00:32:03.540 So normally we don't have any unique mix unless you impose certain smallness condition on the data associated with the problem. 00:32:03.540 --> 00:32:14.540 And then zero diffusivity in my transport equation that's the key challenge we encountered because differentiability leads to very high order regularity required for velocity field. 00:32:14.600 --> 00:32:23.600 And if one considers boundary control then the creation of vorticity on the domain boundary and also compatibility condition may come into play even in the case of non-smooth solutions. 00:32:23.600 --> 00:32:31.600 So here what do I mean by compatibility means initial condition restricted on the boundary has to be the boundary condition when t equals zero. 00:32:31.600 --> 00:32:40.600 So if we have to address the compatibility condition basically you introduce addition layer or addition constraint to the control. 00:32:40.660 --> 00:32:52.660 That's something we don't want. However in the case where we have the transport equation capital ways the Stokes equations where f theta this is the buoyancy term for example. 00:32:52.660 --> 00:33:04.660 The problem is rather challenging in the sense we need to find the proper state space such that the governing system is well posed and also somehow allow the first order. 00:33:04.720 --> 00:33:06.720 First the variational system is well posed. 00:33:06.720 --> 00:33:16.720 And then since we have quite high regularity required for velocity and then that leads to high regularity for initial and boundary data. 00:33:16.720 --> 00:33:24.720 So choosing the proper state space is very extremely important in this case. 00:33:24.720 --> 00:33:26.720 And then computation. 00:33:26.780 --> 00:33:36.780 Now first of all when you solve the Navier-Stokes and then the we need to make sure the divergence free condition holds so that the mass is conserved for the transport equation. 00:33:36.780 --> 00:33:42.780 So find the appropriate numeric scheme for mass conservation in compressed workflows is very important. 00:33:42.780 --> 00:33:50.780 And also during mixing occur small scales of the mass conservation is very important. 00:33:50.840 --> 00:33:58.840 And also during mixing occur small scales like fine filament and the large gradient of the calipher is going to develop. 00:33:58.840 --> 00:34:04.840 So like when you try to capture this mixing process one really need to refine the mesh. 00:34:04.840 --> 00:34:06.840 The computational is really really costly. 00:34:06.840 --> 00:34:18.840 And then furthermore if one want to do the like optimal open loop control then we know that when we solve this optimal control problem this results in a state equation of the mass conservation. 00:34:18.900 --> 00:34:28.900 A state equation about data right and then the joint equation about the log-group multiplier actually you introduce a system to solve the optimal control problem. 00:34:28.900 --> 00:34:35.900 Which joint equation has to be solved backwards in time and we also have a non-linear automatic condition. 00:34:35.900 --> 00:34:46.900 So solving the entire automatic condition for finding the optimal control requires solving state equation both in time and then the joint equation backwards in time and capital waste non-linear automatic condition. 00:34:46.960 --> 00:34:54.960 And then we have to store all the data so computationally rather costly. 00:34:54.960 --> 00:35:08.960 Then so can we somehow construct something that is optimal in certain sense but also computationally less costly? 00:35:08.960 --> 00:35:14.960 So I would like to mainly focus on the feedback control we consider in this work. 00:35:15.020 --> 00:35:20.020 So let's first look at the case where the velocity field is governed by Stokes flow. 00:35:20.020 --> 00:35:32.020 So basically I don't have this non-linear direction term and here I don't assume any sort of force from theta act on the velocity so just a one way coupling. 00:35:32.020 --> 00:35:41.020 So I'm going to consider transport equation with a Stokes equation and here this simple elliptic equation just to address the mixed norm. 00:35:41.080 --> 00:35:58.080 Well we know solving optimal feedback one can use Hamiltonian-Jacobi equation but that very often needs to vary high dimensional like needs really computational dealing with high dimensional equations. 00:35:58.080 --> 00:36:04.080 Now here instead we're going to consider so-called instantaneous control design. 00:36:04.080 --> 00:36:09.080 So later on we will say this is like an approximation of an open loop control. 00:36:09.140 --> 00:36:20.140 So this design actually is closely tied to Resilient Horizon Control RFC or Model Predictive Control MQC with fine time horizon. 00:36:20.140 --> 00:36:33.140 So the idea is let's say we consider first of all the uniform partition of time interval 0 to capital T and then we let this H be T the finite time capital T divided by the number of intervals. 00:36:33.200 --> 00:36:40.200 And then I'm going to use a semi implicit order method in time for discretizing my governing system. 00:36:40.200 --> 00:36:55.200 So for example here I'm going to discretize theta the time derivative theta is just all those methods and then for this non-linear term I use explicit form for theta implicit form for V. 00:36:55.260 --> 00:37:04.260 So similarly here I use implicit form for my SOAPS term V here and also my control term. 00:37:04.260 --> 00:37:09.260 So after discretization now I'm dealing with the stationary system. 00:37:09.260 --> 00:37:17.260 Then the cost of function I just like to optimize the cost of function over one time step. 00:37:17.320 --> 00:37:26.320 The previous setup like you're looking at the optimal control for the cost of function you want to minimize it over the entire interval from there of capital T. 00:37:26.320 --> 00:37:29.320 Now I just want to do one step minimization. 00:37:29.320 --> 00:37:34.320 I solve this optimization problem just by one step in time and then match forward. 00:37:34.320 --> 00:37:42.320 And then through this recursive procedure I'm able to derive something quite interesting. 00:37:42.380 --> 00:37:53.380 So with one step optimization actually you will realize this is able to connect the joint state with the state. 00:37:53.380 --> 00:37:57.380 That's why we're able to close the loop just one step. 00:37:57.380 --> 00:38:09.380 And then via recursive approach one can show that the semi implicit time discretization actually equivalent to the closed loop system looks like this. 00:38:09.440 --> 00:38:17.440 So now keep in mind this H actually is the time step we take initially to discretize the system. 00:38:17.440 --> 00:38:28.440 In the end here when we close the loop if you look at the continuous level so this entire procedure is the discretization of this closed loop system. 00:38:28.440 --> 00:38:33.440 And here now this H is the fixed parameter in my new continuous closed loop system. 00:38:33.500 --> 00:38:38.500 And then in fact we have a feedback control load here. 00:38:38.500 --> 00:38:47.500 U is given by console with gamma and then negative one this H is the step size and then B is the control input operator. 00:38:47.500 --> 00:38:59.500 B star is the joint operator and times I minus step size H times the stock operator taking inverse and then add down the lower projector of theta times greater than eta. 00:38:59.560 --> 00:39:02.560 So this console only depends on state variables. 00:39:02.560 --> 00:39:10.560 But since here this procedure has a lot of approximation so we get suboptimal control. 00:39:10.560 --> 00:39:13.560 So that's my closed loop design. 00:39:13.560 --> 00:39:16.560 Then a couple of questions we need to address here. 00:39:16.560 --> 00:39:19.560 Is the closed loop system stable? 00:39:19.560 --> 00:39:24.560 Well I probably should say whether a system is well posed and then whether it's stable. 00:39:24.560 --> 00:39:28.560 So now let's look at the closed loop. 00:39:28.620 --> 00:39:33.620 As I said here the B in theoretical is the constant input operator. 00:39:33.620 --> 00:39:40.620 But when we show the well posedness and the stability we have to assume the B is sort of identity operator. 00:39:40.620 --> 00:39:45.620 So basically in this case we assume we have to sort of stir the fluid everywhere in the domain. 00:39:45.620 --> 00:39:50.620 This condition is a little bit strong but this is what we can get at this point. 00:39:50.680 --> 00:39:59.680 So now we have an internal control and then if I replace the U by my control law. 00:39:59.680 --> 00:40:03.680 So I have a closed loop system and one can show the well posedness. 00:40:03.680 --> 00:40:15.680 So if the ideal condition theta not in the intersection of L infinity and H1, the velocity is V2 would get the well posedness and then the regularity of the solution in space here. 00:40:15.740 --> 00:40:20.740 So V actually is up to, it's sort of H3. 00:40:20.740 --> 00:40:24.740 Oh by the way I probably forget to mention what this notation means. 00:40:24.740 --> 00:40:31.740 So this notation means divergence free of the sub space of H3. 00:40:31.740 --> 00:40:40.740 So it's quite smooth because this operator is like Laplace inverse, it smooths out the velocity field. 00:40:40.740 --> 00:40:41.740 That's why here it's quite smooth. 00:40:41.800 --> 00:40:48.800 And again this inverse operator is really due to the discretization scheme we used. 00:40:48.800 --> 00:40:52.800 Later I'm going to show you different discretization scheme. 00:40:52.800 --> 00:40:55.800 This guy may not be there. 00:40:57.800 --> 00:41:04.800 Now this is well posedness and in fact when we show the stability, stability actually is quite challenging. 00:41:04.860 --> 00:41:12.860 Well if one look at the total energy, I mean total energy in terms of the mixed norm and the H1 velocity. 00:41:12.860 --> 00:41:17.860 Now the total energy rate of change actually is strictly negative. 00:41:17.860 --> 00:41:22.860 So this guy is bounded by H1 normal velocity and this is a negative sign here. 00:41:22.860 --> 00:41:25.860 The total energy actually is decaying. 00:41:25.860 --> 00:41:27.860 But here we only have information about the velocity. 00:41:27.860 --> 00:41:30.860 We don't have any information about the scalar field. 00:41:30.920 --> 00:41:43.920 So what we did was we analyzed the stability of velocity in terms of H2, H1, even H2 norm and the time derivative even in terms of H2 norm converges to zero. 00:41:43.920 --> 00:41:51.920 And then we look at the transport equation to get the stability of my scalar field. 00:41:51.920 --> 00:41:59.920 So in the end what we can show in scalar field actually in terms of H1 norm decays and converges to certain constants more than initial distribution. 00:41:59.980 --> 00:42:02.980 Of course in this case the control converges to zero. 00:42:02.980 --> 00:42:09.980 So we get the stability but we don't really know the decay rate of the velocity field and the mixed norm. 00:42:09.980 --> 00:42:11.980 That our initial work. 00:42:11.980 --> 00:42:18.980 And later on we actually was able to show the polynomial decay using semi-group theorem and the non-mean analysis. 00:42:18.980 --> 00:42:23.980 We were able to show if we make certain ends of the velocity field, 00:42:24.040 --> 00:42:31.040 if we assume velocity somehow decays of a polynomial order, 00:42:31.040 --> 00:42:36.040 then we can show the mixed norm and you control also the decay of the polynomial rate. 00:42:38.040 --> 00:42:44.040 Now I'm going to briefly represent our numerical experiments. 00:42:44.040 --> 00:42:47.040 So when we solve the closed loop system, 00:42:47.040 --> 00:42:52.040 we have sort of transport half of the flow equation and also we have a unique equation. 00:42:52.100 --> 00:42:59.100 So where we use Helenhood is a class of finite element method with a projection method for solving Stokes equation. 00:42:59.100 --> 00:43:04.100 And time wise we use backward discretization of order two. 00:43:04.100 --> 00:43:11.100 And then for transport equation we use Runge-Kutta discontinuous Galerkin method with a third order in time. 00:43:13.100 --> 00:43:16.100 Then here is a forward simulation. 00:43:16.100 --> 00:43:19.100 We assume initially, well our domain first of all is a unit disk. 00:43:19.160 --> 00:43:26.160 Initially we have bi-color fluids and the initial distribution is hyper-tangent. 00:43:26.160 --> 00:43:27.160 Y over 0.1. 00:43:27.160 --> 00:43:30.160 We just like to smooth out the interface. 00:43:30.160 --> 00:43:33.160 It's for initial data we need it's infinity intersects with H1. 00:43:33.160 --> 00:43:36.160 So we need some smoothness for initial data. 00:43:36.160 --> 00:43:41.160 And then so the parameter H is time step size we take 0.1 00:43:41.160 --> 00:43:43.160 and then the control weight is 10 to the negative 6. 00:43:43.160 --> 00:43:47.160 So that you can see how the mixing occur and the mixing process. 00:43:47.220 --> 00:43:52.220 Really say the fine sort of a filament appear in the new domain. 00:43:52.220 --> 00:43:57.220 And then of course a large gradient of data encountered here as well. 00:43:57.220 --> 00:44:06.220 Then if you look at the mixed norm, mixed norm actually somehow decays following T to negative 0.5. 00:44:06.220 --> 00:44:09.220 And then we try a different H. 00:44:09.220 --> 00:44:12.220 Let's say the first rows when H is 0.01. 00:44:12.220 --> 00:44:16.220 So we have a smaller sort of time step size when we discretize the original system. 00:44:16.280 --> 00:44:19.280 And then second row is H is 0.1. 00:44:19.280 --> 00:44:22.280 The last is 1 at different times to us. 00:44:22.280 --> 00:44:26.280 At this point we can only claim our feedback control is suboptimal. 00:44:26.280 --> 00:44:29.280 But we don't really know it's optimality. 00:44:29.280 --> 00:44:39.280 Although we try different sort of parameters and then it's hard to really say how optimal this feedback law we constructed. 00:44:39.340 --> 00:44:46.340 And I mentioned earlier use different numeric scheme to discretize the governing system. 00:44:46.340 --> 00:44:49.340 We may get different sort of regularity of your control. 00:44:49.340 --> 00:45:01.340 Now, for example, here if we apply explicit Euler discretization for the Stokes term here. 00:45:01.340 --> 00:45:06.340 And then in the end we get the control law without this inverse Stokes term. 00:45:06.400 --> 00:45:09.400 So less smooth compared to the previous approach. 00:45:09.400 --> 00:45:18.400 Then we actually get more chaotic mix and result for different H. 00:45:18.400 --> 00:45:29.400 And then again, in the end, we did the approximation of the mixed norm using different control laws. 00:45:29.460 --> 00:45:38.460 And then with different discretization parameters, we observed the all-decay-winner polynomial rate. 00:45:38.460 --> 00:45:42.460 So that's about the closed-loop system. 00:45:42.460 --> 00:45:52.460 Actually, I think this part I showed very briefly in 2022 when I visited FAU for the first time. 00:45:52.460 --> 00:45:58.460 And then Ricky was suggesting maybe we come first to just focus on the transport equation 00:45:58.520 --> 00:46:05.520 and determine the velocity field which achieve mixing with no dynamics. 00:46:05.520 --> 00:46:10.520 And then we know how to sort of control the Stokes or another Stokes flow. 00:46:10.520 --> 00:46:16.520 Then we approximate the design of velocity field by active control of the flow dynamics. 00:46:16.520 --> 00:46:25.520 So then so far, I mainly focused on let's say how can we optimize velocity along no dynamics for this point. 00:46:25.580 --> 00:46:32.580 And then I have a few questions here related to the open-mean problems I mentioned earlier. 00:46:32.580 --> 00:46:39.580 Now, again, let's say if we have a periodic domain and always initial data mean zero. 00:46:39.580 --> 00:46:43.580 Again, I'll assume we lost the result of this regularity here. 00:46:43.580 --> 00:46:48.580 Now, if S is smaller than 1, and for any T-stack, 00:46:48.640 --> 00:46:55.640 can we always construct feedback flow which only depends on theta and the eta, 00:46:55.640 --> 00:47:00.640 such that perfect mixing can be achieved? 00:47:00.640 --> 00:47:07.640 So like I mentioned earlier, if S is zero, it's possible there exists T-stack, 00:47:07.640 --> 00:47:09.640 the perfect mixing is possible. 00:47:09.640 --> 00:47:14.640 But here I like to ask for any T-stack, can we always construct feedback flow? 00:47:14.700 --> 00:47:15.700 Can we always construct feedback flow? 00:47:15.700 --> 00:47:16.700 Make this possible. 00:47:16.700 --> 00:47:21.700 Then for S greater than or equal to 1, can we always construct feedback flow, 00:47:21.700 --> 00:47:23.700 such that exponential mixing is possible? 00:47:23.700 --> 00:47:28.700 Basically, mixing decay follows this exponential rate. 00:47:28.700 --> 00:47:31.700 So that's the question I put in the end. 00:47:31.700 --> 00:47:35.700 And then I guess that's the end of my talk. 00:47:35.700 --> 00:47:38.700 Let me know if you have any questions. 00:47:44.640 --> 00:47:47.700 Thank you.